The Chomsky Hierarchy

Home , Chomsky hierarchy, Chomsky normal form, Head grammar, Indexed grammar, Nested stack automaton

Figure E.I-l shows the inclusion relations that hold among the formal lan• guages considered in this section (this is essentially the Chomsky Hierarchy). Tables E.I-l and E.I-2 are a summary of closure and decision properties for these classes of languages.

559 560 ApPENDIX E-I

non-Turing-acceptable languages

recursively enumerable sets = Turing-acceptable languages

non-deterministic linear bounded ..._------1 automaton languagest

context free languages = non-deterministic pda languages

deterministic pda languages

regular languages = right-linear languages = deterministic or non-deterministic finite automaton languages

t The context sensItIve languages are the non-deterministic linear bound automaton languages-{e}. It is not known whether the deterministic linear bounded automaton languages are equal to the non-deterministic lba languages or are a proper subset.

Figure E.I-l: Inclusion relations of the classes of formal languages THE CHOMSKY HIERARCHY 561

Operation comple- n with a regu- Class of languages U n mentation • * lar language Regular yes yes yes yes yes yes Det. pda no no yes no no yes Context free yes no no yes yes yes Context sensitive yes yes ? yes not yes Recursive yes yes yes yes yes yes Recursively yes yes no yes yes yes enumerable t The context sensitive languages do not contain e and therefore are never equal to EO Table E.I-1: Summary of closure properties of classes of formal languages

Question xE L(G) L(G) = L(G!) ~ L(GJ) = L(GJ) n

Class of languages L(G)? = 0? EO? L(G2 ) = 0? L(G2)? L(G2 )? Regular yes yes yes yes yes yes Det. pda yes yes yes no ? no Context free yes yes no no no no Context sensitive yes no yes t no no no Recursive yes no no no no no Recursively no no no no no no enumerable t This case is trivially decidable (the answer is always no) because the context sensitive languages do not contain e.

Table E.I-2: Summary of decidability properties of classes of formal languages Appendix E-II Semantic Automata

Since the earliest days of modern generative linguistics the theory of au• tomata and their relation to formal grammars has been a cornerstone of syntactic linguistic theories; recently semantic applications have been devel• oped by Johan van Benthem for 'quantifier automata'. In this section the main idea is outlined and illustrated, but for proofs of the advanced theorems the interested reader is referred to van Benthem (1986), chapter 8. A generalized quantifier is interpreted as a functor D EAB which assigns to each domain E a binary relation between its subsets A and B (see Chapter 14). In the procedural view of quantifiers the determiner D of the N P takes as input a list of members of A marked for their membership in B and either accepts or rejects the list. In its mathematical formulation a determiner D is presented with finite sequences of O's and 1 's, where 0 stands for members of A - Band 1 for members of An B, respectively. The output of D is either 'yes' or 'no', depending on whether or not DEAB is true for the sequence read.

Examples: Every AB is recognized by the finite state machine in Fig. E.II-I. Every AB is true iff IA - BI = 0, so the Every automaton should accept all and only those sequences consisting of only 1 'so In all figures So is the initial state, and the square state always marks an accepting state, so in Fig. E.II-1 So is also the accepting state. For instance, for every man walks the l's represent men who walk and the O's men who do not walk. The automaton accepts only 1 's, i.e., whenever A ~ An B. The negated quantifier not every AB is obtained by reversing the accepting and rejecting states of the every automaton. The not every automaton accepts sequences that consist of at least one

563 564 ApPENDIX E-II

1 o

Figure E.Il-I: Every automaton

Figure E.II-2: Not every automaton

A that is not in B. Reversal of accepting and rejecting states in a finite state automaton for a determiner gives in general the automaton for the (externally) negated determiner. The automaton for at least one (some), the dual of every, should accept any sequence containing at least one 1, i.e., one A that is also in B. It exchanges 1 and 0 in the initial state of the every automaton and reverses its accepting and rejecting states. To complete the traditional Square of Opposition the no automaton, the negation of the at least (some) automaton, reverses its accepting and reject• ing states, and accepts only sequences consisting of only O's. SEMANTIC AUTOMATA 565

Figure E.II-3: At least one (some) automaton

o 1

Figure E.II-4: No automaton

Comparison to the tree of numbers There is a clear relation between the finite state automata for these de• terminers and their tree of numbers representation (see Chapter 14). In a tree of numbers a node (x, y) corresponds to x O's and y l's in the input for the automaton, since x = IA - BI and y = IA n BI. The tree for every shows that only the right most pair of numbers on each line is not in the quantifier. The trees for at least one (some) and no are related in exactly the same way. Thus we see that negation of a quantifier reverses the tree pattern a.nd switches accepting and rejecting states. From a given tree the corresponding automaton can be derived by a reduction algorithm. Let us look at the example for at least one first. Nodes in the tree are first identified with states, connected by adding O's or 1 's, and accepting states are indicated by squares. This can be shown to be equivalent to the simpler automaton in Fig. E.II-3 by the reductions in Fig. E.II-6. 566 ApPENDIX E-II

o 1

1 Is:l_ lL-Ql

Figure E.II-5: The at least one tree of numbers turned into an automaton

The procedure which turns a tree into an automaton is specified as:

(i) look at the downward triangle generated by a node (x, y)

(ii) turn it into a +/- pattern

(iii) if it is identical to the triangle pattern of the node above and to the right, it should correspond to the same state as that one

(iv) if it is identical to the node up and left, it is the same state as that one

(v) the minimum number of states should be the number of distinct trian• gle patterns in the tree.

This procedure may not be completely general for arbitrarily complex de• terminers, but it works well for the limited cases we consider here. Fig. E.II-7 is a more complicated example for exactly two AB. The square states are the accepting states again, and the bold face states represent the four states needed for the automaton, corresponding to the four different SEMANTIC AU'l'OMATA 567

triangle patterns in Fig. E.II-8. The exactly two automaton is shown in Fig. E.II-9.

(1) mark all accepting states with [±] -all rejecting states with El

(2) the configuration:~ o 1 + + corresponds to the acceptance for ever after-state (3) the configuration:~ o 1 -- corresponds to the rejection for ever after-state ( 4) the ,"nfigu

corresponds to the rejection with J-state Figure E.II-6 568 ApPENDIX E-II

50 ,0 o 1

Figure E.II-7: The tree/automaton for exactly two SEMANTIC AUTOMATA 569

0,0: 0,1:

0,2: O'3:~

Figure E.II-8 °

Figure E.II-9

Higher order quantifiers The determiner most is not recognized by any finite state machine; this follows from the Pumping Lemma. The language for most is context free, so it can be recognized by pushdown automaton. The idea is that the stack stores the values read, and that complementary pairs 0,1 or 1,0 of the top stack symbol and the symbol being read are erased as they occur. When the entire sequence has been read the stack should only contain 1 'so This is a slightly liberal version of a pushdown automaton accepting with a non• empty stack, but it can be turned into a normal pushdown automaton with some additional encoding. 570 ApPENDIX E-II

It turns out that higher order quantifiers may need context free languages, but, for instance, the higher order quantifier an even number of is recognized by a finite state machine. The following theorem characterizes the first-order definable determiners.

THEOREM E.II.1 (van Benthem (1986)) All and only all first-order definable quantifiers are computable by permutation-invariant and acyclic finite state machines. _

Permutation-invariance means that any permutation in the order of 1 's and O's of an accepted or rejected list is accepted or rejected, respectively, and an automaton is acyclic when it contains no non-trivalloops between states. A proportional determiner like at least two-thirds is essentially context free. A pushdown automaton for it keeps track of two top stack positions, comparing them with the next symbol read. If it reads 1 with 1,0 or 0,1 on the top two positions of the stack, it erases that top and continues. Likewise if it reads o with 1,1 on top. In the other cases the symbol read is stored on top of the stack. When the entire sequence is read, the automaton checks if the stack contains only 1 's, and if so, it recognizes the sequence. Many other determiners can be recognized by finite state or pushdown automata. It turns out that natural language determiners venture rarely, if at all, beyond the context free realm. For proofs and more substantial arguments the interested reader is referred to van Benthem (1986). Exercises

1. Design finite state automata for:

(a) at"least three AB (b) all but one AB (c) an even number of AB (d) almost all AB (in the sense of 'with at most finitely many excep• tions ') Review Exercises

1. Consider the following grammar G:

s -> N seems certain S -> I am right N -> that S

(a) List three members of L(G).

(b) Abbreviating so that ~ = {s,c,i,a,r,t}, construct a pushdown automaton which accepts L( G).

2. For each language below, construct a non-deterministic finite state automaton which accepts all and only the specified strings. In all cases the input alphabet is {O, I}.

(a) The terminal language of the following grammar:

S -> OA S -> IB A OA A -> IB B lA B -> 1 (b) The set of all strings two or more symbols long whose first and last symbols are identical. (c) The set whose regular expression is (01)*1*1.

3. Find a regular expression for:

(a) The language of problem 2a above. (b) The intersection of the languages of 2b and 2c.

571 572 REVIEW EXERCISES

(c) The set of all strings of 1's and O's in which every substring of consecutive 1's is of even length. e.g. include: but not: 011001111000 00100 0000 00110111 1111 01 0111111 111

4. Construct a Turing machine which operates with the alphabet {#, a, b} which when started in qo will move to the right until it encounters an a, change the a into a b, then move left until it encounters a different b, change that b into an #, and halt. If there is no a to the right of the initially scanned square, or if there is such an a but no b to its left, the machine will never halt. PART A CHAPTER 1 573

Solutions to selected exercises

PART A

CHAPTER 1

1. (a) tj (b) fj (c) f; (d) t; (e) fj (f) tj (g) tj (h) fj (i) fj (j) tj (k) fj (I) tj (m) fj (n) t; (0) fj (p) fj (q) f; (r) t

2. (a) yeSj (b) nOj (c) yeSj (d) {{S}}

3. (a) Rule: 1. 5 E A 2. If x E A, x + 5 E A Property: A = {x I x is positive and x is a multiple of 5} (b) Property: B = {x I x + 3 is a positive multiple of 10} or B = {x I x is a positive integer whose last digit is 7} (c) Rule: 1. 300 E C 2. If x E C and x < 400, x + 1 E C (d) Rule: 1. 3 E D 2.4 ED 3.IfxED,x+4ED or 1. 3 E D 2. If xED and x is odd, x + 1 E D 3. If xED and .r is even, x + 3 E D Property: D = {x I x is a positive multiple of 4 or x + 1 is a positive multiple of 4} (e) Rule: 1. 0 E E 574 SOLUTIONS

2. If x E E, then x + 2 E E 3. If x E E, then -x E E (f) Property: F = {x I x = 2~ where n is a non-negative integer} 5. (c) {0}; (d) {{0},0}; (e) {0, {{a}}, {{b}},{ {a,b}}, {0}, {0, {{a}}, {0, {b}},{0, {a, b}}, {{a}, {b}}, {{a}, {a,b}},{{b}, {a,b}}, {0,{a},{b}}, {0,{a},{a,b}}, {0,{b}, {a, b}}, {{a}, {b}, {a,b}}, {0, {a}, {b}, {a,b}}}

6. (a) {a,b,e,2}, (b) {a,b,e,2,3,4}, (c) {a,b,e,{e}}, (d) {a,b,{a,b}, {e,2}}, (e) {b,e}, (f) {a,b}, (g) {a,b}, (h) {e}, (i) 0, (j) 0, (k) 0, (1) {e, 2, 3, 4}, (m) 0, (n) {2}, (0) {a, b, {e}}, (p) 0, (q) {{a, b}, {e, 2}} 7. (a) {a,b,e,2}, (b) {a,b,e,2}, (c) {a}, (d) {2}, (e) {2}, (f) {a,b,e,2, 3,4,{e}}, (g) {2,3,4,{a,b},{e,2}}, (h) {2,3,4,{a,b},{e,2}}, (i) 0, (j) U, (k) {b,e,2}, (1) {2}, (m) U, (n) U 8. (a) (i) {a,b,e,d}; (ii) {e}; (iii) {a,b,e,d}; (iv) 0; (v) {e,d}; (vi) 0; (vii) {a,b} (b) no; (c) yes

9. (b) 1. An (B - A) 2. An (B n A') Compi. 3. (B n A') n A Comm. 4. Bn(A'nA) Assoc. 5. Bn(AnA') Comm. 6. Bn0 Compi. 7. 0 Ident.

11. (b) 1. (AUB)-(AnB) 2. (A U B) n (A n B)' Com pI. 3. (A U B) n (A' UB') DeM. 4. «A U B) n A') U «A U B) n B') Distr. 5. (A n A') U (B n A') U (A n B') U (B n B') Distr. (twice) 6. 0 U (B n A') U (A n B') U 0 CompI. (twice) 7. (BnAI)U(AnB') Ident. (twice) 8. (B - A) U (A - B) Compi. (twice) 9. (A - B) U (B - A) Comm, (twice) (c) (X U Y) - (X n Y) = (Y U X) - (Y n X) by the commutativity of union and intersection. PART A CHAPTER 2 575

(d) (i) 0, (ii) A', (iii) A, (iv) B - A, (v) AU B (e) 1. (A-B)+(B-A) 2. «A - B) U (B - A)) - «A - B) n (B - A)) Def. of A + B 3. (A + B) - «A - B) n (B - A)) Def. of A + B 4. (A + B) - «A n B') n (B n A')) Campi. 5. (A+B)-(AnA'nBnB' ) Assoc., Comm. 6. (A + B) - 0 Campi., Ident. 7. (A+B)n0' Campi. 8. (A + B) n U Campi. 9. (A+B) Ident. (f) 1. (A + B) ~ B 2. (A + B) U B = B Cons. Prin. 3. «A U B) - (A n B)) U B = B Def. of A + B 4. «A U B) n (A n B)') U B = B Campi. 5. «A U B) n (A' U B')) U B = B DeM. 6. «A U B) U B) n «A' U B') U B) = B Distr. 7. (A U (B U B)) n (A' U (B' U B)) = B Assoc. (twice) 8. (A U B) n (A' U U) = B Idemp., Ident. 9. (AUB)nU=B Ident. 10. AuB = B Ident. 11. A~B Cons. Prin.

CHAPTER 2

1. (a) (i) {(b,2),(b,3),(e,2),{e,3)}; (ii) {(2, b), (2, e), (3, b), (3, e)}; (iii) {(b, b), (b, e), (e, b), (e, e)}; (iv) {(b,2),(b,3),{e,2),(e,3),(2,2),(2,3),{3,2),(3,3)}; (v) 0 (since An B = 0); (vi) same as A x B (b) (i) True; (ii) False; (iii) False, (e,e) E (A x A); (iv) True; (v) True; (vi) True; (vii) True (c) (i) dom(R) = A, ran(R) = {b,2,3}; (ii) R' = {(b,e),(b,3),{e,b),{e,e)}, R-1= {{b,b),(2,b),{2,e),(3,e)}; (iii) No. 576 SOLUTIONS

2. In relations from A to B each of a, b, and c can be paired with 1, with 2, with both 1 and 2, or with neither, i.e., in four possible ways. Therefore there exist 4 X 4 X 4 = 64 distinct relations. In functions from A to B each of a, b, and c can be paired with 1 or 2, Le., in two possible ways. Therefore there are 2 X 2 X 2 = 8 distinct functions. Six ofthese are onto (only {( a, 1), (b, 1), (c, I)} and {(a, 2), (b, 2), (c, 2)} are not onto). Since none of these are one-to-one and onto, none have inverses that are functions. There are 8 X 8 distinct relations from B to A of which 3 x 3 = 9 are functions. None are onto, six are one-to-one, and none have inverses that are functions.

3. (a) R2 0 Rl = ((1,2), (1,4), (1,3), (2,2), (2,4), (2,3), (3,4), (4,2), (4,3), (4,4)}, R 1 oR2 = {(3,4), (3,1), (1,1), (1,2), (1,4), (2,4), (2,3), (2,1), (1,3)} (b) Rl1 0 Rl = {(I, 1), (1,2), (1,4), (2,1), (2,2), (2,4), (3,3), (3,4), (4,4), (4,3), (4,2), (4, I)}

CHAPTER 3

1. (a)(ii) irreflexive, non-symmetric, non-transitive No.te: In considering transitivity, it can be misleading to consider only triples of distinct elements, as this example shows. For if a is the brother of b (aBb) and bBc, and a, b, c are all distinct, a must ind€ed be the brother of c. But consider a pair of brothers aBb and bBaj now it is false that a is the brother of a, as one would have to conclude if the relation were transitive. 2. (a) Irreflexive (no utterance forms a minimal pair with itself), sym• metric, nontransitive (e.g., (cat, bat) and (bat, bag) are minimal pairs but not (cat, bag)), and nonconnected; (d) Irreflexive or reflexive (depending on how the term 'allophone' is interpreted), symmetric, transitive (if "phonemic overlap" is excluded, otherwise nontransitive) and nonconnected. If one takes the view that it is reflexive, symmetric, and transitive, then A is an equivalence rela• tion that partitions the set of English phones into equivalence classes corresponding to the ("taxonomic") phonemes of English; (e) Reflexive, symmetric, transitive, and non connected (in general). Each equivalence class contains all the sets that have the same num• bers of members; PART A CHAPTER 4 577

3. (a) Rl and Rt1 : reflexive, antisymmetric, nontransitive, nonconnected; Ri: irreflexive, nonsymmetric, nontransitive, nonconnected; R2 and R'i 1 : irreflexive, asymmetric, transitive, connected; R;: reflexive, anti• symmetric, transitive, connected; R3 and R;:l: nonreflexive, symmet• ric, nontransitive, nonconnected; R;: nonreflexive, symmetric, non• transitive, nonconnected; R4 and R41 : reflexive, symmetric, transitive, nonconnected; R~: irreflexive, symmetric, intransitive, nonconnected; R4 (= R41 ) is an equivalence relation. The partition induced in A is {{1,3}, {2,4}} (b) {(I, 1), (2, 2), (3, 3), (4,4), (2, 3), {3, 2}} (c) 15

4. The fallacy lies in ignoring the if-clause in the definition of symmetry: il aRb, then bRa. The "proof" takes aRb for granted, so a counterex• ample can be constructed by finding a relation R which is symmetric and transitive on a certain set S where for some a, a does not bear the relation R to any member of S. Such an example is the following: Let S be the set of all humans and let R be defined by 'aRb if and only if a and b have the same parents and those parents have at least two children.' Then an only child does not bear R to anyone, including himself. (A simpler R might be 'has the same oldest brother as,' but it could be objected that the relation is simply not defined, rather than failing to hold among people who have no older brothers.)

5. (a) R = {(I, 1), (2,2), (3,3), (5,5), (6,6), (10,10), (15,15), (30,30), (1,2), (1,3), (1,5), (1,6), (I, 10), (1,15), (1,30), (2,6), (2, 10), (2,30), (3,6), (3,15), (3,30), (5,1O), (5,15), (5,30), (6,30), (10,30), {15,30}}, which is reflexive, antisymmetric, transitive, and nonconnected. (b) 1 is minimal and least; 30 is maximal and greatest.

CHAPTER 4

1. There is one-to-one correspondence of every set with itself-for exam• ple, the identity function. Thus, the relation of equivalence of sets is reflexive. If I is a one-to-one correspondence from A to B, then 1-1 is a one-to-one correspondence from B to A. Therefore, the relation is symmetric. If I and 9 are one-to-one correspondences from A to Band 578 SOLUTIONS

from B to C, respectively, then 9 0 f is a one-to-one correspondence from A to C. (This can be easily shown by an indirect proof.) Thus, the relation is transitive.

2. The set can be denoted {lOt, 102 , 103 , 104 ,00.}.

4. The cardinality is No. A 1-1 correspondence between the set of all sentences of English and the natural numb~rs could be established as follows: First, arrange all the sentences into groups according to the number of symbols in their written form, and order these groups lin• early starting with the group of the shortest sentences. Within each group, arrange the sentences alphabetically (using some arbitrary con• vention for the punctuation marks and space). This procedure puts all the sentences into a single linear order, and thus establishes a 1-1 correspondence between the sentences and the natural numbers.

5. The hotelkeeper uses his intercom to ask each guest to move into the room whose number is twice the number of his present room. That leaves all the odd numbered rooms empty, so each football player can double the number on his shirt and subtract one to find his room number.

6. (a~ The turtles can be numbered as follows: o 1 2 3 4 5

Since the turtles can be effectively listed, so can the corresponding mo• notheistic sects, and the cardinality of the set of such sets is therefore No; (b) Taking the preceding enumeration, it is clear that each sect corre• sponds to a subset of the set of all natural numbers; the atheistic sect corresponds to 0, the monotheistic sets to singleton sets. The set of all sects therefore can be put in one-to-one correspondence with P(N) and thus has cardinality 2No.

7. (a) Let A={1,2,3,00.} and B={O}. (b) Let A={0,1,2, ... } and B={a,b}. PART A REVIEW PROBLEMS 579

The set {(O,a),(O,b),(I,a),(I,b),(2,a),(2,b), ... } is mapped 1-1 onto {O,I,2,3, ... } by f(n,a)=2n, f(n,b)=2n+ l. (c) Let A = {O,2,4,6, ... } and B = {I, 3,5,7, ... }. (d) Let A = {O,I,2,3, ... } and let B be the set of "primed" integers {O', 1',2',3', ...} disjoint from A. A x B is equivalent to A x A, which has cardinality No.

REVIEW PROBLEMS, PART A

1. (a) A4; (b) 0; (c)As; (d)A4

2. Sample answers: (a) Let aRb be a + b = 5 or 1a - b 1= lor a X b = 24 or 'a + b is odd'. (b) Let aRb be a > b or 'a - b is positive'.

3. (a) Antisymmetric, transitive, reflexive (b) Yes. If n EN, n = {O,1,2,3, ... ,n - 1 }. If x E n, x is a natural number smaller than x. Then since all the members of x are natural numbers smaller than x, they are also smaller than n and therefore members of n. Hence x is a subset of n. It is a proper subset, i.e., not equal to n, because x was given as a member of n, and the members of n do not include n itself.

4. (a) The set of all members could be represented graphically in a family tree diagram:

where left-to-right order of branches from a single node represents oldest-to-youngest order among brothers. Then the nodes can be ef- 580 SOLUTIONS

fectively enumerated proceeding top-to-bottom and left-to-right: o

4 5 6 7 8 9 10 11 etc.

(b) Assume that the set of all clubs is denumerably infinite, so that the clubs could be listed as Co, CI , C2 , C3 , •••• Then define a new club C* by the membership requirement:

mani E C* if and only if manj f/. Ci

Since C* is thereby made distinct from every club III the putative listing, the listing could not have been complete, i.e., no complete effective enumeration is possible.

5. (asterisked entries are discussed below)

PO SO WO (a) yes yes no (b) yes yes no (c) yes yes no* (d) no* no no (e) yes yes yes (f) yes no* no (g) yes yes yes* (h) yes yes no* ( c) It is easy to be misled into thinking that the negative rational numbers with 0 are well-ordered by the relation ;::::, because the whole set does have a first element, namely O. But well-ordering requires that every subset have a first element, and there are in this case infinitely many subsets that do not. Consider, for example, the subset consisting of all the negative rational numbers less than -1. There is no first (i.e. largest) number in that set, since for any rational number x which is PART A REVIEW PROBLEMS 581

less than -1 it is possible to find another rational number y which is larger than x but still smaller than -1; (d) The set is not even partially ordered, because two distinct strings may have the same length-i.e., antisymmnetry does not hold; (f) The ordering looks like this:

0,2,4,6,8, ...

1,3,5,7,9, ... with no relation holding between any evens and any odds. This sort of ordering satisfies the partial ordering definition, but fails to meet the linear-ordering requirement that R hold in some direction between every pair of elements. (g) The ordering looks like this:

0,2,4,6,8, 10, ... , 1,3,5, 7,9, ...

This is a well-ordering; every subset has a first element, which will be the smallest even number in the subset, if there are any, and otherwise the smallest odd number in the subset; (h) The ordering looks like this:

... ,9,7,5,3,1,0,2,4,6,8, ...

This is not a well-ordering, since any subset which includes infinitely many odd numbers will lack a first member.

6. If I A I ::; I B I and I B I ::; I C I, there are functions f: A -+ Band g: B -+ C which are onto Band C, respectively. We prove that go f is an onto function (from A to C). Assume that it is not onto. Then there is some z E C such that for no x E A, (g of)(x)=z, i.e., g(f(x)) = z. But 9 is onto C, and thus there is some y E B such that g(y) = z. Thus it must be that there is no x E A such that f( x) = y. But this contradicts the assumption that f is onto B. Therefore, go f is onto C, and 1A I ::; I C I. 582 SOLUTIONS

PART B

CHAPTER 6

1. (Other proposition letters may be used, other interpretations may be argued for, and other equivalent symbolizations are possible for any given interpretation.)

(a) j = John is in that room, m = Mary is in that room, j V m (b) a = the fire was set by an arsonist, e = there was an accidental explosion in the boiler room, (a V e) & ""' (a & e), or equivalently, a ...... rve (c) r = it rains, p = it pours: (1) If the statement means that every rainstorm is a big one, then "when" must be translated as "if and only if", since it can't pour without raining: r ...... p (2) In the famous salt advertisement, the second "it" refers to salt, which presumably can also pour when it isn't raining, r ---+ p (d) s = Sam wants a dog, a = Alice prefers cats, s & a. (The difference between "but" and "and" relates to the content of the conjoined sentences, not their truth values) (e) I = Steve comes home late, s = Steve has had some supper, r = we will reheat the stew: (1) If one interprets the "if" as "if and only if" , then the propo• sition can be symbolized as (l & '" s) ...... r (2) Under the interpretation that the stew may be reheated in any case, then the symbolic form is (l & rv s) --+ r (f) c = Clarence is well educated, r = Clarence can read Chuvash, c---+r (g) m = Marsha goes out with John, b = John shaves off his beard, d = John stops drinking, m --+ (b V d) (The statement might also be taken to mean m +-+ (b&d).) (h) s = the stock market advances, c = public confidence in the (fflonomy is rising, s ...... c (i) n:::: negotiations commence, b = Barataria ceases all acts of ag• gression against Titipu, n ---+ b PART B CHAPTER 6 583

2. (a) j = John is going to the movies, b = Bill is going to the movies, t = Tom is going to the movies, j & b & ,...., t (b) 8 = Susan likes squash, t = Susan likes turnips, ,...., (8 V t) or equivalently, ,...., 8 & ,...., t ( c) p = Peter is going to the party, f = Fred is going to the party, i = I am going to the party, ,...., (p V f) ---'>""" i or equivalently, (,...., p& ,...., f) ---'>""" i (d) I = Mary has gotten lost, a = Mary has had an accident, h = Mary will be here in five minutes, ( rv I & rv a) ---'> h or equivalently, rv(lVa)---'>h (e) b = a bear frightened the boys, w = a wolf frightened the boys, bv w (f) p = a party would have amused the children, 8 = a softball game would have amused the children, p& 8. (Perhaps p V 8 could be argued for, but under the natural interpretation that either one would have amused them, then one is saying both that a party would have amused them and that a softball game would have amused them.) 3. (b) False; (d) True; (f)False 4. (a) and (b) are logically equivalent.

5. (b) p: 1, q: 0, or p: 0, q : 1; (d) p: 1, q: 0, r: °or 1,8: 1; (e) p: 1, q: 0, r : 1,8: 1, or p: 0, q: 1, r : 1,8: 1 6. (a) tautology; (b) contingent; (c) tautology; (d) contradiction; (e) con• tingent

7. (a) p ---'> q can be defined as rv(p& rvq) (b) p & q can be defined as rv ( ,...., pV rv q) (c) p ...... q can be defined as (p -> q) & (q -> p)

8. (a) 1. rv p V (p & q) 2. (,,-,pVp)&(rvpVq) Distr. 3. (pV rv p) & ("-' p V q) Comm. 4. T&(rvpvq) Compi. 5. (rvpVq)&T Comm. 6. ""pVq Ident. 7. p->q Condo 584 SOLUTIONS

(b) "'Pi (c) Fi (d) "'Pi (e) T

9. (a) 1. p---+q (d) 1. P ---+'" q 2. q---+r 2. r---+q 3. "'r 3. '" r ---+ s 4. "'q 2,3, M.T. 4. P Auxiliary Premise 5. "'P 1,4, M.T. 5. "'q 1,4, M.P. 6. "'r 2,5, M.T. 7. s 3,6, M.P. 8. p-+s 4-7, C.P. (f) 1. pV(q&r) 2. ",t 3. (p V q) ---+ (s V t) 4. "'P 5. (p V q) & (p V r) 1, Distr. 6. pVq 5, Simpl. 7. svt 3,6, M.P. 8. s 2,7, D.S. 9. pVr 5, Simpl. 10. r 4,9, D.S. 11. r&s 8, 10, Conj.

(h) 1. '" P ---+ q 2. r ---+ (s V t)

3. s ---+ '" r 4. p ---+ '" t 5. r Auxiliary Premise 6. svt 2,5, M.P. 7. "'s 3,5, M.P. 8. t 6,7, D.S. 9. "'P 4,8, M.T. 10. q 1,9, M.P. II. r---+q 5-10, C.P. PART B CHAPTER 6 585

(i) I. p-+(q&r) 2. q-+s 3. r -+ t 4. (s&t) -+-u 5. u 6. p Auxiliary Premise 7. q&r 1,6, M.P. 8. q 7, Simpl. 9. s 2,8, M.P. 10. r 7, Simpl. II. t 3,10, M.P. 12. s&t 9,11, Conj. 13. -u 4,12, M.P. 14. u& -u 5,13, Conj. 15. f"VP 6-14, Indirect Proof (I) I. P 2. (p & q) V (p & r) 3. (pVq)-+f"Vr 4. pVq 1, Addn. 5. -r 3,4, M.P. 6. p&(qVr) 2, Distr. 7. qVr 6, Simpl. 8. q 5,7, D.S. 9. p&q 1,8, Conj. 10. (p & q) V ( - p & '" q ) 9, Addn. II. pf-+q 10, Bicond. 10. (a) valid, B = the butler killed the baron, /( = the cook killed the baron, C = the chauffeur killed the baron, S = the stew was poisoned, Q = there was a bomb in the car: I.BV/(VC 2. (1( -+ S)&(C ~ Q) 3.",S& ""B 4. '" B 3, Simpl. 5. /( V C 1,4, D.S. 6. /( -+ S 2, Simpl. 7. "" S 3, Simpl. 8. "" /( 6,7, M.T. 9. C 5,8, D.S. 586 SOLUTIONS

(b) Valid; (c) Invalid (recall that 'p only if q' is represented logically as p --> q); (d) Valid; (e) Invalid (let it be false that the segment is voiceless and let all the other elementary propositions be true).

11. (a) R is an equivalence relation since (1) it is reflexive: x ..... x is always true (2) it is symmetric: whenever x ..... y is true, y ..... x is true (since they have the same truth tables) (3) it is transitive: if x ..... y is true; y ..... z is true, then x ..... z is true; otherwise, if x .;.. z, then for some assignment of truth values to elementary statements there would be some y which is both true and false. (b) {p,(pV p),(p&(qV rvq)),(pV (q& rvq))} {(p V q), (rv q -> p), ("-' p -> q)} {(pV "-' p), (p ~ (q -> p)), (p V (qV "-' q))}

12. (a) TRUE FALSE 1. p-> (q-> r) p& rvq "-'(rvpVr) 2. "-'pVr 3. "-'p 4. r 5. P 6. 6'1 q -> r 6,2

7. 7'1l r ------8. 8'121 8'121 P 8,122 "-'q

9. 9'122 q

All the subtableaux close, so the tableau closes, and the argu• ment is shown to be valid. PART B CHAPTER 7 587

(c) TRUE FALSE 1. «p -> q) & (s V t)) «p -> q) V "" (s -> q)) (t -> q) 2. (p -> q) 3. (s V t) 4. (p -> q) ------5. ""(s -> q) This tableau remains open.

13. (a) (i) KKApqAqrAps, (ii) CKNpCNpqq, (iii) CApqKErsp

(b) (i) pv «"'p& "'q) -> (p&(q +-+ r))) ( ii) ('V ( ( ( «p +-+ q) +-+ r) & s) V p) & q) V r) & s (iii) "'««p +-+ q) & r) V s) -> t) (c) NApq ¢> KNpNq, NKpq ¢> ANpNq

CHAPTER 7

1. (a) (Vx)(B(x) V W(x)) : B(x) - 'x is black', W(x) - 'x is white'

(c) (Vx)(D(x) -> Q(x)): D(x)-'xisadog',Q(x)-'xisaquadruped' (e) (Vx)(3y)L(x,y): L(x,y) - 'x loves y' (g) (3y)(Vx)L(x,y): L(x,y) - 'x loves y' (i) (Vx)(L(x, x) -> (x = j» : L(x,~) - 'x loves y', j - 'John' (k) (Vx)«W(x)&L(g,x)) -> (K(g,x)V M(g,x)): W(x) - 'x is a woman', L(x, y) - 'x loves y', g - 'you', K(x, y) - 'x kisses y', M(x, y) - 'x loses y'

(m) (Vx)«P(x)&M(x,h») -> L(x,h) : P(x) - 'x is a person', h - 'New York', L(x,y) - 'x loves y', M(x, y) - 'x lives in y' (0) (Vx)(",L(x,h) ->r-vN(x,h): L(x,y) - 'x loves y', h -'New York', N(x,y) - 'x knows y'

(q) (Vx )(Vy)«F(y, x) & G(x, y,g) -> (3z)(H(z, y) & I(g, z, x») F(x,y) - 'x is a finger of y', G(x,y,z) - 'x gives y to z', H(x,y) - 'x is the whole hand of which y is a finger', [(x, y, z) - 'x takes y from z', g - 'he' 588 SOLUTIONS

(5) «3x)N(x) --? (Vx)A(x)) : N(x) - 'X is noisy', A(x) - 'x is annoyed' (t) ",(3x)N(x)&A(j): N(x) - 'x made noise', A(x) - 'x was annoyed', j - 'John'

(v) (Vx)(A(x) --? B(x)) : A( x) - 'x causes bad accidents', B(x) - 'x is a drunk driver under 18'

(x) (Vx)(A(x) --? B(x)) : A(x) - 'x is drunk', B(x) - 'for x to drive is risky'

2. (a) ",(3xj(P(x) & (Vy)(Q(y) --? A(x,y»)

(h) (Vx(Q(x) --? (3y)(P(y)&A(y,x»)

(d) (3x)(P(x) & (Vy)(Q(y) --?'" A(x, y»), although 'some people' might also be taken to mean not 'at least one person' but 'at least two persons'. (e) L(x,y) - 'x likes y',(Vx)(L(x,Mary» ...... "-'f(x, Mary» (g) T( x, y) - 'x attempted y',

(Vx)«P(x) & (3y)(Q(y) & A(x, y») --? (3z)(Q(z) &T(x, z») 3. (a) x in P(x) bound, last x and the y free; (h) x free, y and z bound; ( c) everything bound; (d) first x bound, everything else free; (e) ev• erything bound 4. (a) (1) Yes. Applying (7-24) to (Vz) in the translation gives

(Vz)«Vx)(3y)F(y,x)&(O(z) --? fez»~). Then ('Ix) and (3y) can- be moved outside by the following se•

quence of equivalences. Note first that P&Q ¢:> "-' (P --? "-' Q).

Thus, (Vx)P(x)&Q ¢:> ,,-,«Vx)P(x) --? "'Q) ¢:>

",(3x)(P(x) --? "'Q) (by Law 11) ¢:>

('Ix) ,,-,(P(x) --? "-'Q) (by Law 1111) ¢:> (Vx)(P(x)&Q). The steps for (3y) are similar. (2) No. Applying (7-24) in reverse to (Vz) in (2) gives

(3y)(Vx)(F(y,x)& (Vz)(O(z) - fez»~) which says 'There is something which is everything's father, and all odd numbers are integers.' (3) Yes. Applying Law 6 to (1). PART B CHAPTER 7 589

(b) (1) No. (1) is equivalellt by Law 9 to B(a) --> (Vx) '" (M(x) --> H (x», which is equivalent by the steps noted in the answer to (a,I) to B(a) --> (Vx)(M(x)& "'H(x», which says 'If Adam is a bachelor, then everything is a man and a non-husband,' or loosely, 'If Adam is a bachelor, then everyone is a bachelor.' (2) Yes. Apply Law 10 to (2), then Law 1. (3) No. 'It's not the case that if Adam is a bachelor then all men are husbands.' (N.B. This is equivalent to: 'Adam is a bachelor and not all men are husbands. ') (4) Yes.

5. (a) (Vx)(I(x) --> (3y)(I(y)&L(y,x»), (Vx)(3y)(I(x) -> (I(y)&L(y,x))), ",(3x)(I(x)& ",(3y)(L(y,x» (and many others; such listings are never exhaustive)

(b) «Vx)(P(x) --> O(x» V (3y)(I(y) & ",O(y»), (3y)«Vx)(P(x) --> O(x» V (I(y)& ",O(y»), By various applications of commutativity and Laws 9-12: (Vx)(3y)«P(x) -> O(x» V (I(y) & ",O(y»), (3y)(Vx)«P(x) --> O(x» V (I(y) & ",O(y») (Note that this is one case in which the order of universal and existential quantifiers is immaterial; the reason is that there are no two-place predicates relating x and y.)

(c) «3x)(P(x)& '" O(x» --> (Vy)«P(y)&G(y, 7» -> O(y»)). It is natural to try an alphabetic change of variable and a reinterpre• tation of -> in terms of '" and V, in hopes of leading to an ap• plication of Law 3. However, once we get to: «Vx) '" (P( x) & '" O(x» V (Vx)«P(x) &G(x, 7» --> O(x»), we see that only Law 4, which is not an equivalence, will apply. With two variables, we can apply Laws 9 and 12 to get: (Vx)(Vy)«P(x)& '" O(x» -> «P(y)&G(y, 7» ---+ O(y»)

(d) (Vx)(H(x) ---+ M(x» -> M(s), (3x)«H(x) ---> M(x» ---> M(s», ",M(s) ---> (3x)(H(x)& 'VM(x», etc. 590 SOLUTIONS

6. (a) «(3x)A(x) & (3x)B(x)) --+ C(x)) 1. «(3y)A(y)&(3z)B(z)) --+ C(x)) alph. variant 2. (",«3y)A(y) --+ ",(3z)B(z)) --+ C(x)) put in terms of "', --+ 3. (",(\iy)(A(y) --+ ",(3z)B(z)) -; C(x) Law 12 4. (",(\iy)(A(y) --+ (\iz) "'B(z)) --+ C(x)) Law 1 5. (.-v (\iy)(\iz)(A(y) --+ ",B(z)) --+ C(x)) Law 9 6. «3y)(3z) .-v (A(y) -> .-vB(z)) -> C(x)) Law 1 (2x) 7. (\iy)(\iz)('" (A(y) -> ",B(z)) --+ C(x)) Law 12 (2x) 8. (\iy)(\iz)«A(y) & B(z)) --+ C(x))

(b) (\ix)A(x) <-+ (3x)B(x) 1. ",«(\ix)A(x) --+ (3y)B(y)) --+ ",«3w)B(w) -> (\iz)A(z») alph. variants and in terms of "', --+ 2. ,,-,«3x)(3y)(A(x) --+ B(y)) -> .-v (\iw)(\iz)(B(w) --+ A(z))) Laws 9, 10, 11, 12 3. ",«3x)(3y)(A(x) --+ B(y)) --+ (3w)(3z) ",(B(w) -> A(z))) Law 1 4. ,,-,(\ix)(\iy)(3w)(3z)«A(x) --+ B(y)) -> "'(B(w) --+ A(z))) Laws 10 and 12 5. (3x)(3y)(\iw)(\iz) "'«A(x) --+ B(y)) --+ "-'(B(w) --+ A(z))) Law 1 6. (3x)(3y)(\iw)(V'z)«A(x) --+ B(y))&(B(w) -> A(z)))

7. (a) 1. ",(3x)(P(x)&Q(x)) 2. (3x)(P(x)&R(x)) 3. P(w) & R(w) 2, E.1. 4. (V'x) ",(P(x)&Q(x)) 1, Quant. Neg. 5. ",(P(w)&Q(w)) 4, V.1. 6. '" P( w) V "'Q( w) 5, DeM. 7. P(w) 3, Simpl. 8. ",Q(w) 6,7, D.S. 9. R(w) 3, Simpl. 10. R(w)& ",Q(w) 8,9, Conj. 11. (3x)(R(x)& "-'Q(x)) 10, E.G. PART B CHAPTER 7 591

(e) 1. (VX)(P(X) ~ Q(x» 2. R(a) 3. Pea) 4. Pea) ~ Q(a) 1, V.l. 5. Q(a) 3,4, M.P. 6. R(a) & Q(a) 2,5, Conj. 7. (3x)(R(x)&Q(x» 6, E.G.

(f) 1. (Vx)«P(x) V Q(x» ~ R(x» 2. (Vx)«R(x) V Sex»~ ~ T(x» 3. P(v) Auxiliary Premise 4. (P(v) V Q(v)) ~ R(v) 1, LV. 5. P(v) V Q(v) 3, Addn. 6. R(v) 4,5, M.P. 7. (R(v) V S(v» ~ T(v) 2, V.l. 8. R( v) V S(v) 6, Addn. 9. T(v) 7,8, M.P. 10. P(v) ~ T(v) 3-10, C.P. 11. (Vx)(P(x) ~ T(x» 10, V.G.

8. (c) D(x) = x is a duck, O(x) = x is an officer, P(x) = x is (one of) my poultry, W(x) = x waltzes

1. (\fx)(D(x) ~ "V W(x» 2. (Vx)(O(x) -> W(x» 3. (Vx)(P(x) -> D(x» 4. P(v) -> D(v) 3, V.l. 5.D(v) --->"VW(v) 1, V.l. 6.P(v) ->"-'W(v) 4,5, H.S. 7. O(v) -> W(v) 2, V.l. 8. "VW(v) ~,,-,O(v) 7, Condo 9.P(v) ->,,-,O(v) 6,8, H.S. 10. (\fx)(P(x) -> ,,-,O(x» 9, V.G. 592 SOLUTIONS

9. (a) TRUE FALSE D={a, ... } 1. '" (3x )F( x) ('v' x) '" F( x ) 2. (3x)F(x) 3. "'F(a) 4. F(a) 5. F(a) ------

(b) TRUE FALSE D = {a, b} 1. ('v'x)(3y)R(x,y) (3y)('v'x )R(x, y) 2. (3y )R( a, y) 3. ('v'x)R(x,a) 4. R(a, a) 5. R(a, a) ------6. R(b, a) 7. (3y)R(b, y) 8. R(b,a) ------9. R( b, b) 10. R(a, b)

So a counterexample consists of a model with a universe of dis• course D = {a,b} and an extension R = {(a,a),(b,b)}

6. (3y)R(y, b) 7. R(a, b) 8. R( b, b)

9. R(a,b) PART B CHAPTER 7 593

10. (a) 1. VI ~ V2&V2 C V3 Auxiliary Premise 2. V2 C V3 1, Simpl. 3. V2 ~ V3 &V2 :f:. V3 2, Def. of C 4. V2 ~ V3 3, Simpl. 5. VI ~ V2 1, Simpl. 6. VI ~ V2&V2 ~ V3· 4,5, Conj. 7. VI ~ V3 6, (7-54) 8. Vt = V3 1, Auxiliary Premise 9. Vt ~ V3 &V3 ~ VI 8, (7-52) 10. V3 ~ VI 9, Simpl. 11. V3~Vt&VI~V2 5,10, Conj. 12. V3 ~ V2 11, (7-54) 13. V2 ~ V3 & V3 ~ V2 4,12 Conj. 14. V2 = V3 13, (7-52) 15. V2 :f:. V3 3, Simpl. 16. V2 = V3&V2 :f:. V3 14,15, Conj. 17. Vt:f:. V3 8-16, Indirect Proof 18. Vt ~ V3 &Vt :f:. V3 7,17, Conj. 19. VI C V3 18, Def. of C 20. (Vt ~ V2&V2 C V3) -+ Vt C V2 1-19, Conditional Proof 21. ('v'X, Y, Z)«X ~ Y & Y C Z) -+ Xc Z) 20, D.G.

11. (a) Assume x E (A - B). Then x E A and x rt B, from which it follows that x E A. Therefore, if x E (A - B), then x E A. Thus, (A - B) ~ A. (b) If A = B, then (A-A)U(A-A) = 0u0 = 0. If(A-B)U(B-A) = 0, then both A - B = 0 and B - A = 0 (otherwise the union could not equal 0). A - B = 0 means .-v (3x)(x E A&x rt B), which by Quant. Neg., DeM., and Condo is equivalent to ('v'x)(x E A -+ x E B), i.e., A ~ B. Similarly, B - A = 0 is equivalent to B ~ A. Thus, A = B. (e) If x E peA) U pCB) then x E peA) and x E pCB). x E peA) iff x ~ A and x E pCB) iff x ~ B. If x ~ A and x ~ B, then x ~ An B, and thus x E peA n B). The converse is proved by taking these steps in the opposite order. 594 SOLUTIONS

CHAPTER 8

1. Assume that atomic statement is defined as in (8-7). Base: Every atomic statement is a Pwff (Polish well-formed formula). Recursion: For all a and /3, if a and /3 are Pwff's, then so are (a) Na, (b) Aa/3, (c) Ka/3, (d) Ca/3, (e) Ea/3 Restriction: Nothing else is a Pwff.

2. f(O) = 2,J(n) = (J(n - 1))2; f(O) = 2 f(l) = (J(0))2 = 22 = 4 f(2) = (J(1))2 = 42 = 16 f(3) = (J(2))2 = 16 2 = 256 f( 4) = (J(3))2 = 2562 = 65,536

4. The power set of the set with zero members, 0, is {0}, which has one member. Since 2° = 1, this establishes the base. To prove the induction step, let Ak+l, a set with k + 1 members, be formed from Ak, a set with k members, by the addition of some element x not in Ak; i.e., Ak+l = AkU{X}. LetB1 ,B2, ... ,B2k bethe2k membersofP(Ak). P( Ak+d contains all these sets plus the sets formed by taking B2k U{ x}. This makes an additional 2k set. Thus, P(Ak+d has 2(2k) = 2k+1 members. The desired result now follows by mathematical induction.

6. The induction fails in going from 1 to 2 as the reader can verify by letting n take on the value 1 in the induction step. PART B CHAPTER 8 595

7. 1. p---+( q---+p) (AI) 2. (( '" q ---+ '" p) ---+ (p---+ q ) )---+ ( '" p---+ ( ( '" q ---+ '" p) ---+ (p---+ q ) ) ) 1, (R2) (Subst. ("'q---+ "'p)---+(p---+q) for p and "'p for q) 3. ( '" p---+ '" q) ---+ ( q---+ p ) (A3) 4. ( '" q---+ '" p) ---+(p---+q ) 3, (R2) (Subst. p for q and q for p) 5. '" p---+ ( ( '" q---+ '" p) ---+ (p---+ q ) ) 2,4, (R1) 6. (p---+( q---+r) )---+( (p---+q )---+(p--+r)) A2 7. ( '" p--+ ( ( '" q ---+ '" p) ---+ (p---+ q ) ) )-+ « '" p-+( '" q-+ rv p) )-+( '" p-+(p-+q))) 6, (R2) (Subst. "'p for p, ("'q---+ "'p) for q, and (p---+q) for r) 8. ( '" p---+ ( '" q---+ '" p))---+ ( '" p---+ (p---+q) ) 5,7, (R1) 9. '" p---+ ( '" q--+ '" p) 1, (R2) (Subst. "'P for p and "'q for q) 10. '" p---+ (p---+q) 8,9, (R1)

8. The alphabet is (p, " N, A, K, C, E); the axioms are the elementary propositions as defined in (8-7); the productions are: x ---+ Nx x,y ---+ Axy x, y ---+ Kxy x,y -+ Cxy x, y ---+ Exy where x and yare any strings on the alphabet. An equivalent semi• Thue system has a basic alphabet as above, an auxiliary alphabet {Q,R}, and axiom set {Q}, and productions: Q -+ NQ Q ---+ AQQ Q ---+ KQQ Q ---+ CQQ Q ---+ EQQ Q ---+ R R ---+ R' R --+ p

9. A = {J,K},B = {a},S = {J,J(}; 596 SOLUTIONS

P: aJ /3 ...... aJaa/3 aJ/3 ...... a/3 aJ(/3 ...... aJ(aaa/3 aJ( /3 ...... a/3 where a,/3 are any strings on (A U B)*. 11. If the axiom had not been independent, it would be provable from the remaining axioms. But then its deletion from the set of axioms would have no effect on the total system: anything that could be proved before could still be proved. If that is not the case, the axiom must have been independent.

12. (c) '0' is interpreted as +1; 'is a natural number' as 'is a power of -2'; 'successor of x' as '-2 times x'. Thus the sequence is viewed as having the form: (-2)°, (-2)1, (_2)2, (_2)3, ...

13. (a) In this model, the "lines" are not the sides or diagonals of the rectangle, but rather pairs of vertices, by Axiom 1.

a b ~ c d

(b) Consider an arbitrary point p. By Axiom 2 there is at least one other point; call it q. By Axiom 3, there is a line containing p and q; call it L 1 • By Axiom 4, there exists a point not in L 1 ; call it r. By Axiom 3 again, there is a line containing p and r; call it L 2 • Thus pis in at least two lines, L1 , and L2 .

(c) Assume the empty set is one of the lines. Consider any point p: by (b) above, p is in at least two distinct lines Ll and L 2 • But then Axiom 5 is violated, because p is a point not in the empty line, but Ll and L2 are both lines containing p and disjoint from the above empty linp.

(d) This was proved in the course of proving (b) above. PART B REVIEW PROBLEMS 597

(e) No, because Axioms 2 and 3 together contradict the new axiom.

REVIEW PROBLEMS, PART B

1. If P +-+ Q is true, then either P and Q are both true or P and Q are both false. In either case P V '" Q is true.

2. Other valid derivations are possible in all the following examples: (a) 1. p 2.q 3. p& q 1,2, Conj. 4. (p & q) V r 3, Addn.

(b) 1. p +-+ q 2. (p -+ q)&(q -+ p) 1, Bicond. 3. (q -+ p)&(p -+ q) 2, Comm. 4.q -+ P 3, Simpl. 5. '" q V P 4, Condo

(c) l.p-+(q ~ ' .....If .. V r) 2. p& "'r 3.p 2, Simpl. 4. q -+ """"r 3,1, M.P. 5. q -+ r 4, Compl. 6. '" r & p 2, Comm. 7. '" r 6, Simpl. 8. '" q 7,5, M.T.

(d) 1. p V q 2. ""p V r 3. ""q 4.q V P 1, Comm. 5.p 4,3 D.S. 6. P -+ r 2, Condo 7. r 5,6, M.P. 598 SOLUTIONS

(e) 1. p& (q -> (r V ",,,,s)) 2.q 3.p 1, Simpl. 4. q -> (r V """" s) 1, Comm. Simpl. (2 steps) 5. r V """"s 2,4, M.P. 6. """"s V r 5, Comm. 7.s V r 6, CampI. 8. p& (s V r) 3,7, Conj.

(f) 1. "" (p V "" q) 2. r V p 3. ""p & '" "'q 1, De M. 4. ""p 3, Simpl. 5.p V r 2, Comm. 6. r 5,4, D.S.

7. t"V t"Vq 3, Comm. Simpi. (2 steps) 8. q 7, CampI. 9. q&r 8,6, Conj.

(g) 1. p -> q 2. p -> (q -> r) 3. q -> (r -> s) 4. p Auxiliary premise 5. q 4,1, M.P. 6. q -> r 4,2, M.P. 7. r 5,6, M.P. 8. r -> s 5,3, M.P. 9. s 7,8, M.P. 10. P -> s 4-9, Conditional proof PART B REVIEW PROBLEMS 599

(h) l. r -+ (p V 8) 2. q -+ (8 V t) 3. f'V8 4. f'Vp& ",t Auxiliary premise 5. "'P 4, Simpl. 6. "'p& f'V8 5,3, Simpl. 7. '" (p V 8) 6, DeM. 8. "'r 7,1, M.T. 9. ",t 4, Camm. Simpl. 10. "'8& ",t 3,9, Conj. 1l. f'V(8 V t) 10, DeM. 12. "'q 11,2, M.T. 13. "'r& "'q 8,12, Adjunction 14. ("'p& "'t) -+ (",r& f'Vq) 4-13 Conditional proof

(i) l. p -+ q 2. "'q&r 3. p Auxiliary premise 4. q 3,1, M.P. 5. f'Vq 2, Simpl. 6. q& f'Vq 4,5, Conj. 7. "'P 3-6, Indirect Proof

(j)l. P V q 2. r& "'P 3. f'Vq Auxiliary premise 4. p 1,3, D.S. 5. f'Vp&r 2, Comm. 6. "'P 5, Simpl. 7. p& "'P 4,6, Conj. 8. q 3- 7, Indirect Proof 600 SOLUTIONS

(k) 1. P H- ("'q - r) 2. ""r& ",(s - q) 3. p Auxiliary premise 4. (p - (""q - r» 1, Bicond. &«""q - r) - p) 5. p - ("'q - r) 4, Simpl. 6. "'q - r 3,5, M.P. 7. "'(s - q) 2, Comm. Simpl. 8. s& ""q 7, Cond., Compl., DeM. 9. ""q 8, Comm. Simpl. 10. r 9,6, M.P. 11. ""r 2, Simpl. 12. r& ""r 10,11, Conj. 13. ""p 3-12, Indirect Proof

3. 1. r & (p V q) Premise 2. ""(p& r) Premise 3. "'(q& r) Premise 4.r 1, Simpl 5.""pV"'r 2, DeM. 6. "'r V "'P 3, Comm. 7. r - "'P 6, Condo 8. '" p 4,7, M.P. 9-12. ""q analogous to 5-8 13. "'p& ""q 8,12, Conj. 14.",(p V q) 13, DeM. 15.p V q 1, Comm., Simpl.

Since 14 is the negation of 15 and both are derived from the premises, the premises are inconsistent.

6. (a) (Vx)(hx - qx) & (3y)(qy & ",hy) or (Vx)(hx - qx)&(3x)(qx& ""hx) or

(Vx)(hx - qx) & (3x) f'V (qx - hx) or (Vx)(hx - qx) & '" (x)(qx - hx) (or yet others); (b) Let I xy: x is identical with y, Pxy: x is a phonemic transcription of y, Ux: x is an utterance, (Vx)(Vy)[(Ux & Uy & '" Ixy) - (Vz)(Vw)«Pzx & Pwy) - '" Izw))J or (Vx)(Vy)(Vz)(Vw)«Ux&Uy& ""Ixy&Pzx&Pwy) -",Izw) PART C CHAPTER 9 601

(or any other logically equivalent expression); 7. (b) D(x) = x is a cab driver, H(x) = x is a head waiter, Sex) = x is surly, C( x) = x is churlish

1. (Vx)«D(x) V H(x)) --+ (S(x)&C(x)) 2. D( v) Aux. Premise 3. D(v) V H(v) 2, Addn. 4. (D(v) V H(v)) --+ (S(v)&C(v)) 1, U.l. 5. S(v)&C(v) 3,4, M.P. 6. S( v) 5, Simpl. 7. D(v) --+ S(v) 2-6, Condo Proof 8. (Vx)(D(x) --+ Sex)) 7, U.G.

PART C

CHAPTER 9

1. (a) The universal set U, the union of all the sets in the collection, is the (two-sided) identity. (b) If U is a member of the collection, then it is the only element with an inverse; viz., itself.

2. (a)

CHAPTER 10

1. Consider the group operation table, or "multiplication table," for ad• dition modulo 4: + mod 4 0 1 2 3 o 0 1 2 3 1 123 0 22301 33012

Closure is shown by the fact that every cell of the table is filled with an element of the set {O, 1,2, 3}. Associativity must be verified by exhaustion (or else by a general argument, which in this case would take even longer, since the best general argument would go by way of proving associativity for addition over all the integers). The fact that o is the unique identity element is evident from the fact that the 0- column and the O-row match the outside column and row respectively, and no other rows or columns do. The fact that 1 and 3 are inverses of each other and that 2 and, of course, 0 are each their own inverses can be seen from the position of the O's in the table.

2. (a) No; e.g. 703 == 10 (mod 11), and 10 is not in the set, so the closure property fails. (b) Yes, if it's associative, which in fact it is. (c) No, a is the identity element; c is its own inverse, but has d as an additional "right-inverse," i.e. cod = a, which violates the uniqueness condition on inverses (Corollary 10.2); in addition, d has no "right• inverse," which violates part of axiom G4. Also, the operation is not associative. (d) Yes. (Remember to check associativity.) (e) No. The identity element is 0, but no sets other than 0 have Inverses. (f) No. (g) Yes. (h) No.

3. (a) The convention for multiplication tables is that the entry in the ath row and the bth column shows a 0 b, not boa. This makes a difference PART C CHAPTER 10 603

for non-commutative operations like this one.

b a 0 b I R R' R" H V D D' f f R R' R" H V D D' R R R' R" f D D' V H R' R' R" f R V H D' D a R" R" f R R' D' D H V H H D' V D f R' R" R V V D H D' R' f R R" D D H D' V R R" f R' D' D' V D H R" R R' f

(b) ({f,R,R',R"},o), ({f,R',H, V},o),({f,R',D,D'},o); (c) The sets are {f,R'}, {f,H}, {f, V}, {I,D}, {I,D'}. The opera• tion tables all look alike: II~ ; X X f

(d) ({I,R',H, V},o) and ({I,R',D,D'},o). It turns out that any cor• respondence which maps f onto f is an isomorphism in this case, e.g., f -+ f, R -+ R', H -+ D, V -+ D', or f -+ f, R' -+ D, H -+ D', V -+ R', etc .. (e) For the subgroup ({f,R,R',R"},O), the only non-trivial automor• phism is f -+ f, R -+ R", R' -+ R', R" -+ R. For the other two subgroups, any of the non-I elements can correspond to any other, so there are actually five different non-trivial automorphisms for each of them, of which the following is one example for ({I, R', H, V}, 0): f -+ f, R' -+ V, H -+ R', V -- H. (f) For the subgroup ({I, R, R', R"}, 0) and any of the subgroups ({I, X}, 0), the only possible homomorphism is f -+ f, R' -+ f, R -+ X, R" -- X. It may be useful to see why some other correspondences do not give homomorphisms. Consider, for instance, the correspondence f(1) = f, f(R) = R', f(R' ) = R', f(R") = R'. This is not a homomorphism because, for instance, R 0 R' = R" but f( R) 0 f( R') = R' 0 R' = f, and f :f f( R"). For either of the other two subgroups, the one with {f,R',H, V} or the one with {f,R',D,D/}, there are several possible 604 SOLUTIONS

homomorphisms with any of the {I, X} subgroups. They all have the following form: Let I and any other one element correspond to I, and let the other two elements correspond to X, e.g., for ({I, R', H, V}, 0) and ({I,D},o), one possible homomorphism is I - I, R' -+ I, H - D, V-D.

4. It must be a subset because the original group had to contain an iden• tity element; by definition the identity element is included, associativ• ity is automatic for subsets of groups, and since the identity element is its own inverse, all the group axioms are satisfied.

5. (a) No. The second and third group axioms are satisfied, but 2,3, and 4 lack inverses, and 203 = 0, which is not a member of the set. (b) Multiplication is associative, the set is closed under it, 1 is the identity element, 2-1 = 4, 3-1 = 5, 6-1 = 6. (c) ({I}, x mod 7), ({1,6}, x mod 7), ({I, 2,4}, x mod 7). (d) ({O, 1,2,3,4, 5}, + mod 6). Correspondence:

0- 1 0 -+ 1 1 -+ 3 1 -+ 5 2-+ 2 or 2 -+ 4 (no others) 3 -+ 6 3 -+ 6 4 -+4 4 -+ 2 5 -+5 5 -+ 3

(e) Condition: n must be a prime number.

6. 5 must be a group to be a subgroup of 5'. The only further condition to be met is that the set of 5 be a subset of the set of 5", and that follows from the transitivity of 'is a subset of,' plus the fact that the subset relation must hold between the sets of 5 and 5' and between those of 5' and 5".

7. (a) Examples: the set of all positive rationals of the form lin, or the set of all positive integers, or the set of all rational numbers equal to or greater than 1. (b) It is a semigroup and a monoid but not a group, because 0 has no inverse.

8. Symmetric difference can easily be shown to be commutative and asso• ciative either by set-theoretic equalities or by Venn diagrams. It is not PART C CHAPTER 10 605

idempotent since A + A = 0. 0 is the two-sided identity element, and every set is its own inverse. (P(A), +) is an Abelian group of order 4.

10. (a) 1. a+b=a+c Premise 2. -a + (a + b) = -a + (a + c) By D2, a has an additive in• verse -a and addition is well• defined and unique 3. (-a + a) + b = (-a + a) + c By D2, + is associative 4. 0 + b = 0 + c Def. of inverse, 0 is the addi• tive identity 5. b = c Def. of additive identity (b) 1. a+O=a Def. of identity 2. ao(a+O)=aoa Uniqueness of multiplication (i.e. if a+O and a are the same element, then there is a single element which is that element multiplied by a 3. ao(a+O)=aoa+aoO Distributive law 4. aoa+aoO=aoa 2,3, Transitivity of = 5. -(a 0 a)+(a oa+a 00)= -(a oa)+a oa Existence of add. inverse, Uniqueness of addition 6. (-(a oa)+a oa)+a 00 = -( a 0 a) + a 0 a Associativity of + 7. O+aoO=O Def. of inverse 8. a 00 = 0 Def. of identity The proof that 0 0 a = 0 is similar. 0 0 a = 0 also follows directly from a 0 0 = O.

(c) (1) Associativity of + (2) Distributive law (3) Def. of inverse (4) By the theorem of problem b. (5) Distributive law (6) Def. of in verse (7) By the theorem of problem b. (8) From 4, 7, by symmetry and transitivity of =. 11. Given: a < band b < c. Then a+b < b+c, by the addition law. Since 606 SOLUTIONS

-b = -b, (a+b)+(-b) < (b+c)+(-b) by the addition law a + (b + (-b)) < « -b) + b) + c by assoc. and comm. a+O

13. (a) To show: every positive integer n has the property that for all m,

(i) To show that 1 has that property:

am 0 a l = am 0 a (by def.) = am +1 (by def.) (ii) To show that if k has the property, then k + 1 must have it: am 0 ak = am+k conditional premise am 0 ak+l = am 0 (a k 0 a) by def. =(amoak)oa by assoc. of 0 in integral domains = am +k 0 a by the conditional premise = a(m+k)+1 by def. = am+(k+l) by assoc. of + in ordinary arithmetic From (i) and (ii) the conclusion follows by the principle of math• ematical induction. (b) We will use induction on n. pen): for all m, (am)n = (an)m (i) for n = 1: (am)l = am (by def.) = (al)m (since a = a l , by def.) (ii) To show that P(k) -.. P(k + 1): (am)k (ak)m conditional premise (am )k+l (am)k 0 am by def. (ak)m 0 am by the conditional premise (akoa)m anobn=(aob)n (theorem proved in text) (ak+l )m by def.

Since P(l) holds and P(k) -.. P(k + 1), it follows from the prin• ciple of mathematical induction that P( n) holds for all n. PART C CHAPTER 11 607

CHAPTER 11

i. 11-1 and 11-2 are lattices; in 11-3 there is no lub for {a,b}, nor any glb for {c, d}.

2. (i) (a), (b), and (c) are posets; (d) and (e) are not transitive. (ii) (a) is a semilattice; (b) and (c) are not lattices.

4. Let the poset A = (A, ~ ) be a meet semilattice. Set a /I. b = inf{ a, b}. Then the algebra Aa = (A, /I.) is a semilattice.

(a) For all a, a /I. a = a (idempotent), since inf{a, a} = a. (b) For all a, b, a/l.b = b/l. a (commutative), since inf{ a, b} = inf{b, a}. (c) Let d = inf{a,inf{b,c}} and e = inf{inf{a,b},c}. Then d ~ a and d ~ inf{b,c}. Therefore, d ~ band d ~ c. Since d ~ a and d ~ b, d ~ inf{a,b}. And since d ~ c, d ~ inf{inf{a,b},c}; i.e., d ~ e. Similarly, e ~ d; thus, d = e. Hence, a /I. (b /I. c) = (a /I. b) /I. c (associativity).

6. Suppose (D1) holds. Then a V (b /I. c) (a V (a /I. c)) V (b /I. c) ( L4 ) = aV«a/l.c)V(b/l.c)) (L2) aV«c/l.a)V(c/l.b)) (Ll) aV(c/l.(avb)) (Dl) aV«aVb)/l.c) (Ll) (a /I. (a Vb)) V « a Vb) /I. c) (L4) «a V b) /I. a) V «a V b) /I. c) (L1) (aVb)/I.(avc) (Dl) So (D2) also holds. A similar proof shows that if (D2) holds, so does (Dl).

7. By Theorem 11-8, a* has a unique complement in a complemented distributive lattice. Since a* /I. a = 0 and a*Va = 1, a is the complement of a*, i.e., a = (a*)*.

8. Interchange /I. and V, 0 and 1 in the proof of Th. 11-11. 608 SOLUTIONS

9. Th. 11.8: a" = a* II 1 def. of 1 and II = a*lI(aVb*) def. of compI. (C1) (a" II a) V (a" lib") D1 o V (a* II b*) C2 = a* II b* def. of 0 and V Th. 11.9: d = dllc RC2, def of V and II d II (b V d') RC2 (d II b) V (d II d') D1 = d II d' RCI, def of V

CHAPTER 12

1. Ifallb=O then a* a* V (a II b) (a* Va) II (a" Vb) = I II (a* Vb) = a" V b; hence a" :::: b Ifavb=l then a* a* II (a V b) (a* II a) V (a* II b) = OV(a*lIb) = a" II b; hence a* ~ b So b = a*.

2. (Idempotent Law) For all a E B, a U a = a and a n a = a Proof: 1. a = aU 0 B4 2. a=aU (ana") BS 3. a=(aU a)n(aua*) B3 4. a = (a U a) nIBS S. a = a U a B4 an a = a can be proved similarly

3. (i) If a = 0, the set of join-irreducible elements S( a) = {x I x ~ a} = 0, and 0 is the lub of the empty set. (ii) Let P( n) be the statement that every element is the join of some join-irreducible elements if the number of elements x ~ a in the finite PART C CHAPTER 12 609

Boolean lattice L is n. P( n) is trivially true if a is join-irreducible. If a is not join-irreducible and not 0, then a = x V y, where x < a and y < a. So n(x) < n(a) and n(y) < n(a). By induction on n, it follows that x and yare joins of join-irreducible elements: x = VS(a) n( x) and y = VS(a) n(y), so a = Vn(x) V Vn(y).

4. Suppose a ~ b V c, then a = (bVc)!\a = (b!\a)V(c!\a) (distributivity) (b!\ a) or (c!\ a) since a is join-irreducible So a ~ b or a ~ c.

5. (i) If L has 1, then a is its own relative pseudo-complement, since by definition for all x E L, x ~ a =? a iff x !\ a ~ a, so a ~ a =? a. (iii) For all x E L, x ~ a =? b iff a !\ x ~ b; since a !\ b ~ b we infer b ~ a =? b (provided a =? b exists).

6. Since the lattice if finite, it has O. Given a, b, define

Let Co be the lub of C, so Co = Cl V C2 V ... V Ck. Then, if a !\ c ~ b, also c ~ co. Conversely, if c ~ Co

a!\c~a!\co a !\ (Cl V C2 V ... V Ck) (a!\ cd V •.. V (a!\ Ck) (distributivity)

But a !\ Ci ~ b for each i, 1 ~ i ~ k. Therefore, a !\ Co ~ b. Hence a !\ C ~ b. Thus Co satisfies the conditions on a =? b. So the lattice is a Heyting lattice. 610 SOLUTIONS

REVIEW EXERCISES, PART C

2. For instance, P = ({O, 1,2,3,4}, +mod 5)

3. Care must be taken in this problem to remember that x + (y - 3) is defined as a single operation on two elements x and y. Thus to check associativity it must be verified that

«x + (y - 3)) + (z - 3») = (x + (y + (z - 3)) - 3)) which turns out to be true because both sides can be reduced to (x + +y + z) - 6. The identity element is 3 and the inverse of x is 6 - x.

4. (a) The only finite one is ({ O}, + ). All others consist of all the mu tiples of anyone integer, e.g. {O, 2, - 2,4, -4, ... }, {O, 10, -10,20, - 20, ... }, etc. (b) ({O},+) certainly is not. All of the others are. To set up an isomorphism between the group with all integers and the group with all multiples of a,let n -- an. This correspondence is clearly one-one and preserves addition, since

5. There are only two: 0 -- I 0-- I 1 -- R 1 -- R" 2 -- R' 2-- R' 3 -- R" 3 -- R 6. A non-Abelian group of order 6.

7. (a) Every string is a conjugate of itself since x = x ,-.. e e ,-.. x. Conjugacy is symmetric by the definition. To prove transitivity, let x and y be conjugate and also y and z. Then for some t, u, v, w, x = ,-.. ...-...... -...... -.... d t u, Y = u t, Y = v W, and z = W v. Case 1: let u = v an t = w. Then x = t ~ u = w "--'v = Zj thus x and z are conjugate because they are identical. Case 2: let u be shorter than v; that is, there is some r such that u '-"r = v. Since y = v ~w = u ~r '-"w = u ~t, it follows that r '-"w = t. Therefore, x = t ~u = r '-"w ~u, and Z = w ~v = w ~u ""-"rj thus x and Z are conjugate. Case 3: let u be longer than Vj that is, for some s, v ,-.. s = u. Since y = PART C REVIEW PROBLEMS 611

u '-"'t = v r-- 8 """' t = V r-- w, it follows that w = 8 r-- t. Therefore, x = t "-"u = t "-"v "-"8, and z = w "-"v = 8 r--. t r--. v ; thus, x and z are conjugate. This exhausts the possible cases. This relation partitions A* into equivalence classes, each class containing all the strings that are conjugates of each other. A string of length n may be the only string in its equivalence class (aaaa, for example, is conjugate only with itself) or there may be as many as n strings in the class (abca for example, is conjugate with itself and with bcaa, caab, and aabc). (b) Let x and y be conjugates. Therefore, x = uv and y = vu, for u and v. The string u is a string such that x r-- U = U y, since ,,-., r--,,-., X U = U v u = u y.

8. (1) (a) all; (b) all but G4; (c) Yes (2) (a) none: G1 not, because 2 - 4 ::/: a non-neg. integer, G2 not, because (9 - 5) - 4 = 0, 9 - (5 - 4) = 8, G3 not, because there is no e such that for all x, both x - e = x and e - x = x, G4 not, because G3 not; (b) all but G4; (c) no; D1 not, D2 not (3) (a) all; (b) all but G4 (5 has no inverse); (c) no: D4 not, because 5·2= 10 (mod 25) and 5·7 = 10 (mod 25), but 2 ::/: 7 (mod 25) (4) (a) only G2: G1 not, because 1 + 10 = 0 (mod 11), and 0 rt A; G3 not, because 0 rt A; G4 not, because G3 not; (b) all; (c) no: D1 not, D2 not (5) (a) only G2; (b) all; (c) no: D1 not, D2 not (6) (a) only G2: G1 not, because 2/3 + 2/3 = 4/3,4/3 ¢ A, G3 not, because 0 ¢ A, G4 not, because G3 not; (b) only G1 and G2; (c) no: D1 not, D2 not, D3 not (7) (a) all; (b) G3 not, because for no x does 38 x = 3, G4 not - only G1 and G2; (c) D3 not, D4 not, because 384 = 38 .S = 0, but 4::/:5

9. ( a) (a + b) . (c + d) [( a + b) . c] + [( a + b) . d] [c· (a + b)] + [d· (a + b)] D3 (commut.) (c . a + c . b) + (d . a + d . b) D.S (a·c+b·c)+(a·d+b·d) D3 (b) -0 + 0 = -0 because :r + 0 = x for all x (G3) -0 + 0 = 0 because -x + x = 0 for all x (G4) hence -0 = 0 because -0 + 0 must have a single value (G 1) 612 SOLUTIONS

(c) l. a· b = 0 premise 2. a;iO auxiliary premise 3. a·O = 0 Problem 1O(b) in Chapter 10 4. b=O 1,3 cancellation law (D4) 5. a;iO:)b=O condo proof 6. rva;iOVb=O 5, conditional law 7. a=Ovb=O double neg.

(d) l. -a + -( -a) = 0 Inverse law 2. -a + a = 0 Inverse law 3. -(-a)=a 1,2, problem 10(a) in Chapter 10

10. (a)Yes. Ixl· Iyl = Ix· yl and we can show the new system is also a group. (b) No. -2 ~ 2, -4 ~ 4,8 -+ -·8 but -2· -4 = 8 while 2·4;i -8. (c) No. 2 -+ 4, 3 ~ 6, 6 -+ 12 but 2 . 3 = 6 while 4 . 6 ;i 12. (d) Yes. l/x· l/y = l/x· y and the new system is a group. (e) Yes. x2 . y2 = (x· y)2 and the new system is a group.

11.* (a) Reflex: for all a, a-I. a = e & e E Sj Sym: given a-I. bE S, does it follow that b-l . a E S7 (b-l.a).(a-l.b) b-1.(a.a-1).b b-1 ·c·b b- 1 ·b e Since (b-1·a)·( a-1 .b) = e, b-1·a is the inverse of a-I ·b. Therefore b-1 . a must also be in S. Trans: given a-I. bE S, b-1 • c E S then (a- 1 .b).(b-1 .c)ES = a-1.(b.b-1).c = a-I. e . c = a-I. c so a-I . c E S It is an equivalence relation. One of the equivalence classes is {x I XES}. The others can vary from case to case. For exam• ple, for G = the symn;J.etries of the square and S = {I, H}, the equivalence classes are: El = {J,H} E2 = {R', V} E3 = {D,R} E4 = {D', R"} PART C REVIEW PROBLEMS 613

But for the same G and S = {I, H, V, R'}: E1 = {I, H, V, R'} E2 = {D, D', R, R"} (b) Reflex: a + (-a) = 0 and 0 is even, Sym: if a + (-b) = c and c is even, then b + (-a) = -c which is also even, Trans: if a + (-b) is even and b + (-c) is even, then a + (-c) is even because it is the sum of two even numbers, a + (-b) and b + (-c), Equivalence classes:

E1 = {x I x is odd} (because the difference of any two odd numbers is even) E2 = {x I x is even };

(c) Reflex: a + (-a) = 0 not odd: no, Sym: if a + (-b) = c is odd, so is b + (-a) = -c, Trans: if a + (-b) is odd and b + (-c) is odd, a + (-c) is not odd, a + (-c) = (a + (-b)) + (b + (-c)) = odd + odd = even: No not an equivalence relation;

12. (a)

1axis 31 / 7 / I R 120 0 clockwise R 2400 clockwise Dl flip in axis 1 D2 flip in axis 2 D3 flip in axis 3

1 axis 11 614 SOLUTIONS

y x·y 1 R R' Dl D2 D3 1 1 R R' Dl D2 D3 R R R' 1 D2 D3 Dl X R' R' 1 R D3 Dl D2 Dl Dl D3 D2 1 R' R D2 D2 Dl D3 R 1 R' D3 D3 D2 Dl R' R 1

(b) {I, R, R'}, {I, DIl, {I, D2}, {I, D3}, {I} (C) 1 R ~' } _ I R' --+1 Dl D, } D2 D2 --+ Dl D3 D3

14. (a) a ~ b iff a=> b = 1. (c) b 1\ a ~ a 1\ b, and c ~ a => b iff a 1\ c ~ b, so a ~ b => (a 1\ b).

PART D

CHAPTER 13

1. (a) tree (ii) (b) interpretation (ii)

("-'(p&q) V p), 2.3 1,2.3 ~ ~ "-'(p&q), 2.1 p 0,2.1 1 I I (p&q),2.2 1,2.2 A A p q 1 1 PART D CHAPTER 13 615

2. (b) tree (ii) (e) interpretation (ii) Conn (Neg (Conn (p,q))), p, 2 1 ~ v Neg (Conn (p, q)), 2 p (1,1)->1 (0,1) ~ (1,0)->1 I '" Conn (p, q), 1 (0,1)-+1 0 A (0,0)--0 ~ & p q 1--0 1 0--1 ~ (1,1)--1 (1,1) (1,0)--0 (0,1)--0 (0,0)-0 3. (a) A possible model making all four formulas true is: M = (D,F) where D = {a,b} F(P) = {a} F(R) = {(a,a),(b,a)} with assignment given g(x) = a and g(y) = b. Note that although (iii) is logically equivalent to (iv), the inter• pretation process is quite different. (b) (i) Every student kissed someone. (ii) There is someone whom every student kissed. (iii) & (iv) If someone/anyone is a student then everyone kissed him, i.e., Everyone kissed every student.

5. Take a very simple model where your choice of statement for "y is true. The entire statement-schema could be instantiated by e.g., (( '" (p--q)) V r), but the choice of statement constants is yours.

7. (b) t, FALSE, (d) t, TRUE (convert 5/ x, 8 / y), (e) t, FALSE(convert 5 / y, 8 / x), (g) t, TRUE 616 SOLUTIONS

CHAPTER 14

1. E' E ~e ~

Conservativity: only a and b matter to DAB. Extension: E and E' are irrelevant to DAB. Quantity: only the cardinalities a, b, c, e matter to DAB. Variation: 0 C D(A) C p(E) (proper subsets). Cumulative effect: D(A) is completely specified by the set of all pairs (a, b), the cardinalities of A and B.

2. (a)

0,2 0,3

(b) 0,0

10~'11 1 0,2 2 1 1 2 0,3

(d) PART D CHAPTER 14 617

(e)

3,0

(f)

0,3

(g) Some partes) of the tree must be shaded. (h) Any node to the right of a node in D(A) is itself in D(A). (i) If (a,b) belongs to D(A) then (O,b) belongs to D(A).

3. right left T ! T ! several + +/7 at most 3 + + none + + at least n + + some + + these + neither + every + + all + + each + + infini tely many + + a finite number of + + most + many + See section 5 for most and many.

4. (a) (i) Some men walk fast -+ some men walk. Several students failed the final test -+ several students failed a test. Infinitely many numbers are even primes -+ infinitely many numbers are prime. (ii) If some men are over 60 and some men are over 20 then some men are over 40. 618 SOLUTIONS

(b) lEI ~ 2,D(A) = {X ~ EIAnx =10 and A -X =l0} D(A) means 'some but not all A's.' If A ~ B, D(A) ~ D(B), so D is left monotone increasing. But D is not right monotone increasing since E f/. D(A) for all A. Another example is 'not every'.

5. Model should show mapping 7r : A -t A' where at least one a E A n X, 7r( a) f/. A n X, if X is the set of entities owned by John.

6. Suppose QE meets Variation, i.e. X E QE' and Y f/. QE' for some X, Y ~ E'. rv QE =def {Z ~ EIZ f/. Qd. By Extension, Z E QE' iff Z E Q E. Hence X f/. rv Q E' and Y f/. rv Q E', so rv Q E satisfies Variation. For Q rv similarly.

7. (a) We know that if Q is monotone increasing, Q '" and", Q are monotone decreasing, and if Q is monotone decreasing, Q rv and '" Q are monotone increasing. Since QN =,..... (Q "') it follows by double negation that if Q is increasing, QN is. (d) If Q = QN, X E Q .... (E - X) f/. Q. SO if for some Y C E, (E - Y) E Q then Y f/. Q and if Y f/. Q then (E - Y) E Q.

8. Since D is definite, D(A) is the principal filter generated by some B ~ E. DAE by definition of principal filter, so by theorem 14.2 DAA. Note that DAB-tDAA is the relational definition of positive strength, also called the property of quasi-reflexivity.

9. (a) (i) DAB .... D(A n B)E: DAB ...... DA(A n B) (conservativity) .... D(A n (A n B» (AnB) (intersective) .... D(AnB)(AnB)(AnB = An(An B») ...... D(A n B)E (Theorem 14.2) 0

(ii) DAB <-+ DBA: DBA <-+ D(A n B)E (by 9.(a)(i» and DAB <-+ D(A n B)E (by (a)(i») so DBA <-+ DAB. 0

(b) Some students walk -t some walkers are students -t some walking students are individuals. (There exist walking students.)

10. (a) DAB & DBC, so A ~ Band B ~ C by theorem 14.3, so A ~ C. By Conservativity DA(A n B), so DAA, since An B = A. But then also DA(A n C), since An C = A. So by Conservativity DAC. 0 (b) If D is connected, rv D is antisymmetric by defini tion of connect• edness. With the result of (a), '" D is transitive. So D is almost• connected by definition of transitivity. 0 PART D CHAPTER 15 619

(c) Suppose DEAB and D transitive. By Conservativity DEA(AnB). Take E' 2 E and A' ~ E' such that IA'I = IAI and A n A' = An B. Clearly DEA(A n A'), and with Extension DE,A(A n A'). With Con• servativity DE,AA'. Take 7r a permutation of E' which leaves A n A' and E'-(AUA') intact, but interchanges A-A' and A'-A. Quantity gives DE,7r(A)7r(A'), i.e. DE,A' A. Since D is transitive, DE,AA' and DE/A' A imply DE,AA, and Extension gives DEAA. 0

CHAPTER 15

1. g"( x) = d3 , since Pd3 is true at i 2 , which is the only index accessible from i l if we drop all reflexive pairs from R, and Qd3 is true at i l , the only remaining accessible index for i 2 .

2. Evaluating C3x )Px at io gives 0, so the entire antecedent is false at io, and the whole formula hence must be true at io.

3. (a)

reflexivity 'P true on io, i l , i2

(b) transitivity ~-~~~ 70 II 12 'P true on io, iI, i2

PART E

CHAPTER 17

1. (a) (ii), (iii), (iv); (b) The set of all strings containing a total of n l's, where n == 2 (modulo 3)

2. (c)

3. (a) o o

(b) PART E CHAPTER 17 621

(c)

(d)

0,1

(e)

"-_____)0

(f) 622 SOLUTIONS

4. (c)

5. (a) c

(b)

6. (a) Duplicate the states of the original machine, and duplicate all the transitions except those leading to So. Add a new state 5b. Wherever the original machine had a transition from a state 5, into So, labeled a, let the new automaton have a transition from 5, into 5b, labeled a. If the original automaton had a loop transition on So, add a loop transition on 5b instead. For every other transition from So to a state

5J , add a transition from 5b to 5J , with the same label. If So was PART E CHAPTER 17 623

a final state, make 5b a final state as well. The resulting automaton starts in So, but for the rest of the computation, 5b plays the role previously played by So.

7. (a) €Jo 0

(b) o

(c)

(d) A deterministic equivalent of Cis:

The complement construction then gives us D: 624 SOLUTIONS

D accepts any string containing a positive even number of 1 'so

8. (a)

(b) 1

(c)

9. (a) the (man is u men are) here (b) the old * (man is U men are) here ( c) the old * (man is U men are) here (and the old * (man is U men are) here)* PART E CHAPTER 17 625

(d) 0*10*10* (e) (0 U 1)*101(0 U 1)* or (corresponding to the automaton given as the answer to the original problem) (0*11*00)*0*11*01(0 U 1)*

10. (a)

(b) o

"-_____~1

(c)

(d)

(e) o "-_____)1 626 SOLUTIONS

(f) 0,1

11. (a) S -- aa (b) S--e (c) S -- aA A--a B--b S -. ab S -.as S --bS A--aB B-.bB S -- ba S-.a A--bA S -- bb S --bS S--b

(d) S--a A--bA C--aC (e) S--e A--aS B--bA S--b A--aB C--bD S--aA A--a B--aC S--aC A--a C --b S--bC A--bB C--aB S --bA B--bB D--aD C--bS B --b D--a C --b (f) S -- bA A--bS C--bB S-.aB A--aC C-.aE B--bC D--bE B--aD E--bD B --a E--b

12. (a) (a, qo, qd (c) (b,qo,qo) (e) (a, qo. qt) (a, q3, q2) (b, qo, qt) (a,qo,qd (a, ql, qo) (b, q3, qo) (a, ql, q2) (a,ql,q2) (b, ql, q2) (b, qo. q3) (b, ql, q2) (b, ql, qt) (b, q2, qt) F = {qo} F = {q2} (b,q2,q2) (a, q2, q3) F = {q2}

CHAPTER 18

1. (a) S--aSa (b) S--AA (c) S--aBB A-+aS B-+bS S -- aBa A -- aAb S -- bAB A -- bBAA B -+ aBBB B-+bB A--ab S--bBA A-+bABA A-+bAAB PART E CHAPTER 18 627

2. Any rules which are already of the form A - a or A --+ BC can be left unchanged. Remove all rules of the form A --+ B by the procedure out• lined in Sec. 18.5. For each rule of the form A--+(\:1(\:2".(\:n, where n 2: 2 and one or more of the (\:1 are terminal symbols, replace each oc• currence of such an (\:1 by a new non-terminal AI that occurs nowhere

else and add the rule AI - (\:1 (which is of the allowed form). The right sides of all rules now consist either of a single terminal symbol or else a string of two or more non-terminals. Each remaining rule of the form A --+ BI B2 ... Bn (n 2: 3) is now replaced by the rules A --+ BI C I, CI - B 2C2,···, Cn - l - Bn-IBn , where CI , C2 , ... , Cn - l are new non• terminals that occur nowhere else in the grammar. All rules are now of the required form. Clearly for every derivation of a terminal string in the original grammar there is a derivation of that string in the new grammar and conversely.

3. (b) If L is context free, then for some sufficiently long string w E L, there exist U,V,X,y,z such that w = uvxyz, v and y not both e, and UVlxylZ E L for all i 2: O. Let u = aP, v = aq , x = aT, y = as, Z = at, and call b = p + r + t and c = q + s. Then b + ic must be prime for all i 2: O. When i = bc2 , b + ic = b + (bc 2 )c = b + bc3 = b(l + c3 ) = b( 1 + c)( 1 - c + c2). Since c 2: 1, (c + 1) 2: 2 and thus b + bc3 is divisible by 2 or some large integer. This establishes that b + bc3 could not be prime unless it happened to equal 2. However, in this case b = 1 and c = 1, and it is obviously false that 1 + i . 1 is prime for all integral values of i. Therefore an (n prime) is not type 2.

4. If L is generated by a cfg G, construct G' by reversing the right sides of all rules of G. G' generates LR.

5. M = (I\,L:,r,~,s,F) 1\ = {qo, ql} ~ = {( qo, ab, e) ---. (qo, A) L: ={a,b,c,d} (qo,cd,A)-(ql,e) r ={A} (ql,cd,A)-(ql,e)} s = qo F ={qd 628 SOLUTIONS

6. ~={(qO,e,e)--->(q1,$) (q1, a, $) ---> ( q1 , A$) (q1, b, $) ---> (q1, B$) (q1, a, A) ---> (q1, AA) (q1, b, B) ---> (qb BB) (q1, a, B) ---> (q1, e) (qb b, A) ---> (q1, e) (q1, e, $) ~ (q1, e)} F={qd

7. The machine will store the number of a's preceding the b. Then it will make a choice of whether to start checking off 1 or 2 a's from the storage tape with each new a of the input tape. Once made, the choice must stay consistent for the rest of the computation. ~ = {(qo, a, e) ---> (qo, A) ( qo, b, e) ---> (q1, e) (qo, b, e) ---> (q2, e) ( q1 , a, A) ---> (qb e) (q2, aa, A) ---> (q2, e)} F = {q1, q2}

8. No. Problem 7 provides a counterexample.

9. Choose akbkek in L. Then if L is context free, there exist U,V,X,y,z, (Ivyl ~ 1) such that uvxyz = akbkek and uvnxynz E L for all n ~ o. But if so, then neither v nor y can contain more than one type of letter since "pumping" would produce a string with b's preceding a's, or e's preceding b's. Therefore, the only remaining possibilities are: 1) both v and y in a*, 2) both v and yin b*, 3) both v and yin e*, 4) v in a* and y in b*, 5) v in a* and y in e*, or 6) v in b* and y in e·, provided, of course, that not both v and yare empty. But in cases 1-4 "pumping" v and y produces a string not in L since the a's, b's, and e's will not be in the required proportion of i a's, j b'S, and max(i,j) e's. In case 5, if y = e, then it is like case 1; if v = e, then it is like case 3; if neither v nor y is empty then n = 0 ("pumping" zero times) produces PART E CHAPTER 19 629

a string not in L. Case 6 is similar. Therefore,' no such u,v,x,y,z exist, and L is not context free.

CHAPTER 19

1. 15 = {(qo,O) -+ (qO, R) 2. 15 = {( qo, # ) -+ ( ql, 1) (qO' 1) -+ (ql, R) (ql, 1) -+ (ql, R) ( ql , 0) -+ (qo, R) ( ql, # ) -+ ( q2, 1) ( ql, 1) -+ ( q2, L) (q2, 1) -+ (q2, R) ( q2, 1) -+ ( q3, 0) (q2,#)-+(qo,1)} (q3,0)-+(q4,R) s = qo (q4, 1) -+ (q3, 0) (q4, 0) -+ (qo, R)} s = qo 3. (a) qo a # # a (b) The machine moves back and forth a qo # # a along the tape; each time it encounters an a ql a # a # it changes it to a and reverses direction; ql a a # a (c) No; (d) No; (e) No; (f) Same as (b) ql # a a # a a ql a a # a a qo a a # a a a qo a # a a a a qo # a a a a ql a a a a ql a a a a ql a a a a 4. (a) M = ({qo,q!l,{a,#},b,qo) 15 = {( qo, #) -+ (qO, R) (qo,a) -+ (qll R) ( ql , # ) -+ ( ql , R) ( ql , a) -+ ( ql , R) } (b) M = ({qo,q!l,{a,#},b,qo) 15=0 5. (a) Total, since the sum of any two even numbers is even. (b) Partial. There are some prime numbers whose sum is prime (2+5 = 630 SOLUTIONS

7,2+ 11 = 13), but unless one of the primes is 2, both primes will be odd, and their sum will then be even, and hence divisible by 2, i.e. not prime. ( c) Partial.

6. Reading the digits from right to left, multiply the first digit by 20 ( = 1), the second by 21, and so on; in general, the nth digit is to be multiplied by 2n. Then add the resulting numbers to get the result, e.g. 101 => (1 x 2°) + (0 x 21) + (1 X 22) = 1 + 0 + 4 = 5.

7. G = ({[,]'A,D,S,a},{a},R,S) R={S-[AJ, [-t [D, DA-AAD, DJ -], [ -e, J -e, A-a}

8. Given arbitrary M, modify it so that it first erases its input tape and then proceeds as M would (on the resulting empty tape). Call this machine M'. Then M' accepts its input (in fact, all inputs) iff M accepts e. Thus, a TM which decided whether an arbitrary TM accepts at least one input could be applied to M' to decide, in effect, whether M accepts e. Since the latter is impossible, the problem is undecidable.

CHAPTER 20

1. (a) S - aSBc 5 -t aBc cB-Bc aB -t ab bB-bb (b) Replace the rule S -> e by S - ABC in the grammar of (19-5). ApPENDIX E II 631

(c) 5 --+ aa Aa --+ aA A' A --+ aA' 5 --+ bb Ab --+ bA A' B --+ aB' 5 --+ aA5' Ba --+ aB B' A --+ bA' 5 --+ bB5' Bb --+ bB B' B --+ bB'

5' --+ aA5' AA --+ aA' A' A" --+ aa 5' --+ bBS' AB --+ aB' A' B" --+ ab 5' --+ aA" BA--+bA' B' A" --+ ba 5' --+ bB" BB --+ bB' B' B" --+ bb

APPENDIX E II

L (c) o o

An even number of automaton. Note that there is a non-trivial loop between So and 51, so this automaton is cyclic. (d) o

Almost all automaton. Note that this automaton is not permutation invariant. 632 SOLUTIONS

REVIEW PROBLEMS, PART E

1. (a) I am right; That I am right seems certain; That that that I am right seems certain seems certain seems certain. (b) For example: J( = {qo,qd,I: = {s,e,i,a,r,t},r = {t},F = {qo} 6 = {(qo,t,e)-t(qo,t) (qo,iar,e)-t (ql,e) ( ql , se, t) -t ( ql , e) }

2. (b)

0,1 (c) eo! _____~1

3. (a)j (GI*1l)*O*1l or 0*1l(0*1l)*; (b) 111 * or 1* 1l or 11 * 1; (c) (0*(11}*0*)* PART E REVIEW PROBLEMS 633

4. J( = {qQ,qI,q2,q3},L: = {a,b,#},s = qQ 6 = {(qQ, b) -> (qQ, R) (qQ, #) -> (qQ, R) (qQ, a) -> (ql , b) ( ql , b) -> ( q2, L) (q2,a)->(q2,L) ( q2, # ) -> ( q2 , L) (q2,b)-> (q3,b)} Bibliography

PART A

Introductory Textbook: Halmos, P.R.: 1960, Naive Set Theory, Princeton, N.J., Van Nostrand.

Set Theory, Paradoxes and Foundations: Anderson, A.R. (ed.): 1964, Minds and Machines, Englewood Cliffs, N.J., Prentice-Hall. Barwise, J. and J. Etchemendy: 1987, The Liar: A n Essay on Truth and Circularity, New York, Oxford University Press. Cohen, P.: 1966, Set Theory and the Continuum Hypothesis, New York, Benjamin. Copi, LM.: 1971, The Theory of Logical Type, London, Routledge & Kegan Paul. Dauben, J.W.: 1979, Georg Cantor: His Mathematics and Philosophy of the Infinite, Cambridge, MA., Harvard University Press. Frankel, A. and Y. Bar-Hillel: 1973, Foundations of Set Theory. 2nd edi• tion, Amsterdam, North-Holland. Godel, K. 1940, The Consistency of the Axiom of Choice and of the Gener• alized Continuum Hypothesis with the Axioms of Set Theory, Princeton University Press. Hofstadter, D.R.: 1979, Gadel, Escher, Bach: An Eternal Golden Braid, New York, Vintage Books. Kunen, K: 1980, Set Theory: An Introduction to Independence Proofs, Am• sterdam, North-Holland. Martin, R.L.: 1984, Recent Essays on Truth and the Liar Paradox, New York, Oxford University Press. Moore, G. H.: 1982, Zermelo's Axiom of Choice. Its origins, development and influence, New York/Heidelberg/Berlin, Springer Verlag. Quine, W. van Orman: 1963, Set Theory and Its Logic, Cambridge, MA., Harvard University Press. Quine, W. van Orman: 1966, The Ways of Paradox, New York, Random House. 635 636 BIBLIOGRAPHY

Russell, B.: 1919, Introduction to Mathematical Philosophy, London, Allen and Unwin. Smullyan, R.: 1978, What Is the Name of This Book? Englewood Cliffs, N.J., Prentice-Hall. Zadeh, L.A.: 1987, Fuzzy Sets and Applications: Selected Papers, New York, Wiley. Part B

Introductory Textbooks: Copi, LM.: 1965, Symbolic Logic, 2nd edition, New York, Macmillan. Jeffrey, R.C.: 1967, Formal Logic: Its Scope and Limits, New York, McGraw• Hill. Kalish, D. and R. Montague: 1964, Logic, New York, Harcourt, Brace and World. Mates, B.: 1972, Elementary Logic, 2nd edition, New York, Oxford Uni• versity Press. Thomason, R. H.: 1970, Symbolic Logic, New York, Macmillan. An innovative method for learning predicate logic on a Macintosh computer: J. Barwise and J. Etchemendy, Tarski's World. Available from Kinko's Courseware 1987.

Mathematical Logic, Axiomatization and Metatheory: Barwise, J. (ed.): 1977, Handbook of Mathematical Logic, Amsterdam, N orth-Holland. Bell, J.L.: 1978, Boolean-valued Models and Independence Proofs in Set Theory, Oxford, Clarendon Press. Bell, J.1. and M. Machover: 1977, A Course in Mathematical Logic, Ams• terdam, North-Holland. Beth, E.: 1962, Formal Methods: An introduction to symbolic logic and to the study of effective operations in arithmetic and logic, Dordrecht, Reidel. Beth, E.: 1970, Aspects of Modern Logic, Dordrecht, Reidel. Boolos, G. and R. Jeffrey: 1980, Computability and Logic, 2nd edition, Cambridge, England, Cambridge University Press. Chang, C.C., and H.J. Keisler: 1973, Model Theory, New York, American Elsevier and Amsterdam, North-Holland. Davis, M.: 1965, The Undecidable, Hewlett, N.Y., Raven Press. Davis, M. and E.J. Weyuker: 1983, Computability, Complexity and Lan• guages. Fundamentals of Theoretical Computer Science, New York/Lon• don, Academic Press. BIBLIOGRAPHY 637

Gamut, L.T.F.: 1991, Logic, Language, and Meaning (2 vols), Chicago, University of Chicago Press. G6del, K.: 1962, On Formally Undecidable Propositions, New York, Basic Books. Kleene, S.C.: 1967, Mathematical Logic, New York, Wiley. Kneale, W. and M. Kneale: 1962, The Development of Logic, Oxford, Clarendon Press. Kreisel, G., and J.L. Krivine: 1967, Elements of Mathematical Logic: Model Theory, Amsterdam, North-Holland. Landman, F.: 1986, Towards a Theory of Information. The Status of Par• tial Objects in Semantics, Dordrecht, Foris. Malitz, J.: 1979, Introduction to Mathematical Logic. Set-theory, Com• putable Functions, Model Theory, New York, Springer Verlag. Peano, G.: 1973, Selected Works of Giuseppe Peano, ed. and transl. by H.C. Kennedy, Toronto, University of Toronto Press. Quine, W. van Orman: 1972, Methods of Logic, 3rd edition, N.Y., Holt Rinehart and Winston, Inc. Rasiowa, H. and R. Sikorski: 1970, The Mathematics of Metamathematics, 3rd edition, Warszawa, Panstwowe Wydawn. Robinson, A.: 1965, Introduction to Model Theory and to the Metamathe• matics of Algebra, 2nd edition, Amsterdam, North-Holland. Shoenfield, J.R.: 1967, Mathematical Logic, Reading, M.A., Addison-Wesley.

Smullyan, R.: 1961, Theory of Formal Systems, Princeton, N.J., Princeton University Press. Tarski, A.M.: 1956, Logic, Semantics, Mathematics, Oxford, Oxford Uni• versity Press. Tarski, A.M. and R.M. Robinson: 1953, Undecidable Theories, Amsterdam, North-Holland. Tarski, A.M. and J.C.C. McKinsey: 1948, A Decision Method for Elemen• tary Algebra and Geometry, 2nd edition, Berkeley, U.C. Press.

Part C

Abbott, J. C.: 1969, Sets, Lattices and Boolean Algebras, Boston, Allyn & Bacon. Birkhoff, G.: 1961, Lattice Theory, Providence, R.I., A.M.S. Colloquium, vol. 25. Birkhoff, G. and S. MacLane: 1977, A Survey of Modern Algebra, New York, Macmillan. 638 BIBLIOGRAPHY

Gill, A.: 1976, Applied Algebra for the Computer Sciences, Englewood Cliffs, N.J., Prentice-Hall. Gratzer, G.: 1968, Universal Algebra, New York, Van Nostrand. Gratzer, G.: 1971, Lattice-theory. First concepts and distributive lattzces, San Francisco, Freeman and Co. MacLane, S.: 1971, Categories for the Working Mathematician, Berlin, Springer. MacLane, S. and G. Birkhoff: 1979, Algebra, 2nd edition, New York, Macmillan. Scott, D.S.: 1972, Continuous lattices, in: Toposes, Algebraic Geometry and Logic, F.W. Lawvere (ed.), 97-136, Berlin, Springer.

Part D

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Cooper, R.: 1983, Quantification and Syntactic Theory, Dordrecht, Reidel.

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PARTE

Textbooks: Arbib, M.A.: 1969, Theories of Abstract Automata, Englewood Cliffs, New Jersey, Prentice-Hall. Brainerd, W.S. and L.H. Landweber: 1974, Theory of Computation, New York, John Wiley & Sons. Cohen, D.I.A.: 1986, Introduction to Computer Theory, New York, John Wiley & Sons. Gill, A.: 1963, Introduction to the Theory of Finite-State Machines, New York, McGraw-Hill. Ginsburg, S.: 1962, An Introduction to Mathematical Machine Theory, Reading, Massachusetts, Addison-Wesley Publ. Co. Ginsburg, S.: 1966, The Mathematical Theory of Context-Free Languages, New York, McGraw-Hill. Gross, M. and A. Lentin: 1970, Introduction to Formal Grammars, New York, Heidelberg, and Berlin, Springer-Verlag. Harrison, M.A.: 1965, Introduction to Switching and A utomata Theory, New York, McGraw-Hill Book Co. Hopcroft, J.E. and J.D. Ullman: 1979, Introduction to Automata The• ory, Languages, and Computation, Reading, Massachusetts, Addison• Wesley Publ. Co. BIBLIOGRAPHY 643

Lewis, H.R. and C.H. Papadimitriou: 1981, Elements of the Theory of Computation, Englewood Cliffs, New Jersey, Prentice-Hall. Minsky, M.L.: 1967, Computation: Finite and Infinite Machines, Engle• wood Cliffs, New Jersey, Prentice-Hall. Nelson, R.J.: 1968, Introduction to Automata, New York, John Wiley & Sons. Rogers, H., Jr.: 1967, Theory of Recursive Functions and Effective Com- putability, New York, McGraw-Hill. Salomaa, A.: 1969, Theory of Automata, Oxford, Pergamon Press. Salomaa, A.: 1973, Formal Languages, New York, Academic Press.

Collections: Davis, M. (ed.): 1965, The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Hewlett, New York, Raven Press. Dowty, D.R., L. Karttunen, and A.M. Zwicky (eds): 1985, Natural Lan• guage Processing: Theoretical, Computational and Psychological Per• spectives, New York, Cambridge University Press. Luce, R.D., R.R. Bush, and E. Galanter (eds.): 1963, Handbook of Mathe• matical Psychology, Vol. 2, New York, John Wiley & Sons. Luce, R.D., R.R. Bush, and E. Galanter (eds.): 1965, Readings in Mathe• matical Psychology, Vol. 2, New York, John Wiley & Sons. Manaster-Ramer, A. (ed.): 1987, Mathematics of Language, Amsterdam, John Benjamins. Moore, E.F. (ed.): 1964, Sequential Machines: Selected Papers, Reading, Massachusetts, Addison-Wesley Publ. Co. Oehrle, R.T., E. Bach, and D. Wheeler (eds.): 1988, Categorial Grammars and Natural Language Structures, Dordrecht, Reidel. Shannon, C.E. and J. McCarthy (eds.): 1956, Automata Studies, Princeton, New Jersey, Princeton Univ. Press. Savitch, W.J., E. Bach, W. Marsh, and G. Safran-Naveh (eds.): 1987, The Formal Complexity of Natural Languages, Dordrecht, Reidel.

Other Books and Articles: Ades, A.E. and M.J. Steedman: 1982, 'On the order of words', Linguistics and Philosophy 4, 517-558. Aho, A.V.: 1968, 'Indexed grammars - an extension of context-free gram• mars', Journal of the Association for Computing Machinery 15 No.4, 647-67l. Adjukiewicz, K.: 1935, 'Die syntaktische KonnexiUit', Studia Philosophica 1,1-27. English translation in Storrs McCall (ed.)(1967), Polish Logic, Oxford, Oxford Univ. Press. 644 BIBLIOGRAPHY

Bar-Hillel, Y.: 1953, 'A quasi-arithmetical notation for syntactic descrip• tion " Language 29, 47-58. Bar-Hillel, Y., C. Gaifman, and E. Shamir: 1960, 'On categorial and phrase structure grammars', Bulletin of the Research Council of Is• rael9, 1-16. Reprinted in Bar-Hillel, Y.: 1964, Language and Infor• mation: Selected Essays on their Theory and Application, Reading, Massachusetts, Addison-Wesley Publ. Co. van Benthem, J.: 1986, Essays in Logical Semantics, Dordrecht, Reidel. van Benthem, J.: 1988, 'The Lambek calculus', in Oehrle, Bach, and Wheeler (eds.), pp. 35-68. Chomsky, N.: 1956, 'Three models for the description of language', IRE Transactions on Information Theory 2, No. 33, 113-124. A corrected version appears in Luce, Bush, and Galanter (eds.)(1965). Chomsky, N.: 1957, Syntactic Structures, The Hague, Mouton & Co. Chomsky, N.: 1959, 'On certain formal properties of grammars', Informa• tion and Control 2, No.2, 137-167. Chomsky, N.: 1963, 'Formal properties of grammar', in Luce, Bush, and Galanter (eds.), pp. 323-418. Chomsky, N.: 196.5, Aspects of the Theory of Syntax, Cambridge, Massa• chusetts, MIT Press. Chomsky, N. and G.A. Miller: 1958, 'Finite-state languages', Informa• tion and Control 1, 91-112. Reprinted in Luce, Bush and Galanter (eds.)( 1965). Chomsky, N. and G.A. Miller: 1963, 'Introduction to the formal analysis of natural languages', in Luce, Bush and Galanter (eds.)(1963), pp. 269-321. Davis, M.: 1958, Computability and Unsolvability, New York, McGraw• Hill. Friedman, J., D. Dai, and W. Wong: 1986, 'The weak generative capac• ity of parenthesis-free categorial grammars', Proceedings of the 11th International Conference on Computational Linguistics. Friedman, J. and R. Venkatesan: 1986, 'Categorial and non-categoriallan• guage', Proceedings of the 24th Meeting of the Association for Com• putational Linguistics. Gazdar, G.: 1985, 'Applicability of indexed grammars to natural lan• guages', Report No. CSLI-85-34, Center for the Study of Language and Information, Stanford University. Gazdar, G. and G.K. Pullum: 1985, 'Computationally relevant properties of natural languages and their grammars' , New Generation Computing 3, 273-306; also appeared as Report CSLI-85-24, Center for the Study of Language and Information, Stanford University. BIBLIOGRAPHY 645

Ginsburg, S.: 1975, Algebraic and A utomata- Theoretic Properties of Formal Languages, Amsterdam, North-Holland. Ginsburg, S. and B.H. Partee: 1969, 'A mathematical model of transfor• mational grammar', Information and Control 15, 297-334. Ginzburg, A.: 1968, Algebraic Theory of Automata, New York, Academic Press. Hayashi, T.: 1973, 'On derivation trees of indexed grammars - an extension of the uvwxy theorem', Research Institute for Mathematical Sciences, Kyoto University, Vol. 9, pp. 61-92. Joshi, A.K.: 1985, 'How much context-sensitivity is necessary for charac• terizing structural descriptions - tree adjoining grammars', in Dowty, Karttunen, and Zwicky (eds.). Joshi, A.K., L. Levy, and M. Takahashi: 1975, 'Tree adjunct grammars', Journal of the Computer and System Sciences 10, No.1, 136- 163. Kuroda, S.Y.: 1964, 'Classes of languages and linear bounded automata', Information and Control 7, 207-223. Lambek, J.: 1958, 'The mathematics of sentence structure', American Mathematical Monthly 65, 154-170. Langacker, R.W.: 1969, 'On pronominalization and the chain of command', in D.A. Reibel and S. A. Schane (eds.), Modern Studies in English: Readings in Transformational Grammar, Englewood Cliffs, New Jer• sey, Prentice-Hall. Marsh, W.E.: 1985, 'Some conjectures on indexed languages', paper pre• sented to the Association for Symbolic Logic Meeting, Stanford Univ., July 15-19. Peters, P.S., Jr.: 1973, 'On restricting deletion transformations', in Gross, Halle and Schiitzenberger, The Formal Analysis of Natural Languages, The Hague, Mouton. Peters, P.S., Jr. and R.W. Ritchie: 1973, 'On the generative power of transformational grammars', Information Sciences 6, 49-83. Pollard, C: 1984, Generalized Phrase Structure Grammars, Head Gram• mars, and Natural Language, Ph.D. Dissertation, Stanford University.

Pullum, G.K. and G. Gazdar: 1982, 'Natural languages and context-free languages', Linguistics and Philosophy 4, 471-504. Reprinted in Sav• itch, Bach, Marsh, and Safran-Naveh (eds.). Rabin, M.O. and D. Scott: 1959, 'Finite automata and their decision prob• lems', IBM Journal of Research and Development 3, No.2, 114-125.

Roach: 1987, 'Formal properties of head grammars', in A. Manaster-Ramer (ed.). 646 BIBLIOGRAPHY

Shieber, S.M.: 1985, 'Evidence against the context-freeness of natural lan• guage', Linguistics and Philosophy 8, 333-343. Reprinted in Savitch, Bach, Marsh, and Safran-Naveh (eds.). Vijay-Shankar, K.: 1987, A Study of Tree Adjoining Grammars, Ph.D. Dissertation, University of Pennsylvania. Wasow, T.: 1978, 'On constraining the class of transformational languages' , Synthese 39, 81-104. Reprinted in Savitch, Bach, Marsh, and Safran• Naveh (eds.). Weir, D.J. and A.K. Joshi: 1988, 'Combinatory categorial grammars: gen• erative power and relationship to linear and context-free rewriting sys• tems', Proceedings of the 26th Meeting of the Association for Compu• tational Linguistics. Weir, D.J., K. Vijay-Shanker, and A.K. Joshi: 1986, 'The relationship between tree adjoining grammars and head grammars', Proceedings of the 24th Meeting of the Association for Computational Linguistics.

General Collections and Readers: Bernacerraf, P. and H. Putnam (eds.): 1964, Philosophy of Mathematics: Selected Readings, Englewood Cliffs, N.J., Prentice-Hall. Copi, I.M. and J.A. Gould: 1967, Contemporary Readings in Logical The• ory, New York, Macmillan. van Heijenoort, J.: 1967, From Frege to Godel, a Sourcebook in Mathe• matical Logic 1879-1931, Cambridge, MA., Harvard University Press.

Kline, M.: 1972, Mathematical Thought from Ancient to Modern Times, New York/Oxford, Oxford University Press. Martinich, A.P. (ed.): 1990, The Philosophy of Language, 2nd ed. New York/Oxford, Oxford University Press. Index

Abelian group 255 auxiliary 186 monoid 262, 432 basic 186 Absorption 280 non-terminal 435 Abstract collection 3 terminal 435 concept 7 Alphabetic variant 148 relation 28 Alternation 133 Accessibility relations 412-421 Analogue, of arithmetic 21 Ackermann, W. 218, 222 Analysis tree 343 Acoustic phonetics xx Anaphora 387, 425 Acyclic automaton 570 Anti-euclideanness 389 Addition 21,31, 77,411 Anti-persistent determiners 379 cardinal 73 Antisymmetric relation 41,45,206- law 265 211 rule of inference for statement weak linear order 209 logic 117 weak partial order 206 Ades, A. 551 Antisymmetry 41,45,206-211,389- Adjective, autological 26 392, 438 heterological 26 Argument 113 Adjukiewicz, K. 547 Aristotelian syllogistic logic 331 Adjunction 540-545, 553 Arithmetic analogue 21 Agentless passive 361 Arithmetic, cardinal 73 Aho, A. 534, 538 language of 91 Algebra 21, 247-307 Asher, N. 410 Boolean 295-301 Assertability condition 240 compositionality 332 Assignment 344 generalized quanitifiers 386 Associative law 18, 19,21,110,197, Heyting 301-304 279,295,419 universal 279 operation 212, 213, 249 Algorithm 515 Asymmetric relation 40, 45 Almost connectedness 389 strict partial order 207 Alphabet 431 Asymmetry 389, 442

647 648 INDEX

'At least one' automaton 565 relation 30, 39, 40, 47, 204, Atom (of Boolean algebra) 296 206, 391 Atomic formula 138, 151 Binding 371 statement 97, 183 Birkhoff 52, 248, 289, 300 Autological adjective 26 Bolyai, J. 88 Automata 92, 431-570 Bonevac, D. 410 Automorphism 251, 376 Boolean algebra 295-301, 476 Auxiliary alphabet 186 phrasal conjunction, disjunc- Auxiliary premise 119 tion, and negation 350 Axiom 179-232 Bound variable 139 of choice 217 Bounded lattice 287 of extension 169 Branch 438, 494 schemata 187 Bresnan, J. 360, 362 Axiomatic system 183-189 Brouwer, L. 304 extended 186 Brouwerian lattice 303 Axiomatization 87,90, 179-232 Calculus 81 of set theory 215 predicate 140 of the lambda calculus 347 statement 135 Cancellation law 264 Bach, E. 360, 361, 362, 551 Cantor's set theory 7 Bar-Hillel, Y. 547,549 Theorem 62, 349 Barendregt, H. 349 Cantor, G. 62,64, 81 Barwise, J. 371,382,385,386,394, Cardinal number 55-71 410 infinite 59 Basic alphabet 186 addition 73 Basic result in recursive function arithmetic 73 theory 516 multiplication 73 Bell, J. 229,299,301 Cardinality 9, 11,55-71, 298, 392 Bernays, P. 222 set-theoretic 8 Beth, E. 122 Carnap, R. 364 Beth tableaux 121-128,163-168,227 Cartesian product 27-28,60,464 construction rules for 127 Categorial grammar 547-551 Biconditional 103 Categorical system 205 law 110 Category theory 252, 279 Bigelow, J. 403 Causative verbs 363 Bijection 251, 284 Cell of a partition 46 Binary, operation 255 Center embedded structure 539 production 190 Chain 276 INDEX 649

Choice, axiom of 217 string 211 Choice function 230 Concept, abstract 7 Chomsky, N. 92, 461, 479 Conclusion, of an argument' 113 Chomsky hierarchy 448-450 Conditional 102, 237 Chomsky normal form (CNF) 503 law 110,419 Chomsky's program 92 proof 118, 158,219 Church, A. 315, 336, 348, 350 Conjunction 93,100, 147,237,418 Church's hypothesis 515-516 phrasal 101, 350-352 Church-Rosser theorem 348 rule of inference for statement Circularity 389 logic 117 Class, equivalence 45-47, 56, 58 Connectedness 42-43, 266, 389 Clause mates 443 almost 389 Clitic, reflexive 349 strong 389 Closed formula 140 weak linear order 209 Closure 247,458,462,495-497 Connective 98-104, 137, 218, 237, positive 531 316 Coextensive expressions 415,416 sentential 93, 98 Cohen, P.J. 66, 216 Connex relation 42 Commutative group 255 Consequence, logical 108-112 law 18,21,110,419 Conservativity 374-377, 393-396 operation 249 strong 375 Comparable elements 276 Consistency principle 18, 19 Complement 288 Consistency 200, 220 law 18, 19, 110, 419 Constancy 375 of a relation 29, 33, 44 Constant 136, 139 of a set 16, 17 Constituent structure tree 437,439, r~lative 15 440,441 Complementation 497 Constraint on coordination 380 Completeness 200, 201, 225 Context, opaque 404, 405, 406, 407, proof 225 419,420,422,423 Composition 33-36 transparent 404, 411 function 33 Context dependence 373, 375, 392 of lattice isomorphisms 283 Context free 447-450 Compositionality 315-336 grammar 490-503 Compositionality Principle 252 language 490-503, 533 Computability, effective 348 Context sensitive 447-450 Computer 451 grammar 529-531 Concatenation 432-435, 462, 496, language 529-531,533 531 Contextual parameter 425 650 INDEX

Contingency 104-108 Definite determiner 387 Contingent statement 105, 142 Definition, recursive 179-183 Continuity 381 Deletion 553 Continuum hypothesis 65 Demonstrative determiners 388 Contradiction 104-108 DeMorgan's laws 18, 19, 110, 147, . of predicate logic 141 198, 419 Convergence 82 Denotation, of an expression 324 Conversion, lambda 340 Dense relation 51 Cooper, R. 332,336,371,382,385, Denumerable set 59 386, 394 Denumerably infinite set 59 Coordinate, first 28 Deontic logic 411 projection 28 Dependence, quantifier 148 second 28 Derivation schema 112 system 31 tree 343 Copi, I. 201 Descartes, R. (Cartesian) 27-28 Correspondence 31,316 Designator 417, 420, 427 one-to-one 32-36, 55-71, 76 Determiner 93, 371-397,437, 563- Countable set 59 570 Countably infinite set 59 intersective 399 Cresswell, M. 350, 356, 411 logical 349 Cross-serial dependence 501, 545 Deterministic finite automata 455- Curry, H. 205 458 Cycle 553 Deterministic pda language 489,560, 561 D-structure 332 Deterministic pushdown automata Dagger, Quine's 237 488,490,560,561 Dahl, O. 359 Detransitivization 363 Dative movement 363 Diagonal argument 65 Daughter 438 Diagram (poset) 275-278 De dicto interpretation 407 of relation 43 De Jong, F. 396 Venn 12-14 De re interpretation 407 Difference and complement of ar• Decidability 227 bitrary set 15 Declarative sentence 98 Difference, symmetric 26 Dedekind, R. 81, 192, 289 Discourse, domain of 16 Dedekind infinite 193 universe of 16 Deduction, natural 112-121, 152- Disjunction 101, 147, 238,411 163 exclusive 101 Definability, lambda 348 inclusive 12, 101 INDEX 651

Disjunctive Syllogism 117 Equality, between natural numbers Distribution, quantifier 147 76 Distributive lattice 288-293 set-theoretic 17, 18 Distributive law 18, 19, 21, 110, Equivalence class 45, 56, 58 197,264,295,419 extensional 415 Distributive operation 249 logical 108-112, 146 Division 81 of sets 55-58 Domain 29, 30, 34, 36, 140 relation 45-47, 79, 203, 270 integral 264-268 Euclidean geometry and axioms 87 of discourse 16, 140 Euclideanness 389 of entities 324 Euclid 87 subset of 32 'Every' automaton 564 Dominance 438-439 'Exactly two' automaton 568 Double indexing interpretation 426 Exclusive disjunction 101 Double negation, law of 146 Exclusivity condition 440 Dowty, D. 318, 332, 350, 360, 362, Existential generalization 152, 155, 363,413,423 406 Dual atom (of Boolean algebra) 296 instantiation 152, 155 Dual, of a quantifier 382 quantifier 136, 147 Duality 275-278, 383 Expression 21 denotation of 324 Effective computability 348 matrix of 139 enumeration 228 Extended axiomatic system 186 listing 62 Extension 375,377,392,393,394, Element, greatest 50 396,413,414 identity 213 axiom of 169 least 50 Extensional equivalence 415 maximal 50 Extensionality, axiom of 216 minimal 49 of a set 3 Fallacy 115 Emptiness 500 Faltz, 1. 352, 376, 388 question 477 Filter 285-287, 387, 388, 396, 397 Empty list notation 9 maximal 286 Empty set 4, 9, 75, 216 proper 286 Empty string 58, 213, 432 representation of a lattice 286 Enderton, H. 229 Filtering 554 Entity, domain of 324 Final state 453, 456, 485, 486 Enumeration 228 Finite alphabet 67 Epistemic verbs 408-410 automata 453-461,485 652 INDEX

deterministic 455-461, 475 Fuzzy set 6 language (faJ) 462,468 non-deterministic 458-461 Gaifman, C. 549 induction 267 Gallin, D. 348 set 55-62 Gamut, L.T.F. 318 linearly ordered 51 Gazdar, G. 352, 356,501,534,539, First order predicate calculus 93 540, 545 First coordinate 28 Geach rule 550 member 27 Generalization, existential 152, 155 Flattened relational perspective 388 universal 152, 153 sentence 372 Generalized quantifier 356-358, 371- Formal proof 168 397, 563 grammar 431, 435, 471, 479 Generative grammar 92 language 91, 434, 479 syntax 92 semantics 93 transformational theory 431 system 87, 179-232 Generator 452 syntax 179-183 Geometry 278 Fortran 91 Ginsburg, S. 553 Frankel, A. 215 Glottochronology xx Frege, G. 315, 316, 321, 331, 401, Godel, K. 217 402,403,406 Godel's incompleteness theorems 228 Frege's two problems 401-407 Goldblatt, R. 248, 252 Friedman, B. 551 Gradual membership 6 Function 30-33 Grammar, formal 431, 435-436 application, forward, backward generative 92 548 Montague 7 composition 33 transformational xx, 553-557 graph of 31 Graph of function 31 identity 34 Gratzer, G. 248, 301 into 32 Greatest element .50 many-to-one 32 Grelling's Paradox 26 one-to-one 32, 33 Groenendijk, J. 356 onto 32 Group, Abelian 255 partial 31, 32 commutative 255 propositional 136, 140 Group theory 255-263 quaternary 36 single-valued 31 Halfway membership 6 ternary 36 Halting Problem 520-523 value of 34 Halvorsen, P. 332 INDEX 653

Hao Wang 204 Inclusive disjunction 12, 101 Hasse diagram 277 Incompleteness theorem 228 Hayashi, T. 540 Indefinite determiner 385 Head grammar 546-547 Independence 200 Heim, 1. 387 proof 220 Henkin, L. 225, 348 quantifier 148 Heterological adjective 26 Index 412-421 Heyting, A. 301 of evaluation 414-422 Heyting algebra 301-304 Indexed grammar 534-540 lattice 303 Indexicality 425-427 Higginbotham, J. 410 Indirect proof 120 Higher order quantifier 569-570 Individual constant 136, 138, 153 logic 229 Individual variable 136, 138 Hilbert, D. 92, 218, 222 Induction 192-198 Hilbert's program 92 finite 267 Hoeksema, J. 388 rules of 91, 115, 152,218 Hofstadter, D. 229 Inference 403 Homomorphism 251, 269, 283-285, Infinite cardinal number 9 297,333 Dedekind 193 join 285 denumerably 59 meet 285 set 55-71 Hopcroft, D. 461, 464, 490, 528, countable 59 529,531,534 denumerable 59 Horseshoe 237 non-denumerable 69 Hypothetical syllogism 117 vs. unbounded 70 Infinity 55-71,216 Ideal 285-287 Informal proof 168 principal 286 Information-state 304 proper 286 Initial state 453, 486 representation of a lattice 286 Initial symbol 435 Idempotent law 18, 21, 110, 249, Injection 251 279 Input 451 Identity element 213, 250, 255,259 alphabet 485, 486 function 34 Instantiations existential 152, 155 law 18, 19,21, 110 universal 152, 153 relation 35, 39 Integers 75 set-theoretic 8, 9 negative 79 Imperfective paradox 425 positive 6, 79 Implication, rule of 222 Integral domain 264, 268 654 INDEX

ordered 265 semantics 304-307 well-ordered 266 rigid designators 427 well-ordering axiom 267 valuation 305 Intended model 89 Kuroda, Y. 554 Intensional model 413 equivalence 415 Labeling 438-441 Intensionality 401-427 function 441 Interpretation 401, 407, 413, 419 Ladusaw, W. 332, 383, 387 Intersection 12 Lambda abstraction 336-365 and union 11 calculus 346-349 Intersective determiner 399 conversion 340 Into function 32 definability 348 Intransitive relation 41, 45 operator 336 Invalid argument 113 Lambek, J. 205,550 Inverse element 250, 255 Lambek calculus 550 of relation 29 Landman, F. 301 Irrational numbers 64 Langacker, R. 443 Irreflexive relation 39, 45 Language, formal 91 strict partial order 207 logical 98 Irreflexivity 389, 440, 442 meta- 90 Isomorphism 78,202,251,269,270, natural 7, 91, 98 283-285,474 object 90 Isotone mapping 283 of arithmetic 91 oflogic 91 James, W. 72 of set theory 91 Janssen, T. 248,332,350 programming 8, 91 Join homomorphism 285 Lattice 275-293 Joshi, A. 533, 540, 544, 546, 551 Brouwerian 303 distributive 288-293 Kalish, D. 316 filter representation of 286 Kamp, H. 422, 423, 425 Heyting 303 Keenan, E. 3.52, 359, 376, 385, 386, ideal representation of 286 388 modular 288-293 Kernel of a homomorphism 269 pseudo-Boolean 303 Kleene star 462, 496, 531 Laws of statement logic 110 Kleene, S. 464, 516 Leaf 439, 442, 445 Kleene's three-valued logic 239-241 Least element 50,439 Kripke, S. 304 Left identity element 249 Kripke frame 304 inverse 250 INDEX 655

linear grammar 472 Many-to-one function 32 zero 251 Map 31, 251-252 Levy, L. 540 Mapping 31 Lewis, D. 461, 464, 490 isotone 283 Lexical rule 362-365 monotone 283 Lexicon 345 single-valued 32 Lindenbaum algebra 299 Markov, A. 516 Linear Marsh, W. 540 bounded automata 527-529 Massey, G. 234 grammar, right 471 Mathematical induction 192 order 51, 432 Mathematics, discrete 81 strict 209 Matrix of expressions 139 tree, left-, right- 538 Maximal Linearly ordered set 51 element .50 Linguistic object 5 filter 286 theory 435 proper ideal 286 Link, G. 332,388 May, R. 336 Lisp 91, 212 McCawley, J. 359 List notation 4-6 McKinsey, J. 303 Listing, effective 62 Meani.ng 401, 427 Lo Cascio, V. 423 Meaning postulate 362-365 Lobachevsky, N. 88 Meet homomorphism 285 Logic, first-order 91 Member, first 27 higher-order 229 of a set 3 language of 91 second 27 predicate 93, 135-173 Membership 498 term 135 gradual 6 statement 93, 97-121 halfway 6 laws of 110 multiple 6 Logical consequence 108-112 question 477, 531 determiner 349 relation 4 equivalence 108-112, 146 Meta-language 90, 422 language 98 Mildly context sensitive 547 omniscience 420 Miller, G. 461,479 paradox 7 Minimal element 49 Modal logics 304 Machine translation xxx Modality 411-416 Machover, P. 229, 299, 301 Model 323 MacLane, S. 52,248,300 intended 89 656 INDEX

standard 89 Natural deduction 112-121,152-163 theory 92, 179-232 language 7, 91, 98 Modified head grammar 546 number 3, 9, 75, 192 Modular lattice 288-293 Negation 99, 237, 238 law 288 external and internal 381 Modus ponens 117 quantifier 146 Modus tollens 117 Negative, in integral domain 266 Monadic predicate logic 227 integers 79 Monoid 213, 261-263,432 strong 385 Abelian 262 Nested stack automaton 539 Monotone decreasing determiner 381- Newtonian system 87 387 Next state function 456 Monotone increasing determiner 381- 'No' automaton 565 387,397 Node 437-444 Monotone mapping 283 admissibility condition 448 Monotonicity 378-380 Nonconnected relation 42 decreasing 378 Non-denumerably infinite set 69 increasing 378 Non-deterministic finite automata Montague, R. 315, 316, 331, 332, 458-461 336, 337, 349, 350, 351, pushdown automata 488, 489 357,360,364,372 Turing machine 512 Montague grammar 7, 232, 413 Non-distributivity 289 Morphism 251-252, 269-271 Non-empty set 217 lattice, 283-285 Non-modularity 289 Morphological reflexive 363 Non-monotone 380 Morphology 422 Nonreflexive relation 39 Moss, 1. 388 Nonsymmetric relation 40,45 Movement, dative 363 Non-terminal alphabet 435 quantifier 149 symbol 437, 448 Multiple membership 6 Non-triviality 377 Multiplication 21,31,81 Nontangling condition 440 cardinal 73 Nontransitive relation 41 law 265 'Not every' automaton 564 table 261 Notation, list 4-6 empty list 9 n-place relation 30 predicate 6, 12, 15, 16,28,31, n-tuple, ordered 27, 36 33 set of 62 set-theoretic 4 Named object 5 Noun 437 INDEX 657

phrase (NP) 371, 437 Ordered integral domain 265 Null set 4, 10, 19, 21 n-tuple 27, 36 Number, cardinal 55- 71 pair 27-28, 39-47,141 infinite 59 set of 30 irrational 64 quadruples 30 natural 3, 9, 75, 192 set, partially 51 rational 60, 80 triple 27, 30 real 64, 65 Ordering 47-51 theoretic tree 377 Output 451 theory 16 Pair, ordered 27-28,39-47, 141 Object 3, 28 Pairing 216 language 90 Papadimitriou, C. 461,464,490 linguistic 5 Paradox, Grelling's 26 named 5 imperfective 425 paradoxical 16 logical 7 Oehrle, R. 551 Russell's 7, 11,26,63 Omniscience, logical 420 Parallel postulate 87 One-to-one correspondence 32, 55- Parse tree 492-493 71,76,251 Parsing xx function 32 Parsons 332 Onto 251 Partee, B. 332, 350, 352, 355, 356, function 32 364,425,553 Opacity 404, 405, 406, 407-412, 419, Partial function 31, 32 420,422,423 order 51, 275 Open formula 140 recursive function 511 statement 136 Partially ordered set 51 Operation 247 Partition 45-47 set-theoretic 12 cells of 46 Operator, lambda 336 Pascal 91 unary 98 Passive 360-362 Order 47-51, 206-213 Peano, G. 192 linear 51, 209, 391 Peano's axioms 192-198 of a group 261 fifth postulate 267 partial 51,206,275 Permutation, closure 551 strict 47, 207, 209, 391 invariant 570 strong 47 Persistent determiners 379 total 51, 209 Peters, S. 318,332,350,413,553 weak 47, 206, 211 Phrasal conjunction 101, 350-352 658 INDEX

Phrase structure rule 350 Programming language 8, 91, 346, tree 513 489 Polarity reversal 383 Projection, coordinate 28 positive and negative 384 problem 241 Polish notation 133, 232 Prolog 91,212 Pollard, C. 546 Proof, completeness 225 Poset 275-278 conditional 118, 158, 219 Positive closure 531 formal 168 in integral domain 266 independence 220 integers 79 indirect 120 power 268 informal 168 strong 385 Proper filter 286 Possessive determiner 376 ideal 286 Post, E. 516 maximal 286 Post correspondence problem. 523 Proper subset 10 Power, positive 268 Proposition 140 Power set 11,62-65,216,278,299 Propositional calculus 93 Precedence 439-440 function 136, 140 Predecessor 49, 76 Pseudo-Boolean lattice 303 Pseudo-complement 302 Predicate calculus 93, 140 Pullum, G. 501, 540 logic. 93,135-173,141,321-331 Pumping theorem 468-471, 477, 492- term 135 495,543,569 notation 6, 12 Push and copy rule 534 Prefix 433 Pushdown automata 485-490, 560, Premise 113, 419 561 auxiliary 119 deterministic 488, 490, 560, 561 Prenex normal form 146-152 languages 489, 560, 561 Preorder 208 non-deterministic 488, 489 Presupposition 240, 396 store 485 Principal filter 387 ideal 286 Quantification 407 Principle of Finite Induction 267 vacuous 138 Probability and statistics xx Quantifier 95, 316, 371-397 Produces-in-one-move 457, 489, 508 automata 563 Product, Cartesian 27-28, 60 dependence 148 Production 184 distribution 147 schemata 186 existential 136, 147 Program 453 generalized 356-358 INDEX 659

independence 148 language 462-471, 490 laws 146-152 Regularity, axiom of 216 movement 149 Reichenbach, H. 425 negation 146 Rejection for ever after 567 scope 331 with 0 567 universal 136, 147,328,329 with 1 567 Quantity 376, 377, 392, 393, 394, Relation 28-30 396 anti-symmetric 41, 45 weak 376 asymmetric 40 Quasi-order 208 binary 30,39,40,47,204,206, Quasi-reflexivity 618 391 Quaternary function 36 complement 29, 33 relation 30, 36 connected 42-43 Quine's dagger 237 connex 42 Quine, W.V.O. 237,404 dense 51 diagram of 43 Rabin, M. 461 equivalence 45-47, 79, 203, 270 Range 29, 30 identity 35, 39 Rasiowa, H. 304 intransitive 45 Rational number 60, 80, 255, 265, inverse of 29 266, 272 irreflexive 39, 45 Real number 65, 265, 266, 276 n-place 30 Reconstruction of number systems nonconnected 42 75 nonreflexive 39 Recoverability 554 nonsymmetric 40, 45 Recursive definition 179-183 nontransitive 41 function 511 reflexive 39-40 rule 8, 24 subset 28 set 517,529 symmetric 40-41, 45 Recursively enumerable 512 transitive 41-42, 4.5 language 517-518 Relative clause 349, 353-356 Reference 402 Relative complement 15 Reflexive 349, 363 pseudo-complement 302 Reflexivity 39-40, 389 Replacement 216 of relations 39-40 Restriction 396 of weak linear order 209 Reuland, E. 387 of weak partial order 206 Reversal 432 Regular expression 464 Rewrite rules 92 Regular grammar 449 Riemann, B. 88 660 INDEX

Right tableaux (Beth) 121 identity element 249 type 372 inverse 250 value 140, 141 linear grammar 449, 471, 490 Semantics 92 zero 251 formal 93 Rigid designator 417, 420, 427 Kripke 304-307 Ritchie, R. 553 of predicate logic 140-145 Roach, K. ,546 of programming languages 8 Rogers, H. 516 Semi-Thue system 190-192 Rohrer, C. 423,425 Semigroup 261-263 Root 439 Semilattice 278-282 Rooth, M. 352 Sense 402, 403, 405, 413, 426 Ross, J .R. 72, 3,59 Sentence, declarative 98 Rosser, J. 348 Sentential complement 403,408,411, Rule 315-320 416 implication 222 connective 93, 98 inference 91, 115, 152,218,402 tree 543 lexical 362-365 Separation, axiom of 217 rewrite 92 Set 3-71 semantic 317 as a member of a set 3 substitution 112, 222, 218 Cantor's theory 7 syntactic 316 countable 59 well- formedness 92 countably infinite 59 Russell, B. 7,218 denumerable ,19 Russell's paradox 7, 11,26,63 denumerably infinite 59 empty (nUll) 4, 9, 10, 19, 21, Salomaa, A. 534, .540 75,2]6 Schema equivalence 55-,18 axiom] 87 finite 3, 9, 5,1-62 production 186 fuzzy 6 derivation 112 infinite 3, 6, 8, 9, 55-71 Scott, D. 349,461 member 3, 7, 8 Second coordinate 28 non-denumerably infinite 69 member 27 non-empty 217 Self dual 382 of n-tuples 62 Semantic ordered 36 automata 563-570 of ordered pairs 28 component 91, 93 power 11, 62-6.5 rule :317 specification of '1 INDEX 661

subset 10 State 451, 453 theory 140, 169 State diagram 455 axiomatization of 215 final 453 uncountable 69 initial 453 universal 19, 21 Statement, atomic 97, 183 well-defined 4 calculus 93, 135 well-ordered 51 contingent 105, 142 Set-theoretic cardinality 8 logic 93, 97-121, 317-320 complement 15 open 136 difference 15 Statistics, probability and xx equality 17, 18 Steedman, M. 551 identity 8, 9 Stokhof, M. 356 intersection 11 Stone, M.F. 299 notation 4 Stone representation 299, 301 union 11 Strict order 47 universe 16 linear 209 Shamir, E. 549 partial 391, 440 Sheffer stroke 238 total 442 Shieber, S. 502 String Sikorski, R. 304 concatenation 211 Simple determiner 379 empty 58, 213 Simplification (statement logic) 117 Stroke, Sheffer 238 Single root condition 439 Strong Single-valued function 31 connectedness 389 Singleton 4 conservativity 375 Sister node 438 order 47 Situation 456, 457, 489 Subgroup 261-263 Skolem function 230 Sublattice 278-282 normal form 230 Submonoid 263 Square of opposition of syllogistic Subposet 282 logic 380, 564 Subset 10-11 Stack 485 proper 10 alphabet 485, 486 relation 28 Standard model 89 Substitution 402,406,416,553 Standard theory (transformational ru~ of112, 218, 222 grammar) 553 Substring 433 State acceptance for ever after 567 Subtraction 81 with 0567 Successor 49 with 1 567 function 346 662 INDEX

Suffix 433 type 232 Suppes, P. 208 Thomason, R. 201 Surface structure 553 Thue, A. 190 Surjection 251 Time 421-425 Syllogism, disjunctive 117 Top and bottom laws (Boolean al- hypothetical 117 gebra) 296 Symmetric difference 26 Topology 278 relation 40-41, 45 Total order 51, 209 Symmetry 40-41, 389 Transformation 31 Syntactic component 93 Transformational cycle 553 rule 316 grammar 553-557 Syntax 92 Transition function 456 generative 92 relation 460 of formal systems 179-183 Transitive relation 41-42, 45 of predicate logic 135-140 verb 360 System, formal (axiomatic) 87, 179- Transitivity 41-42, 389, 442 232 order, strict partial 207 weak linear 209 Tableau, Beth 121, 128-163, 168- weak partial 206 227 Translation, machine xx Takahashi, M. 540 Transparent context 404, 411 Tarski, A. 303 Tree 437-448 Tau tology 104-108 analysis 343 Tense 421-425 derivation 343 Ter Meulen, A. 387,388 of numbers 565 Term, of predicate logic 135 Tree adjoining grammar 540-545 Terminal alphabet 435 Trichotomy, law of 266 symbol 435, 449 Triple, ordered 27, 30 Ternary function 36 Truth 92 relation 30, 36 -functional property 99 That-clause 403, 405, 408,409,410 table 99-104, 252 Theorem 91 value 99-104, 140 provers 121 Turing, A.M. :148, 505 Theory, category 2.52, 279 Turing acceptable 509 function :346 decidable 510 model 179-2:32 machine .505-523 numben' 16 universal 518-520 of meaning 401, 427 non-deterministic 513 set 3, 140, 169 Turnstile 457 INDEX 663

Two-sided identity element 250 Van Benthem, J. 372, 376, 381, inverse 250 388, 389, 391, 392, 411, zero 251 563, 570 Type raising 550 Variable 6, 136 semantic 372 assignment 324 theory 7, 232, 336-338 bound 139 Type 0 grammar 449, 513-515, 530, individual 136 554 Variation 377 1 grammar 449, 527, 554 Venn diagram 12-14 2 grammar 449, 490 Verb 3 grammar 449, 471-480 causative 363 Type-free character 349 phrase 437 VP-deletion 3.58-360 Ullman, J. 461, 464, 490, 528, 529, transi ti ve 360 531, 534 Verkuyl, H. 396 Ultrafilter 286, 297 Vet, C. 423 theorem 297 Vijay-Shankar, K. 545,546 U nary operator 98 Vocabulary 431 Unbounded vs. infinite 70 Uncountable set 69 Waismann, F. 214 Undecidable questions (cft) 500 Wall, R. 318,332,350,413 Union 11-14,216,496 Wave theory iii Uniqueness 248 Weak Universal, algebra 248, 279 determiner 385 base hypothesis 556 non-cardinal 386 generalization 152, 153 order 47 instantiation 152, 1.53 linear 391 011 determiners 381 partial 53, 438 on nega.tion :382 quantity:376 quantifier 136, 147,328,329 Weir, D. 546. 551 set 19,21 Well-formed formula (wff) 97-98 Turing machine .518-520 WPlI-formedness rules 92 Unrestricted grammar 513-515 'Nell-ordered set 51,210 integral domain 266 Vacuous quantification 1:38 Westerstahl, D. :r/5, 376, 388, :392. Valid argument 113 394, :39G Value of a function 31, 34 \Vheeler, D. 5.51 semantic HO, 141 Whitehead, A.N. 218 truth 99-104, 140 \Villiams. E. :3.')0 664 INDEX

Witness set 397 Wrapping operation 546

Yield 445

Zadeh, 1. 6 Zermelo, E. 215 Zermelo-Frankel axiomatization (ZF) 215 Zero 79,251 ZFC 217 Zorn's lemma 297 Zwarts, S. 389, 391, 392 Studies in Linguistics and Philosophy

20. F. Heny and B. Richards (eds.): Linguistic Categories: Auxiliaries and Related Puzzles. Volume II: The Scope;-Order, and Distribution of English Auxiliary Verbs. 1983 ISBN 90-277-1479-7 21. R. Cooper: Quantification and Syntactic Theory. 1983 ISBN 90-277-1484-3 22. 1. Hintikka (in collaboration with 1. Kulas): The Game of Language. Studies in Game-Theoretical Semantics and Its Applications. 1983; 2nd printing 1985 ISBN 90-277-1687-0; Ph: 90-277-1950-0 23. E. L. Keenan and L. M. Faltz: Boolean Semantics for Natural Language. 1985 ISBN 90-277-1768-0; Pb: 90-277-1842-3 24. V. Raskin: Semantic Mechanisms of Humor. 1985 ISBN 90-277-1821-0; Ph: 90-277-1891-1 25. G. T. Stump: The Semantic Variability of Absolute Constructions. 1985 ISBN 90-277-1895-4; Pb: 90-277-1896-2 26. 1. Hintikka and 1. Kulas: Anaphora and Definite Descriptions. Two Applications of Game-Theoretical Semantics. 1985 ISBN 90-277-2055-X; Pb: 90-277-2056-8 27. E. Engdahl: Constituent Questions. The Syntax and Semantics of Questions with Special Reference to Swedish. 1986 ISBN 90-277-1954-3; Pb: 90-277-1955-1 28. M. 1. Cresswell: Adverbial Modification. Interval Semantics and Its Rivals. 1985 ISBN 90-277-2059-2; Pb: 90-277-2060-6 29. 1. van Benthem: Essays in Logical Semantics 1986 ISBN 90-277-2091-6; Pb: 90-277-2092-4 30. B. H. Partee, A. ter Meulen and R. E. Wall: Mathematical Methods in Linguis• tics. 1990; Corrected second printing of the flrst edition 1993 ISBN 90-277-2244-7; Pb: 90-277-2245-5 31. P. Glirdenfors (ed.): Generalized Quantifiers. Linguistic and Logical Ap- proaches.1987 ISBN 1-55608-017-4 32. R. T. Oehrle, E. Bach and D. Wheeler (eds.): Categorial Grammars and Natural Language Structures. 1988 ISBN 1-55608-030-1; Pb: 1-55608-031-X 33. W. 1. Savitch, E. Bach, W. Marsh and G. Safran-Naveh (eds.): The Formal Complexity of Natural Language. 1987 ISBN 1-55608-046-8; Pb: 1-55608-047-6 34. 1. E. Fenstad, P.-K. Halvorsen, T. Langholm and 1. van Benthem: Situations, Language and Logic. 1987 ISBN 1-55608-048-4; Ph: 1-55608-049-2 35. U. Reyle and C. Rohrer (eds.): Natural Language Parsing and Linguistic Theories. 1988 ISBN 1-55608-055-7; Pb: 1-55608-056-5 36. M. 1. Cresswell: Semantical Essays. Possible Worlds and Their Rivals. 1988 ISBN 1-55608-061-1 37. T. Nishigauchi: Quantification in the Theory of Grammar. 1990 ISBN 0-7923-0643-0; Pb: 0-7923-0644-9 38. G. Chierchia, B.H. Partee and R. Turner (eds.): Properties, Types and Meaning. Volume I: Foundational Issues. 1989 ISBN 1-55608-067-0; Pb: 1-55608-068-9 39. G. Chierchia, B.H. Partee and R. Turner (eds.): Properties, Types and Meaning. Volume II: Semantic Issues. 1989 ISBN 1-55608-069-7; Pb: 1-55608-070-0 Set ISBN (Vol. I + II) 1-55608-088-3; Pb: 1-55608-089-1 Studies in Linguistics and Philosophy

40. C.T.J. Huang and R. May (eds.): Logical Structure and Linguistic Structure. Cross-Linguistic Perspectives. 1991 ISBN 0-7923-0914-6; Pb: 0-7923-1636-3 41. M.J. Cresswell: Entities and Indices. 1990 ISBN 0-7923-0966-9; Pb: 0-7923-0967-7 42. H. Kamp and U. Reyle: From Discourse to Logic. Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory. 1991 ISBN 0-7923-1027-6; Pb: 0-7923-1028-4 43. C.S. Smith: The Parameter 0/ Aspect. 1991 ISBN 0-7923-1136-1 44. R.C. Berwick (ed.): Principle-B(lsed Parsing. Computation and Psycholinguis- tics. 1991 ISBN 0-7923-1173-6; Pb: 0-7923-1637-1 45. F. Landman: Structures/or Semantics. 1991 ISBN 0-7923-1239-2; Pb: 0-7923-1240-6 46. M. Siderits: Indian Philosophy o/Language. 1991 ISBN 0-7923-1262-7 47. C. Jones: Purpose Clauses. 1991 ISBN 0-7923-1400-X 48. R.K. Larson, S. Iatridou, U. Lahiri and J. Higginbotham (eds.): Control and Grammar. 1992 ISBN 0-7923-1692-4

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