Author Index

Author Index

Alagic, S. 104 -Eukasiewics 172 Arbib, M. A. 104 McCuIloch, Warren 126 Boden, Margaret 43 De Morgan, Augustus 11 Boole, George 3 Newman, James R. 178 Cantor, Georg 161 Nilsson, N. J. 126 Descartes, Rene 5 Peano, Giuseppe 34 Euclid 4 Peterson, G. R. 126 Euler, Leonhard 178 Pitts, WaIter 126 Fibonacci 96 Stone, MarshaIl 160 Hill, F. J. 126 Vacroux, A. G. 124 Leonardo of Pisa 96 Venn, John 7 Notation Index

.- 1 An 19 xdiv y 2 A~B 19, 160 xmody 2 g·f 20 Z 2 A 20,41 N 3 /\ 47,112 XEA 3 1\ 156 x$A 3 XA 21 PA 3, 111 Q 23 cp 3 .1 23, 158 AcB 3 Pfn[A, B] 23 D 5 ..:... 23 AxB 5 ~ 25 R 5 R 25 2A 6 Ixl 29 IAI 6, 160, 161 n! 35 &l'A 6 (J 38,48,177 AuB 7,43,201 X* 41 AnB 8 In 41 A-B 8 A·B 43,201 Al + ... + An 8 A* 43,44,201 V 47,113 8 U Ai 155 l'5.is.n V 15 48 A 9 qo 48 rnn 12 15* 49,204 f(A) 16 T(M) 50 f:A -+ B 16 v-+w l lw 2 1···IWn 55 a Hf(a) 16 =• 55 n 19 = 55 212 Notation Index

.. - 59 O'(n) 115 X* 60 $m 115 e 60 plq 122 {} 63 NAND 122 T 65 NAND-gate 122 ISATOM 66 NOR 123 HEAD 67 L 123 TAIL 67 (P L q) 123 CONS 68 NOR-gate 127 CAR 68 COND 129 CDR 68 A=> B 132 EQ 68 = 133 EQUAL 68 p => q 135 COND 68 (Vx) 141 LxJ 78 (3x) 141 P(n, r) 82 R:A~A 146 nPr 82 ~A 146 nCr 83 n : n' mod m 146 (~) 83 id A 146 T 111 [a] 147 F 111 A modulo: 149 NOT 112 AI: 149 AND 112 :=:; 150 OR 112 Pfo 151 $ 112 (A, v, A) 157 15 112 T 158 -,p 112 ~o 162 NOT-gate 114 ScopeT(v) 169 AND-gate 114 v(w) 169 OR-gate 114 nn 170 max 114 c5 w 171 min 114 0'0 177 Zm 115 0'1 177 n mod m 115 T(M) 200 Subject Index

A axiomatization 138 complete 140 absolute value 29 sound 139 accepting states 48 Axiom System for Propositional Logic additive identity 45 138 additive inverse 45 addition of cardinal numbers 165 address of a node in a tree 168 B aleph null 162 Algorithm Analysis 92, 102 Backus-Naur Form (BNF) 59 almost everywhere 154 base 12 alphabet 41 basis step 32, 131 ambiguous tree presentation 94 best possible result 78 analytic geometry 5 biconditional form 130 ancestor 88 bijection 18, 20 antecedent 135 bijective correspondence 19 antisymmetry 150 binary arithmetic progression 13 decision tree 7, 106 Artificial Intelligence (AI) 43 search 60, 104 arity 169, 170 tree 90 arrays 62, 64 tree, complete 92 associative 29 Binomial Theorem 83 operation 44 bipartite graph 186 laws 10 bits 15 atom 65 Boolean 3, 158 average complexity 104 algebra 159 214 Subject Index

Boolean (cont.) conjunctive normal form 121, 127 connection matrix 193 connected graph 180 function 118 connectedness relation 179 semiring 47, 190 connection matrices of graphs 193 bottom 23, 158 connection matrix of an FSA 199 box notation for s-expressions 67 construction function 68 branches 88 constructive proof 78 consequent 135 Context-Free Grammars 53, 54 c context-free language 56 canonical surjection 149 contradic tion 137 Cantor's Diagonal Argument 165 contrapositive 132, 136 cardinal number 161 converse 132 cardinali ty 6, 161 corollary 134 carrier 171 cost of a search 103 carry bit 123 counter 64 carry look-ahead adder 126 counting principles 73 Cartesian power 19,42,62 cycle 178 Cartesian product 5, 62, 63 ceiling of x 78 D characteristic function 21,26 codomain 16,22 data type 62 combination 83 daughter 168 comments 63 decimal notation 54 commutative 10, 29 definition commutative mono id 46 domain of 22 compiler 48 formal 47 complemented lattice 159 inductive 64 complete axiomatization 140 recursive 35 complete binary tree 92 Recursive Definition of Addition 39 complete lattice 157 degree of a vertex 185 complexity De Morgan's Laws 11, 113 average 104 denumerable set 163 of switching functions 125 descendants 88 optimal worst-case 109 derivation tree 55 worst-case 103 derives 55 component of a graph 180 diagonal 146 composite 10 dictionary order 152 of maps 20 difference equation 95 of relations 27,153 directed graph 177 computer circuitry 114 directly derives 55 conc function 43 disjoint concatenate 41 partial functions 151 concatenation 43 sets 8 condition sum of partial functions 151 sufficient 132 union 8,42 conditional 135 disjunction 47 conjunct 121, 129 disjunctive normal form 117 conjunction 112,47 disjunctive normal form theorem 120 Subject Index 215

distributive 29 Fibonacci Sequence 95 distributive lattice 159 finite state acceptor (FSA) 48 distributive laws 10 finite state language (FSL) 50, 200 dividend liS floor of x 78 divides II flowcharts 183 domain 16 flow diagram 1, 185 interpretation of an operator forest of trees 90 domain 171 forest, ordered 90 of definition 22 formal definition 47 operator 169, 170 formal language theory 43 tree 168 Four-Color Problem 183 dotted pair 65 full-adder 124 dynamics 199 full binary tree 92 function 16 Boolean 118 E characteristic 21,26 construction 68 E*-connection matrix 193 disjoint partial 151 edges 177 disjoint sum of partial 151 element end 177 maximal 154 factorial 35 minimal 154 Fibonacci 97 empty graph of a function 24 set 3 graph of a partial function 24 sequence 20 identity 146 string 41 length 42 end function 177 monus 23 enumeration 163 partial 17, 22 equivalence class 147 start 177 equivalence (logical) 135 state transition 199 equivalence relation 146, 147 switching 125 Euler path 178 total 22 EuIer's Theorem 181 transition 48 exclusive or 112 functionally complete operations 122 existential quantifier 141 existence proof 78 existence theorem 78 exponents 12 G exponentiation of cardinal numbers 165 grammar of English 53 exponential graph 175 decay 12 bipartite 186 growth 12 component of 180 connected 180 connection matrix 193 F directed 177 of a function 24 factorial function 35 of a partial function 24 false 111 ordered 175 Fibonacci function 97 theory 175 216 Subject Index graph (cont.) interpretation 137 undirected 177 interpretation of an operator domain 71 unordered 175 intersection 8 greatest lower bound (gIb) 156 inverse 20, 45, 47 group 45 additive 45 geometric progression 14 left 28 of a relation 153 right 28 H inverter 114 isolated vertex 180 half-adder 123 isomorphic sets 160 height of a tree 88 isomorphism 20 Hesse diagrams 156 iterate 43 homomorphic image 47 iteration 37 homomorphism 47 iterative solution 64

I J id 20 join 157 identity 44 additive 45 for an operation 44 K function 146 map 20 Konig's lemma 88, 101 matrix 191 Kleene star 43 multiplicative 45 if and only if 133 if ... then . .. else 69, 128 L image 16,22 immediate successor 88 languCfge over X 43 implication 135 latin square 86 impossibility proof 127 orthogonal 86 inclusion relation 150 lattice 157 inclusive or 112 complemented 159 incomparable elements 150 complete 157 induction 57 distributive 159 induction step 32, 131 law inductive definition 64 associative 10 inequality 25 De Morgan's 11, 113 infinite 4 distributive 10 sets 161 leaf 66,169 infinity 160 least upper bound (lub) 155 initial state 48 leaves 87 inorder 93 left inverse 28 input left subtree 90 excitatory 126 lemma 133 inhibitory 126 length 4,41 internal node 87 function 42 Subject Index 217 level in a tree 91 Modus Ponens 139 lexicographic order 152, 160, 173 mono id 44 linear algebra 186 monus 23 linear map 187 multiplication of cardinal numbers 165 Lisp 43, 60, 65 multiplicative identity 45 list 60 mUltiplication modulo m 47 logarithm 12 mutually exclusive events 73 logarithmic search 108 logic Axiom System for Propositional N Logic 138 predicate 140 n-ary predicate 26 propositional 140 n-ary relation 26 loop 177 natural numbers 2, 32 lower bound 156 semiring of 190 L-tree 66 n choose r 83 N-connection matrix 193 NDA 203 necessary and sufficient conditions 133 M necessary conditions 132 negation 111 machine, state-graph of 48 neuron 126 magic square 85 NIL 43,65 magic sum 86 nodes 66,87, 169, 177 map 16, 183 non-constructive proof 78 composite of 20 nondenumerable 166 identity 20 set 164 linear 187 nondeterminism 203 selector 62 nondeterministic finite state acceptor subway 177 (NDA) 203 matrix 187 non-negative integers 3 Boolean connection 193 nonterminals 54 connection matrices of graphs 193 normal form 11 7 connection matrix ofan FSA 199 conjunctive 121, 127 E*-connection 193 disjunctive 117 identity 191 disjunctive normal form theorem 120 N-connection 193 n permute r 82 product of matrices 190 numbers as string of digits 37 sum of 187, 189 symmetric 194 maximal 158 element 154 o maxterm 122 McCulloch-Pitts Neuron 126 il-algebra 171 meet 157 il-trees 172 microcomputers 124 onto 17 minimal 158 one-to-one 18 circuit 124 operator domain 170 element 154 operator label 170 minterm 121 optimal worst-case complexity 109 modulo sum 115 ordered forest 90 218 Subject Index ordered graph 175 proof 138, 139 orthogonallatin square 86 by contradiction 4, 131 outdegree 169 by exhaustion of cases 131 by induction 32, 131 of correctness 104 p techniques 131 propagation of carry bits 125 palindrome 56 proper graph 180 parallelism 126 proper subset 3 parent 88 proposition III parity 84 propositional connectives 140 parity checker 51 propositionallogic 3, 137 parse 53, 203 propositional variable 137 partial function 17, 22 proving theorems 131 disjoint 151 graph of 24 partially ordered set 150 Q partial order 150, 158 partition 153 quantifiers 140 Pascal 47,62 existential 141 path 88 universal 141 of a directed graph 194 quotient 1 Euler 178 length 194 pattern recognition 126 R Peano's Axioms 35 permutations 80, 82 Rabbit breeding 96 Pigeonhole Principle 77, 79 Ramsey's Theorem 80 planar graph 183 range 16,22 poset 150 rational numbers 23 postorder 93, 174 reachable state 199 Polish (prefix) notation 173 Recurrence Relations 95 powerset 6 recursion 32 predicate logic 140 in programming languages 36 predicates 140 recursion step 36 prefix 152 recursive definition 35 preorder 93, 174 Recursive Definition of Addition 39 prime IQ reductio ad absurdum 4, 131 primes, infinitely many 5 reduction step 36 principle reflexive 146 Pigeonhole Principle 77, 79 regular languages 20 Principle of Induction 57 regular set 58 Principle of Induction Extended 57 relation 25, 146 product of matrices 190 composite of 27, 135 production 55 connectedness 179 programming language semantics 152 equivalence 146, 147 progression inclusion 150 arithmetic 13 inverse 153 geometric 14 n-ary 26 Subject Index 219

Recurrence 95 Seven Bridges of K6nigsberg 178 transitive closure 27 Sheffer stroke 122 transitive reflexive closure 27 Socrates 142 union of 27 sorted array 105 remainder 1, 115 sound axiomatization 149 right inverse 28 Strahler numbering 100 right subtree 90 start state 48 root 66, 87, 168 start function 177 rooted tree 87 state 48 Rule of Inference 139 accepting 48 Rule of Product 74 initial 48 Rule of Sum 73 finite state acceptor (FSA) 48 Rules of Sum and Product Generalized 75 machine, state-graph of 48 nondeterministic finite state acceptor (NDA) 203 reachable 199 s start 48 sample 6 transition function 199 satisfiable formula 143 string scalar multiple 187 empty 41 scope 169 notation 41 search key 102 numbers as strings of digits 37 s-expressions 65 of symbols 41 selector maps 62 reversal 47 semantics 136 semi ring 190 of well-formed formulas 137 Strings over an Arbitrary Set 41 semigroup 47 subset 3 semiring 46, 190 substitution rule 139 Boolean 190 subtree 169 of natural numbers 190 subway map 177 string 190 successor 35 sequence 4 ofa node 168,169 empty 20 sufficient conditions 132 Fibonacci 95 sum of matrices 187, 189 sequential search 60,61, 103 switching circuit 114 series 13 functions 125 sum of 13 symbolic-expression 65 set 2 symmetric 146 complements 9 matrix 194 denumerable 163 syntax 54, 136 difference 8, 166 of decimal numbers 59 disjoint 8 of programming languages 59 empty 3 of well-formed formulas 137 infinite 161 isomorphic 160 nondenumerable 164 T of characters 41 regular 58 tautologies 137 set theory 1 telephone directory 61 220 Subject Index terminal 55 true III node 169 truth table 112 ternary 15 theorem 138, 139 threshold 126 u Logic Unit 126 top 158 undirected graph 177 topological sorting 154 union 7,43 total function 22 union of relations 27 total order 150 unit vector 188 Tournament Method 109 universal quantifier 141 transition function 48 unordered graph 175 transitive 146 unordered pair 194 transitive closure of a relation 27 upper bound 155 transitive reflexive closure of a relation 27 traversals of binary trees 93 v trap state 51 tree 53,87, 167 variables 54 ambiguous presentation 94 vel 113 binary 90 Venn diagrams 7 binary decision 7, 106 vertex 66 complete binary 92 derivation 55 domain 168 w forest 90 full binary 92 well-formed formula 135, 138 height 88 well-formed programs 54 level in 91 wff 138 L-tree 66 weight 126 rooted 87 worst-case complexity 103