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Appendix A Mathematical Background

In this appendix, we review the basic notions, concepts and facts of , theory, , analysis, and theory that are used throughout this book. For further details see, for example, [105, 134, 138] for logic, [61, 56, 88] for , [40, 74, 119] for algebra, [51] for formal languages, and [77, 142] for .

Propositional Calculus P

Syntax • An expression of P is a finite of symbols. Each denotes either an individ- ual constant or an individual variable, or it is a logic connective or a parenthesis. Individual constants are denoted by a,b,c,... (possibly indexed). Individual variables are denoted by x,y,z,... (possibly indexed). Logic connectives are ∨,∧,⇒,⇔ and ¬; they are called the dis- junction, conjunction, implication, equivalence, and negation, respectively. Punctuation marks are parentheses. • Not every expression of P is well-formed. An expression of P is well formed if it is either 1) an individual-constant or individual-variable symbol, or 2) one of the expressions F ∨ G, F ∧ G, F ⇒ G, F ⇔ G, and ¬F, where F and G are well-formed expressions of P. A well-formed expression of P is called a sentence. Semantic

• The intended meanings of the logic connectives are: “or” (∨), “and” (∧), “implies” (⇒), “” (⇔), “not” (¬). • Let {,,⊥} be a set. The elements , and ⊥ are called logic values and stand for “true” and “false,” respectively. Often, 1 and 0 are used instead of , and ⊥, respectively. • Any sentence has either the , or ⊥. A sentence is said to be true if its truth value is ,, and false if its truth value is ⊥. Individual constants and individual variables obtain their truth values by assignment. When logic connectives combine sentences into new sentences, the truth value of the new sentence is determined by truth values of its component sentences. Specifically, let E and F be sentences. Then: – ¬E is true if E is false, and ¬E is false if E is true. – E ∨ F is false if both E and F are false; otherwise E ∨ F is true. – E ∧ F is true if both E and F are true; otherwise E ∧ F is false. – E ⇒ F is false if E is true and F is false; otherwise E ⇒ F is true.

© Springer-Verlag Berlin Heidelberg 2015 297 B. Robiˇc, The Foundations of Theory, DOI 10.1007/978-3-662-44808-3 298 A Mathematical Background

– E ⇔ F is true if both E and F are either true or false; else, E ⇔ F is false. • The following hold: ¬(E ∨ F) ⇔ (¬E) ∧ (¬F) and ¬(E ∧ F) ⇔ (¬E) ∨ (¬F).

First-Order Logic L

Syntax

• An expression of L is a finite sequence of symbols, where each symbol is an individual-constant symbol (e.g., a,b,c), an individual-variable symbol (e.g., x,y,z), a logic connective (∨, ∧, ⇒, ⇔, ¬), a symbol (e.g., f,g,h), a predicate symbol (e.g., P,Q,R), a quantification sym- bol (∀,∃), or a punctuation mark (e.g., colon, parenthesis). (Predicates are also called relations.) • We are only interested in the well-formed expressions of L. To define these, we need two def- initions. First, a term is either 1) an individual-constant or individual-variable symbol, or 2) a function symbol applied to terms (e.g., f(a,x)). Second, an atomic formula is a predicate sym- bol applied to terms (e.g., P(y,f(a,x)). Finally, we say that an expression of L is well formed if it is either 1) an atomic formula, or 2) one of the expressions F ∨ G, F ∧ G, F ⇒ G, F ⇔ G, ¬F, ∀τF, and ∃τF, where F and G are well-formed expressions of L and τ is an individual-variable symbol. A logic expression of L that is well formed is called a formula.

Semantic

• The intended meanings of the quantification symbols are: “for all” (∀), “exists” (∃). For the meanings of logic connectives, see P above. • The truth value of a formula is determined as follows. Let E and F be formulas. Then: – ∀τF is true if F is true for every possible assignment of a value to τ. – ∃τF is true if F is true for at least one possible assignment of a value to τ. – For the truth values of F ∨ G, F ∧ G, F ⇒ G, F ⇔ G, ¬F, see Propositional Calculus P above.

Sets

Basics

• Given any objects a1,...,an, the set containing a1,...,an as its only elements is denoted by {a1,...,an}. More generally, given a property P, the set of those elements having the property P is written as {x|P(x)}. If an x is in a set A, we say that x is a member of A and write x ∈ A; otherwise, we write x ∈ A and say that x is not a member of A. The set with no members is called the and denoted by0. / • For sets A and B, we say that A is a of B, written A ⊆ B, if each member of A is also a member of B. A set A is a proper subset of B, written A  B,ifA ⊆ B but there is a member of B not in A. Instead of  we also write ⊂. • Sets A and B are equal, written A = B,ifA ⊆ B and B ⊆ A. • Given a set A = {xι |ι ∈ I}, the set I is called the index set of A. • By (a1,...,an), or also by a1,...,an , we denote the ordered n- of objects a1,...,an. When n = 2, the n-tuple is called the . Two ordered n- (a1,...,an) and (b1,...,bn) are equal, denoted by (a1,...,an)=(b1,...,bn),ifai = bi for i = 1,...,n. A Mathematical Background 299

Operations on Sets  • The of sets A and B, written as A B, is the set of elements that are members of at least A B one of and .  • A B A B The intersection of sets and , written as , is the set of elements that are members of both A and B. We say that A and B are disjoint if A B = 0./ • The difference of sets A and B, written as A − B, is the set of those members of A which are not in B. • If A ⊆ B, then the of A with respect to B is the set B − A. • The of a set A is the set of all of A and is denoted by 2A. • The Cartesian of a finite sequence of sets A1,...,An is the set of all ordered n-tuples (a1,...,an), where ai ∈ Ai for each i. In this case it is denoted by A1 ×...×An.IfA1 = ...= n 1 An = A, the is denoted by A . By convention, A stands for A.

Relations

Basics

• An n-ary on a set A is a subset of An. When n = 2 we say that the relation is binary, or in short, a relation.IfR is a relation, we write xRy to indicate that (x,y) ∈ R. A 1-ary relation on A is a subset of A, and is called a property on A. • A relation R on A is: – reflexive if xRx for each x ∈ A. – irreflexive if xRx for no x ∈ A. – symmetric if xRy implies yRx, for arbitrary x,y ∈ A. – asymmetric if xRy implies that not yRx, for arbitrary x,y ∈ A. – anti-symmetric if xRy and yRx imply x = y, for arbitrary x,y ∈ A. – transitive if xRy and yRz imply xRz, for arbitrary x,y,z ∈ A.

Ordered Sets

• A preordered set is a pair (A,R), where A is a set and R a on A, such that (i) R is reflexive, and (ii) R is transitive. In this case, we say that R is a preorder on A. Two elements x,y ∈ A are incomparable by R (in short, R-incomparable) if neither xRy nor yRx. • A partially ordered set is a pair (A,R), where A is a set and R a binary relation on A, such that (i) R is reflexive, (ii) R is transitive, and (iii) R is anti-symmetric. In this case, we say that R is a partial order on A. A partial order is often denoted by ≤,,., or any other symbol indicating the properties of this order. • Let (A,) be a partially ordered set. The relation ≺ on A is the strict partial order corre- sponding to  if a ≺ b ⇔ a  b ∧ a = b, for arbitrary a,b ∈ A. We say that ≺ is the irreflexive of . Conversely,  is the reflexive of ≺, since a  b ⇔ a ≺ b ∨ a = b. • Let (A,) be a partially ordered set and a,b ∈ A. When a  b, we say that a is smaller than or equal to (or lower than or equal to) b. Correspondingly, we say that b is larger than or equal to (or higher than or equal to) a. When a ≺ b, we say that a is smaller than (or lower than,or below) b. Correspondingly, we say that b is larger than (or higher than,orabove) a. • Let (A,) be a partially ordered set and a,b,c,d ∈ A. Then we say: – a is -minimal if xa implies x=a for ∀x∈A (nothing in A is smaller than a). – b is -least if b  x for ∀x ∈ A (b is smaller than any other in A). – c is -maximal if cx implies x=c for ∀x∈A (nothing in A is greater than c). – d is -greatest if x  d for ∀x ∈ A (d is greater than any other in A). 300 A Mathematical Background

When the relation  is understood, we can drop the prefix “-”. The least and greatest elements are called the zero (0) and unit (1) element, respectively. • Let (A,) be a partially ordered set, B ⊆ A, and u,v,w,z ∈ A. Then we say: – u is a -upper bound of B if x  u for all x ∈ B. – v is a -least upper bound (or -lub)ofB if v is a -upper bound of B and v  u for every -upper bound u of B. – w is a -lower bound of B if w  x for all x ∈ B. – z is a -greatest lower bound (or -glb)ofB if z is a -lower bound of B and w  z for every -lower bound w of B. When the relation  is understood, we can drop the prefix “-”. • A is a partially ordered set (A,) in which any two elements have an lub and a glb. The lub of a,b ∈ A is denoted by a ∨ b, and the glb by a ∧ b.Anupper semi-lattice is a partially ordered set (A,) in which any two elements have an lub (but not necessarily a glb). • A linearly (or totaly) ordered set is a partially ordered set (A,) such that for all x,y ∈ A either x  y or y  x. In this case we say that  is a linear order on A. • A well-ordered set is a linearly ordered set (A,) such that every non-empty subset of A has a -least element. We say that such a  is a well-order on A. • Associated with every well-ordered set (A,) is the corresponding Principle of Complete Mathematical Induction:IfP is a property such that, for any b ∈ A, P(b), whenever P(a) for all a ∈ A such that a  b, then P(x) for all x ∈ A. When A is infinite, a proof using this principle is called a proof by transfinite induction. Equivalence Relations • A relation R on A is an if (i) R is reflexive, (ii) R is symmetric, and (iii) R is transitive. In this case, the R-equivalence of a ∈ A is the set {x ∈ A|xRa}. Elements of the R- of a are said to be R-equivalent to a.IfC is an equivalence class, any element of C is called a representative of the class C. • A {A | ∈ I} A A = A partition of is any collection i i of nonempty subsets of such that (i) A A A = , ∈ I = A i∈I i, and (ii) i j 0,/ for all i j with i j. So, is the of the sets in the partition. • Any equivalence relation on A is associated with a partition of A, and vice versa. If R is an equivalence relation on A, then the associated partition of A is called the quotient set of A relative to R and is denoted by A/R. The members of A/R are the R-equivalence classes of A. The function f : A → A/R that associates with each element a ∈ A the R-equivalence class of a is called the natural of A relative to R. • An equivalence relation is often denoted by ∼,#,≡, or any other symbol indicating the prop- erties of this relation.

Functions

Basics • A total function f from A into B is a triple (A,B, f ) where A and B are nonempty sets and for every x ∈ A there is a unique member, denoted by f (x),ofB. We call A the domain of f and denote it by dom( f ). The set B we call the co-domain of f and denote it by codom( f ).We usually write f : A → B instead of (A,B, f ). A function is also called a mapping. • In specifying a definition of f : A → B we say that f is well-defined if we are assured that f is single-valued, i.e., with each member of A, f associates a unique member of B. • When the consists of ordered n-tuples, the function is said to be of n . A (total) function of n arguments on a set S is a function f whose domain is Sn. We write f (a1,...,an) instead of f (a1,...,an ). A Mathematical Background 301

• Let f : A → B and C ⊆ A. The of C under f is a set denoted by f (C) and defined by f (C)={ f (x)|x ∈ C}. In particular, f (A) is called the range of f and denoted by ( f ). • A function f : A → B is: – injective if f (x) = f (y) whenever x = y; we also say that such an f is an injection. – surjective if f (A)=B; we also say that such an f is a surjection. – bijective if it is injective and surjective; we also say that such an f is a . • An element a ∈ A is called the fixed of a function f : A → A if f (a)=a. • Let f : A → B and C ⊆ A. Then a function g : C → B is the restriction of f to C if g(x)= f (x) for each x ∈ C. The restriction of f to C is denoted by f |C. In that case f is the of g to A. • Let f : A → B and g : C → D be functions and f (A) ⊆ C. Then the composite function (or composition)of f and g, denoted by g◦ f , is the function g◦ f : A → D defined by (g◦ f )(x)= g( f (x)), for each x ∈ A. • Let A and U be sets, and let A ⊆ U. The characteristic function of A is the function χA : U → {0,1} such that χA(u)=1ifu ∈ A and χA(u)=0ifu ∈ A. • Let A and B be nonempty sets. Then the set of all functions having the domain A and co- domain B is denoted by BA.

Cardinality

• Two sets A and B are said to be equinumerous (or equipollent or of the same power), which is denoted by A # B, if there exists a bijection f : A → B. In that case we say that A and B have the same cardinal . The of a set A is denoted by |A|. The relation # is an equivalence relation. • A cardinal number |A| is smaller than a cardinal number |B|, written |A| < |B|, if there is an injection f : A → B,butA and B are not equinumerous. • Cantor’s states that |A| < |2A|, for any set A. • A set A is – finite if either A = 0or/ A #{1,2,...,n} for some natural n; – infinite if it is not finite; – countable (or enumerable,ordenumerable)ifA # B for some B ⊆ N; when B = N, the set A is said to be countably infinite; – uncountable if it is not countable. • If a set A is infinite, then there is B  A such that B # A. • Any subset of a is countable. The union of countably many countable sets is countable. The Cartesian product of two countable sets is countable. • Let n be a , ℵ0 = |N|, and c = |R| the of continuum. Then: ℵ0 + n = ℵ ℵ + ℵ = ℵ · ℵ = ℵ ℵn = ℵ c + ℵ = c ℵ · c = c 0, 0 0 0, n 0 0, 0 0, 0 , and 0 . • A sequence is a function f defined on N, the set of natural . If we write f (n)=xn, for n ∈ N, we also denote the sequence f by {xn},orbyx0,x1,x2,... When xn ∈ A for all n ∈ N, we say that {xn} is a sequence of elements of A. The elements of any at most countable set can be arranged in a sequence. • The cardinality of BA, the set of all functions mapping A into B,is|B||A|.

Operations and Algebraic Structures

• An n-ary on a set A is a function  : An → A. When n = 2, we say that the operation is binary. In this case we write a  b instead of (a,b) When n = 1, the operation is said to be unary and we write a instead of (a). • A on a set A is: – associative if a  (b  c)=(a  b)  c, for all a,b,c ∈ A. – commutative if a  b = b  a, for all a,b ∈ A. 302 A Mathematical Background

• A is a pair (A,), where  is an associative binary operation on A. • A is a semigroup (A,) satisfying the following requirements: – there exists an element e ∈ A such that ae = ea = a, for all a ∈ A (e is called an identity of A); – for each a ∈ A there exists an element a−1 ∈ A such that aa−1 = a−1 a = e (a−1 is called an inverse of a).

Natural Numbers

• Natural numbers are 0,1,2,.... The set of all natural numbers is denoted by N. The cardinal number of N is denoted by ℵ0 (aleph zero). • A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself. There are infinitely many primes. A natural number greater than 1 that is not a prime is called a composite. • The Fundamental Theorem of states: Any positive ( = 1) can be expressed as a product of primes; this expression is unique except for the order in which the primes occur. α α α ( = ) 1 2 ... r , ,..., Thus, any positive integer n 1 can be written as p1 p2 pr , where p1 p2 pr are primes satisfying p1 < p2 <...

Formal Languages

Basics

• An alphabet Σ is a finite non-empty set of abstract symbols. • A word of length k  0 over the alphabet Σ is a finite sequence x1,...,xk of symbols in Σ.A word x1,...,xk is usually written without commas, i.e., as x1 ...xk. • The length of a word w is denoted by |w|. The word of length zero is called the empty word and denoted by ε. R • If w = x1 ...xk is a word, then the word w = xk ...x1 is called the reversal of w. • Two words x1 ...xr and y1 ...ys over the alphabet Σ are equal, written x1 ...xr = y1 ...ys,if r = s and xi = yi for each i. • Let x and y be words over the alphabet Σ. The word x is a subword of y if y = uxv for some words u and v. The word x is a proper subword of y if x is a subword of x,butx = y. A Mathematical Background 303

• Let x and y be words over the alphabet Σ. The word x is a prefix of y, written x ⊆ y,ify = xv for some word v. The word x is a proper prefix of y, written x ⊂ y,ifx is a prefix of x,butx = y. • The set of all words, including ε, over the alphabet Σ is denoted by Σ ∗. • The set Σ ∗ is countably infinite. • Each subset L ⊆ Σ ∗ is called a formal language (or language in short).

Operations on Languages

• If x = x1 ...xr and y1 ...ys are words, then xy, called the of x and y, is the word x1 ...xry1 ...ys. • For languages L1 and L2, the concatenation (or product)ofL1 and L2 is a language denoted by L1L2 and defined by L1L2 = {xy|x ∈ L1 ∧ y ∈ L2}. • L L0 = {ε}  Ln = Ln−1L L For a language let and, for each n 1, let . The Kleene star of is L∗ L∗ = ∞ Li L the language denoted by and defined by  i=0 . Similarly, Kleene plus of is the L+ L+ = ∞ Li Σ language denoted by and defined by i=1 . In particular, for the alphabet , the language Σ n contains all words of length n over Σ, and Σ ∗ contains all words over Σ.

Orders on Languages

n • Let ≤ be a linear order on the alphabet Σ.Alexicographic order ≤lex on Σ , induced by ≤, is the order in which x1 ...xn

Frontmatter and Chapter 2

Box detour x NB nota bene x A,B,C,... sets 13, 298 x ∈ A x is a member of A 13, 298 A ⊆ BAis a subset of B 13, 298 A the complement of A 14, 298 A ∪ B union of A and B 14, 298 A ∩ B intersection of A and B 14, 298 A − B set theoretic difference of A and B 14, 298 2A power set of A 14, 298 (x,y) ordered pair where x is the first and y the second member 14, 298 A × B Cartesian product of A and B 14, 298 |A| cardinality of A 15, 300 N the set of all natural numbers 302 ℵ0 |N|, the least transfinite cardinal 16, 302 ℵi transfinite cardinal 16  is less than or equal to (used for numbers) 14 ω well-ordered set (N,), the least transfinite ordinal 17 R the set of all real numbers 16 c |R|, the cardinality of continuum 16 iff if and only if 17 Ω set of all ordinal numbers (paradoxical) 17 U set of all sets (paradoxical) 17 R Russell’s set of all sets not containing themselves (paradoxical) 17 Z the set of all 19 PM 25

Chapter 3 f.a.s. formal 31 F a particular f.a.s., the theory developed in this f.a.s. 31 F meta-theory, the theory about F 59 a,b,c,... symbols for individual constants in f.a.s. 32 x,y,z,... symbols for individual variables in f.a.s. 32 f,g,h,... symbols for functions in f.a.s. 32

© Springer-Verlag Berlin Heidelberg 2015 305 B. Robiˇc, The Foundations of , DOI 10.1007/978-3-662-44808-3 306 B Notation Index p,q,r,... symbols for predicates in f.a.s. 32 F,G,H,... symbols for formulas in f.a.s. 32 ∧ and 32, 297 ∨ or 32, 297 ¬ not 32, 297 ⇒ implies 32, 297 ⇔ is equivalent to 32, 297 ∃ there exists 32, 298 ∀ for all 32, 298 R rule of 33 MP Modus Ponens 33 Gen Generalization 33 P  FFis derivable (formally provable) from the set of premises P 34 F FFis a theorem of F, i.e., derivable (formally provable) in F 34 S 35 ι mapping that assigns meaning to F in S 36 P |= FFis a of the set of premises P 38 |= F FFis valid in F 39 P Propositional Calculus 36, 297 L First-order Logic 39, 298 A Formal Arithmetic 41 ZF Zermelo-Fraenkel axiomatic set theory 42, 44 ZFC ZF with of Choice 42 NBG Von Neumann-Bernays-Godel’s¨ axiomatic set theory 43, 45

Chapter 4

M formal axiomatic system for all mathematics 53, 54 DEntsch decision procedure for M 53, 55 γ(X) Godel¨ number of syntactic object X 59 G Godel’s¨ formula 60 CH 59 GCH Generalized Continuum Hypothesis 59

Chapter 5 f ,g,... total functions from Nk to N 72 ζ zero function 73, 74 σ successor function 73, 74 π projection function 73, 74 [...x...] expression containing variable x 77 μx[...x...] μ-operation, the least x such that [...x...]=0 and [...z...]↓ for z < x 73, 75 E ( f ) system of equations defining a function f 76 λx.[...x...] of x, defined by [...x...] 77, 78 →α α-conversion, renaming of variables in a λ-term 78 →β β-contraction, application of a λ-term 78 β sequence (composition) of β-reductions 78 β-nf β-normal form 78 TM 80, 101 TP Turing program 80, 103 Σ input alphabet 80, 103 → production 82, 84 M code of an abstract computing machine M 86 B Notation Index 307

←→ is formalized by 87 CT Computability Thesis, i.e., Church-Turing Thesis 88 ϕ,ψ,... partial functions ϕ(x)↓ ϕ(x) is defined 93 ϕ(x)↓= y ϕ(x) is defined and equal to y 93 ϕ(x)↑ ϕ(x) is undefined 93 dom(ϕ) domain of ϕ 93 rng(ϕ) range of ϕ 93 ϕ # ψ of partial functions 93 p.c. partial computable 95, 126

Chapter 6

TM Turing machine 80, 101 T a TM (basic model) 104 Tn the TM with index (code number) n 116 Γ tape alphabet 101 empty space 101 Σ input alphabet 103 Σ ∗ the set of all words over Σ 302 Q set of states 103 q1 initial state 103 F set of final states 103 TP Turing program 80, 103 δ a Turing program 103 δn the TP with index (code number) n 116 qyes a final state 103 qno a non-final state 103 Δ describing δ 103 V a Turing machine (generalized model) 107 T code of T 115 U 117 OS operating system 121 RAM random access machine 122 ψ(k)( ) T x k-ary proper function of T 125 ψ(k)( ) i x the k-ary proper function of Ti 125 W ψ(k)( ) i domain of i x 125 ε empty word 302 GA generator of A 129 c.e. 130 L(T) proper set of T, i.e., language of T 131 U , a large enough set 131 χA characteristic function of A 131, 300 DA decider of A 132 RA recognizer of A 132 P the set of prime numbers 139

Chapter 7 ind(ϕ) index set of a p.c. function ϕ 146 ind(A) index set of a c.e. set A 146 308 B Notation Index

Chapter 8

D 163 d code of an instance d of a decision problem 163 L(D) language of decision problem D 164 DHalt , “Does T halt on w?” 167 DH Halting problem, “Does T halt on T ?” 167 K0 universal language 167 K diagonal language 167 D complementary problem of a decision problem D 171 D Halt non-Halting problem, “Does T never halt on w?” 170 D  H non-Halting problem, “Does T never halt on T ?” 170 DEmp empty proper set problem, “Is L(T)=0?”/ 173 n-BB n-state 173 DBB Busy Beaver Problem, “Is T a busy beaver?” 173 PCP Post’s Correspondence Problem 175 DPCP Post’s Correspondence Problem 175 CFG context-free grammar 177 CFL context-free language 177 D (ϕ) K1 “Is dom empty?” 179 DFin “Is dom(ϕ) finite?” 179 DInf “Is dom(ϕ) infinite?” 179 DCof “Is A − dom(ϕ) finite?” (where ϕ : A → B) 179 DTot “Is ϕ total?” 179 DExt “Can ϕ be extended to a total ?” 179 DSur “Is ϕ surjective?” 179

Chapter 9 sw switching function 191 ≤m is m-reducible to (set, decision problem, function) 197 ≤1 is 1-reducible to (set, decision problem, function) 201 DEntsch 203 C class of all c.e. sets 208

Chapter 10 o-TM oracle Turing machine 224 T ∗ an o-TM (with no oracle set attached) 226 T ∗ the code of T ∗ 227 ∗ Ti the o-TM with index i (and no oracle set attached) 227 O-TM o-TM with oracle set O attached 224 T O an O-TM 226 O O Ti the -TM with index i 227 o-TP oracle Turing program 224  δ an o-TP (i.e., a transition function of an o-TM) 224  δi the o-TP with index i 227 w.l.g. without loss of generality 228 Ψ ∗ i proper functional of the o-TM Ti (with understood) 228 Ψ O O O i proper functional of the -TM Ti (with arity understood) 228 O-p.c. partial O-computable (function) 229 B Notation Index 309

O-c.e. O-semi-decidable (set) 230 indO(ϕ) index set of the O-p.c. function ϕ 230 indO(S) index set of the O-c.e. set S 230

Chapter 11

≤T is T-reducible to (set, decision problem, function) 236 ≡T is T-equivalent to (set, decision problem, function) 240

Chapter 12

S the T-jump of the set S 247 KS the same as S 247 S(n) the n-th T jump of the set S 249

Chapter 13

D the class of all T-degrees 255 a,b,... T-degrees 255 ≤ is lower than or equal to (T-degree) 256 d  the T-jump of d 256 ( ) d n the n-th T-jump of d 256 0 the T-degree of the set0 / 256 ( ) 0 n the n-th T-jump of 0 256 x ⊆ y the word x is a prefix of the word y 260 x ⊂ y the word x is a proper prefix of the word y 260 ≤-lub the least upper bound of T-degrees 262 ≤-glb the greatest lower bound of T-degrees 262 A ⊕ B join of sets A and B 252 ucone(d) upper cone of d 252 lcone(d) lower cone of d 252

Chapter 14

≤btt is bounded truth-table-reducible to (set, decision problem, function) 273 ≤tt is truth-table-reducible to (set, decision problem, function) 273 x ∈!A add (enumerate) x into the set A 275 x ∈!A ban x from the set A 275

Chapter 15

Σn arithmetical class 285 Πn arithmetical class 285 Δn arithmetical class 285 graph(ϕ) graph of ϕ 292 References

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λ-calculus, 77–79 Herbrand-Godel’s,¨ 77 λ-definable function, 78, 87 Markov’s, 85 λ-term, 77–79 Post’s, 84 α-conversion, 77–79 Turing’s, 82 β-contraction, 77–79 description, 71 β-normal form (β-nf), 79 environment, 71 β-redex, 79 execution, 70, 71 β-reduction, 79 existence, 71 abstraction, 77–79 formally, 90 application, 77–79 intuitively, 4, 70, 89 Church’s numeral, 79 non-existence, 71 defining a numerical function, 79 offline, 233 final, 77–79 online, 233 initial, 77–79 problems about, 175 μ-operation, see Recursive function property, 194–195 quantum, 70 Abstract computing machine, 86, 205, 209 shrewd, 195 basic instruction, 86 Alphabet, 27, 32, 35 finite effect, 86 Aristotle, 11, 21, 23 behavior Arithmetical global, 209 class of sets, 286 local, 209 hierarchy, 283–292 capability, 86 relation, 283–284 code · ,86 set, 284 components, 86 Arithmetization internal configuration, 86, 104 of a theory, 59 resources, xi, 86 of man’s reflection, 6 unrestricted, xi, 86 Aspiration, 63 Ackermann, 26, 73, 99 Axiom Adleman, 89 fertility, 12 Al-Khwarizmi, 6 independence, 11 Aleph zero ℵ0, 15, 59, 172 logical, 33, 38, 39, 52 , 3, 51, 69 of a theory, 10, 56 characterization of the concept, 71 of Abstraction, 13, 15, 17, 18, 20, 45 Church’s, 78 of Choice, 13, 24, 44 convincing, 88 of Class Existence, 45 Godel-Kleene’s,¨ 74 of , 13, 44, 45

© Springer-Verlag Berlin Heidelberg 2015 319 B. Robiˇc, The Foundations of Computability Theory, DOI 10.1007/978-3-662-44808-3 320 Index

of Infinity, 44 definition of the set, 20 of Mathematical Induction, 42, 47 diagonalization, 191 of Pair, 44 Paradox, 17 of Power Set, 44 Theorem, 16 of Regularity, 44 view of infinity, 21 of Separation (schema), 44 Cardinal number, 15 of Substitution (schema), 44 ℵ0, the cardinality of N,15 N of Union, 44 ℵ1, the cardinality of 2 ,16 proper (= non-logical), 33, 35, 38, 40, 52 finite, 16 schema, 39 transfinite, 16, 59 Axiomatic Cellular automaton, 89 evident, 11 Characteristic function, 131, 144, 147, 172, hypothetical, 11, 12 204 Axiomatic method Characterization of the algorithm, 71 axioms, 10 Church’s, 78 basic notions, 10 convincing, 88 Euclidean geometry, 9 Godel-Kleene’s,¨ 74 initial theory, 10 Herbrand-Godel’s,¨ 77 limitations of, 58 Markov’s, 85 Axiomatic set theory Post’s, 84 NBG, 43, 45, 54 Turing’s, 82 ZFC, 42, 44, 54 Chomsky, 179 Axiomatic system, 9, 10 Church, 73, 77–79, 87, 129, 161, 166, 202 Church Thesis, 87 Babbage, 6, 70, 119 Church-Rosser Theorem, 79 analytical engine, 6 Church-Turing Thesis, see Computability Basic notions Thesis of a theory, 10, 33, 34, 38, 43, 45, 46, 143 Class, 43, 45, 286 of computation, 72, 130, 143, 161 arithmetical, 286 of generation, 127 non-proper, 45 Behavior proper, 45 of a Turing program (machine) D global, 146, 147, 152, 209 Class of degrees of unsolvability, 255–267 local, 146, 147, 152, 209 basic properties, 257–267 of any mechanism, 209 cardinality, 257–258 Beltrami, 11 greatest lower bound, 262–263 Bernays, 26, 43 incomparable Turing degree, 259–261 Bolyai, 12 intermediate Turing degree, 263–264 Boole, 23, 39 least Turing degree, 262 Boole’s least upper bound, 262 The Laws of Thought, 23 minimal Turing degree, 266–267 , 23 structure of, 258–263 Brouwer, 21 upper and lower cone, 264–265 Burali-Forti, 17 initial clues of, 169–171 Burali-Forti’s Class of all decision problems Paradox, 17 basic structure, 169–171 Busy Beaver, 174 Coding function, 163 Busy Beaver function, 172, 174 Cohen, 58 Compactness Theorem, 47 c.e. Degree, 269–280 Completeness, 49, 203 Post’s Problem, see Post’s Problem Completeness Problem Canonical system, 83, 177, 231 of Principia Mathematica, 29 Cantor, 13 Completeness Requirement, 73, 77, 78, 82, 87, Cantor’s 97 Index 321

Computability Thesis, 87–92, 97, 128, 135, of mathematical objects, 21 136, 142, 147, 150, 156, 194, 219 rules of, 24, 28, 32, 35 Computable function, 95, 126, 138, 151, 152, Continuum Hypothesis, 16, 58–59 154, 158, 163, 172, 193, 196, 201, 204, Generalized, 59 207 Conway, 89 formalization by Cook, 122 Church, 78 Counting problem, 162 Godel¨ and Kleene, 74 Herbrand and Godel,¨ 77 Data vs. instructions, 119 Markov, 85 Data-flow graph, 83 Post, 84 Davis, 180, 231 Turing, 82 , 49, 184 on a set, 96, 126, 196 Decidability Problem, 51, 69, 184, 202 problems about, 179 Decidable problem, 165, 199, 205–210 Computable problem, 161 Decidable relation, 283–284 , see Decidable set Decidable set, 132, 138, 143–144, 158, 164, Computably enumerable (c.e.) set, 130, 138, 198, 206, 207 see Semi-decidable set Decider, 132, 144, 164, 168, 194, 200, 202, 1-complete, 213 203, 205, 248 T-complete (Turing complete), 270, 271 Decision problem, 161–189, 199–210 m-complete, 213 1-reduction, 201 creative, 273 m-reduction, 197 hyper-simple, 273 code of, 163 hyperhyper-simple, 273 coding function, 163 is semi-decidable set, 136 complementary, 171, 202 simple, 272 contained in another problem, 198 Computation, 69 decidable, see Decidable problem formalization by equal to another problem, 198 Church, 78 instance of, 163 Godel¨ and Kleene, 74 negative, 163, 198 Herbrand and Godel,¨ 77 positive, 163–165, 198 Markov, 85 language of, 163–165 Post, 84 reduction, see Reduction Turing, 82 semi-decidable, see Semi-decidable problem halting, 94 subproblem of, 165–167, 198, 199 intuitively, 4, 89 undecidable, see Undecidable problem offline, 233 vs. set recognition, 164 online, 233 Decision procedure, 51, 56, 69, 70 Computational Complexity Theory, xi, 86, effective, 51, 53 102, 108, 114, 118, 122, 138, 163, 185, Deduction, 10, 23, 27 197 Definition, 10 Computational power, see Abstract computing Degree of undecidability, 171 machine Degree of unsolvability Computational problem, 3, 161, 162 see Turing degree, 240 Computer, see General-purpose computing Density Theorem, 279 machine Derivation, 27, 34, 51, 55, 59, 69 Conclusion, 10, 33 as a sequence of formulas, 56, 69 Configuration in an inconsistent theory, 50 , 107 recognition of, 56 internal, 86, 104, 203 searching procedure, 55 , 49, 60, 63, 203 syntactically correct, 56 Consistency Problem, 50 Deutsch, 118 of Principia Mathematica, 29 Development of a theory, 10, 27, 33 Construction mechanical, 35, 63 322 Index

rules of, 33 of undecidable semi-decidable problems, Diagonal language K, 167 169 Diagonalization, 16, 91, 129, 161, 191–195, of undecidable semi-decidable sets, 169, 199 170 direct, 191–193 of undecidable sets, 168 switched diagonal, 192, 193 Platonic view, 21 switching function, 192, 193 Expression, 27, 28, 32, 82 indirect, 194–195 shrewd Turing machine (algorithm), 194 Finite Extension Method, 275 Diophantine equations, 180 Finitism, 28, 54, 61, 70, 90 DNA-calculus, 89 First Incompleteness Theorem, 57–61, 69 Domain, 26 First-Order Logic L, 24, 39, 50, 53, 54, 60, 298 of function, 93, 125, 158 Fixed-Point Theorem, see Recursion Theorem of interpretation, 26, 35 Formal Arithmetic A, 38, 40, 54, 57, 60, 203, dovetailing, 134 204 Formal axiomatic system, 27, 31, 33, 35, 204 Eckert, 120 M, of the whole of mathematics, 54 EDVAC, see General-purpose computing consistent axiomatic extension of, 57, 60 machine extension of L,40 Effective procedure, 73 of the first order, 40, 46 Effectiveness Requirement, 73, 77, 78, 82, 87, of the second order, 46 97 Formalism, 20, 26, 28, 31 Elgot, 122 its goals, 29 ENIAC, see General-purpose computing Formalization machine of algorithm’s environment, 71 Entscheidungsproblem, 53, 55, 56, 69, 97, of arithmetic, 40 202–204 of computation, 71 function, 116, 129 of logic, 39 Equation of set theory, 42 Diophantine, 180 of the whole of mathematics, 54 system, 76 Formula, 27, 32, 59, 203 Euclid, 5, 11 closed (= sentence), 32, 37 Euclid’s independent of a theory, 50, 57, 58, 60 algorithm, 5 its truth-value, 37, 52 axiomatic method, 9 logically valid, 38 fifth axiom, 11 of the second order, 47, 48 geometry, 9 open, 32, 37 Execution satisfiable under the interpretation, 38 of algorithm, 70 undecidable, 69, 90 of instruction, see Instruction valid in a theory, 38, 52, 58 Existence valid under the interpretation, 38 and construction, 21, 172 Foundations of mathematics, 49 intuitionistic view, 21, 172 problems of, 49 of a computable non-recursive function, 91, Fraenkel, 42 191, 193 Frege, 23, 39 of a general-purpose computing machine, Frege’s 119 Begriffsschrift, 24 of a Universal Turing machine, 117–118 Friedberg, 274–278 of equivalent Turing programs, 151, 153 Function, 14 of fixed-points of computable functions, λ-definable, 78, 87 152, 154 Ackermann, 73, 99 of incomputable functions, 172, 174 bijective, 14, 137 of Turing machines, 114 Busy Beaver, 172, 174 of undecidable problems, 166 characteristic, see Characteristic function Index 323

coding (for a decision problem), 163 Generator, 129, 135, 138, 203 computable, see Computable function Gentzen, 61 defined, 94 Geometry domain of, 93, 158 Euclidean, 11 elementary, 182 fifth axiom (=Parallel Postulate), 11 enumeration, see Enumeration function non-Euclidean equal, 93 elliptic, 12 extension of, 93 hyperbolic, 12 fixed point of, 152, 154, 204, 207, 212 model of, 12 general recursive, 76 Goldbach’s Conjecture, 22, 63, 90 graph of, 292 Grammar , 183 in Markov algorithms, 84 identity, 202, 203 in Post’s canonical systems, 127 incomputable, see Incomputable function of a formal language, 177 injective, 14, 93, 163, 201 ambiguous, 177 Markov-computable, 85 context-free, 177 numerical, 72, 90, 91, 193 context-sensitive, 177 oracle-computable, see Oracle-computable equivalent, 178 function regular, 177 p.c., see Partial computable (p.c.) function problems about, 177 pairing, 140 Graph Theorem, 292 partial, 74, 92–97, 103 Godel,¨ 43, 49, 53, 57–62, 73–77, 87, 129, 203 Post-computable, 84 Godel¨ number, 59, 64, 71 proper (of a Turing machine), 125, 146, 246 Godel’s¨ property of, see Property Completeness Theorem, 52 range of, 93, 129, 130 First Incompleteness Theorem, 46, 57–61, recursive, see Recursive function 69 surjective, 14, 93 formula, 60 switched, see Diagonalization Second Incompleteness Theorem, 46, 60–62 total, 72, 90, 97, 205, 207 transition, see Turing machine Halting Problem, 161, 166–169, 172, 191, 195, Turing-computable, 81, 87 199, 202, 203, 219, 245–252 zeros of, 182 Hartmanis, 122 Function computation, 125–126 Heisenberg, 88 Functional, 228, 247 Herbrand, 54, 73, 76, 129 proper (of an oracle Turing machine), 228, Heyting, 21 247, 249 Hierarchy arithmetical, 283–292 General recursive function, 76, 87 jump, 245–252, 288–291 General-purpose computing machine, Hilbert, 12, 26, 49, 202 119–124, 138 Hilbert’s architecture, see RAM elementary geometry, 12 compiler, see Procedure Program, 49, 53, 58, 204 existence of, 119 finitism, 54, 61, 70 operating system, see Operating system intention, 53 Generalization, 33, 39, 40, 52 its fate, 54, 204 Generating (=enumeration) problem, 162 legacy, 63 Generation formalization by IAS, see General-purpose computing machine Church and Kleene, 129 Incomputability proving, 191–215 Post, 127 by diagonalization, see Diagonalization Turing machine, 129 by Recursion Theorem, see Recursion of pairs of natural numbers, 135 Theorem of sequences of symbols, 55, 203 by reduction, see Reduction 324 Index

by Rice’s Theorem, see Rice’s Theorem Instruction Incomputable function, 95, 126, 138, 172, 174, basic, 70, 86 204 execution on a set, 96, 126 continuous, 70 Incomputable problem, 161 discrete, 70 Ambiguity of CFG grammars, 177 predictable, 70 Busy Beaver function, 174 random, 70 Busy Beaver Problem, 174 intuitively, 4 Classification of manifolds, 183 Internal configuration, 86, 203 Correctness of algorithms (programs), 176 Interpretation Decidability of first-order theories, 184 domain of, 26, 35, 36 Domino snake problem, 186 of a theory, 12, 26, 28, 35, 36 Domino tiling problem, 186 standard, 38, 40, 42 Equality of words, 182 Intuition and experience, 11, 12 Equivalence of CFG grammars, 178 Intuitionism, 20 Existence of shorter equiv. programs, 176 and existence of objects, 21 Existence of zeros of functions, 182 and the Law of Excluded Middle, 21 Halting of Turing machines, 173 Mortal matrix problem, 180 Jump hierarchy, 245–252, 288–291 Other properties of CFGs and CFLs, 178 Post’s Correspondence Problem, 175 Kleene, 74, 87, 129, 149, 231 Properties of computable functions, 179 Kucera,ˇ 280 Properties of TM languages, 173 Kuratowski, 14 Satisfiability of first-order formulas, 185 Shortest equivalent program, 210 Lachlan, 279 Solvability of Diophantine equations, 180 Language Termination of algorithms (programs), 175 diagonal K, 167 of first-order formulas, 185 expression of, 47 Word problem for groups, 181 formal, 28, 137 Word problem for semi-groups, 181 natural, 24, 28, 82 Incomputable set, see Undecidable set of a Turing machine, see Proper set of TM Independent formula, 50, 57, 58, 60 problems about, 177 Index programming, 177 of a partial computable (p.c.) function, symbolic, 24, 27, 32, 35 145–147, 151, 153 of the first order, 32, 46 of a proper function (of a TM), 125 of the second order, 32, 46 of a Turing machine, 115, 145, 149 universal K0, 167, 169 of a Turing program, 145, 149 Law of Excluded Middle, 13, 18, 21, 54, 163 Index set, 146–147, 205–207, 211 Leibniz, 6, 59 of a partial computable (p.c.) function, 146 arithmetization of man’s reflection, 6 of a partial oracle-computable function, 230 calculus ratiocinator, 6 of a semi-decidable (=c.e.) set, 147 lingua characteristica universalis, 6 of an oracle-semi-decidable set, 230 Lexicographical order, 91 property of, 206–207 Lobachevsky, 12 Induction Logical validity (of formulas), 38 mathematical, 42, 47 Logicism, 20, 23 transfinite, 54, 61, 62 Lowenheim-Skolem¨ Theorem, 47 Inference, 10, 27 rules of, 23, 28, 33, 35, 52, 56 Machine, 101 Infinity Manifold Aristotle’s view of, 21 classification of, 183 Cantor’s view of, 21 homeomorphic, 183 in intuitionism, 21 topological, 183 Inspiration, ingenuity, 63, 205 Markov, 82, 84–85 Index 325

Markov algorithm, 84–85 discrete, 70 Markov-computable function, 85 predictable, 70 Mathematical induction, 42 random, 70 Mathematical structure, 35 objectively, 70 Matiyasevic,ˇ 180 subjectively, 70 Matrix mechanics, 88 Non-finitism, 61 Mauchly, 120 Nondiamond theorem, 279 Mechanical manipulation Notation of symbols, 24, 34 standard formal, 24 benefits, 34 Number system Melzak, 122 decimal positional, 5 Metamathematics, 11, 28, 38 Roman, 5 Metatheory, see Metamathematics Method, 191 Operating system, 121, 138, 156 Minsky, 122 activation record, 156 Model, 69 bootstrap loader, 121 Model of a theory, 35, 38, 52, 58 data region, 121 standard, 40, 42, 57, 60 dynamic allocation of space, 121 Model of computation, 72–86, 89, 97, 101, 205 executable program, 158 after functions, 72 file system, 121 after humans, 80 global variables, 121 after languages, 82 heap, 121 cellular automaton, 89 input-output management, 121 Church’s λ-calculus, 78 loading, 121 convincing, 88 local variables, 121, 156 data-flow graph, 83 memory management, 121 DNA-calculus, 89 multiprogramming, 121 equivalent, 87, 88, 97, 122, 124, 149, 153, networking, 121 195 procedure-linkage, 121 Godel-Kleene’s¨ recursive functions, 74, 90 program initiation, 121 Herbrand-Godel’s¨ general recursive protection, 121 functions, 77 residency, 121 instance of, see Abstract computing machine runtime stack, 121, 156 Markov algorithm, 85 security, 121 Post machine, 84 system call, 158 RAM, 89, 122 task, 156 RASP, 89, 122 Optimization problem, see Search problem reasonable, 86, 97 Oracle, 219–222 register machine, 89, 122 Oracle Turing machine, 220–224 Turing machine, 82, see Turing machine o-TM, 223–231 unrestricted, 86 coding, 226–227, 246–247, 249 Model theory, 46 enumeration of, 226–227 Modus Ponens, 33, 39, 40, 52 index of, 227 Muchnik, 274–278 operational semantics, 223 oracle tape, 223–224 Natural numbers N, 54, 57, 71, 116, 137, 193, program of, 224–226 204, 207 with a database, 232 ℵ0, the cardinality of N, 15, 193 with a network of computers, 232 N ℵ1, the cardinality of 2 , 16, 191 with oracle set, 224 von Neumann’s construction, 15 o-machine, 220–222 Nature nondeterminism, 221 at macroscopic level, 70 oracle set, 221 at microscopic level, 70, 88 special instructions, 220–221 continuous, 70 special states, 220–221 326 Index

Oracle-computable function, 229 Post machine, 83–84, 128 computable, 229 computation on, 84 on a set, 229 halting, 84 incomputable, 229 starting, 84 partial, 229 step, 84 on a set, 229 control unit, 83 Oracle-decidable set, 230 input word, 83 in a set, 230 Post program, 83 Oracle-semi-decidable set, 230 as a graph, 83 in a set, 230 queue, 83 Oracle-undecidable set, 230 tape, 83 Order window, 83 lexicographical, 91 Post Thesis, 128, 130 shortlex, 91 Post’s Problem, 270 Ordered pair, 14 Friedberg-Muchnik’s solution, 278 , 17, 18, 63 Post’s Program, 271 finite, 17 priority-free solution limit, 63 Kucera,ˇ 280 transfinite, 17 Post’s Theorem, 144 Ordinal type, 18 Post-computable function, 84 OS, see Operating system Predicate Calculus, see First-Order Logic L Outwitting an incomputable problem, 187 Premise, 10, 33, 56 Presburger, 60, 204 Padding Lemma, 143, 145–146, 158 Presburger Arithmetic, 60, 204 generalized, 230 Principia Mathematica, 25, 29, 63, 128 Paradox, 8, 17, 21, 27, 38, 63 Completeness Problem of, 25, 29 Burali-Forti’s, 17, 18 Consistency Problem of, 25, 29 Cantor’s, 17 decidability of, 127 fear of, 18 imperfections in, 25 Russell’s, 18, 24, 45 importance of, 25 safety from, 63 Priority Method, 269, 274–280 Parallel Postulate, see Geometry candidate, 275 Parameter, 147, 153 Finite-Injury Priority Method, 277 Parametrization Theorem, 143, 147–149, 153, Infinite-Injury Priority Method, 277 154, 158, 247 priority, 276 Partial computable (p.c.) function, 95, 126, priority , 279 138, 146, 147, 205–208 requirement, 275 on a set, 96, 126 fulfilling, 275 property of, 205–208 injury, 277 Partial recursive function, 74, 153 receiving attention, 275 also see Recursive function, 93 Problem kinds, 162, 165, 210 relative to a set, 231 computational, see Computational problem Pascal, 6 computable, see Computable problem Peano, 12, 23, 24, 39 incomputable, see Incomputable problem Peano Arithmetic PA, 40, see Formal counting, see Counting problem Arithmetic A decision Peano’s decidable, see Decidable problem axioms of arithmetic, 12, 40 semi-decidable, see Semi-decidable Formulario Mathematico, 24 problem symbolic language, 24 undecidable, see Undecidable problem Peirce, 23 decision (=yes/no), see Decision problem Penrose, 118 general Platonic view, 10, 20 solvable, 165 Post, 51, 82–84, 87, 88, 127, 167, 220, 231 unsolvable, 165 Index 327

generating (=enumeration), 162 Quantum non-computational, 161 algorithm, 70 search, see Search problem phenomena, 70 Procedure, 156–158 theory, 88 activation, 157 activation record, 157 Rado,´ 174 actual parameter, 157 RAM, 89, 122–124, 138 addressability information, 157 accumulator, 122 local variables, 157 equivalence with Turing machine, 122, 124 register contents, 157 halting, 123 return address, 157 input memory, 122 return-values, 157 instructions, 123 call, 157 main memory, 122 callee, 157 output memory, 122 caller, 157 processor, 122 recursive, 157 program, 122 runtime algorithms and data structures, 157 program counter, 122 stack, 157 random access to data, 122 Processor registers, 122 capability starting, 123 limited, 70 von Neumann’s architecture, 122 unlimited, 70 Randomness intuitively, 4, 70 objective, 70 Production, 82, 84 subjective, 70 Program, 101 , 93, 129, 130 problems about, 175 RASP, 89 recursive, 153, 154 Real numbers R with procedures, see Procedure c, the cardinality of R, 16, 58, 172, 193 Proof, 10 completenes of R,48 constructive, 172 Rechow, 122 formal (=derivation), 27, 34, 51, 55 Recipe, 3, 4, 70, 71, 77 informal, 10, 28 Recognition of consistency of a theory, 60 of arithmetical theorems, 204 absolute, 60 of mathematical truths, 69 relative, 60 of sets, 130–133 Property Recognizer, 131, 132, 138, 144, 147, 165, 203, decidable and undecidable, 205–210 248 intrinsic, 205–210 Recursion (=self-reference), 142, 143, 149, of abstract computing machines, 209 161 of algorithms, 194–195 in a general-purpose computer, 156–158 of c.e. sets, 205, 208–209 in a partial recursive function, 73–75 of index sets, 205–207 in definition of a Turing program, 153–154, of p.c. functions, 205–208 158 of sets, 137 in execution of a Turing program, 154–156, of Turing programs (machines), 209 158 trivial and non-trivial, 205–210 primitive, 73–75 , 10, 127 Recursion Theorem, 143, 149–158, 204, 212 Propositional Calculus P, 23, 50, 51, 60, 127, Recursive function, 73–75, 90, 91, 149, 193 297 Ackermann, 73, 99 Putnam, 180 initial, 73–75 projection π, 73–75 Quantification symbol successor σ, 73–75 existential ∃, 32, 46 zero ζ, 73–75 universal ∀, 32, 46 partial, 74 328 Index

primitive, 73 Semi-decidable problem, 165, 199, 203 rule of construction Semi-decidable set, 132, 138, 143–144, 147, μ-operation, 74–75, 90 158, 165, 199, 205, 208–209 composition, 73–75 existence of undecidable one, 169 primitive recursion, 73–75, 153 is computably enumerable (c.e.) set, 136 Recursive set, see Computable set property of, 205, 208–209 Recursively enumerable (r.e.) set, see Semi-Thue system, 181 Computably enumerable (c.e.) set Sentence, 32 Reduction, 196–204, 271–273 Set of a computational problem, 196–197 m-complete, 213 additional conditions, 197 1-complete, 213 basic conditions, 196 arithmetical, 284 informally, 197 Burali-Forti’s paradoxical Ω, 18, 20, 24 of a decision problem c.e., see Computably enumerable (c.e.) set to checking the (non)triviality, 207 Cantor’s definition of, 13, 20 to the set-membership problem, 164 Cantor’s paradoxical U, 18, 20, 24, 42 of a decision problem or a set complement, 169 1-reduction ≤1, 201–204 computable, see Decidable set T-reduction ≤T , 235–237 creative, 273 m-reduction ≤m, 197–201 decidable, see Decidable set btt-reduction ≤btt , 273 equinumerous, 15 tt-reduction ≤tt, 273 homeomorphic, 183 resource-bounded, 197 hyper-simple, 273 strong, 197, 273 hyperhyper-simple, 273 Register machine, 89 incomputable, see Undecidable set Rice, 205 inductive, 45 Rice’s Theorem, 204–210 infinite Riemann, 12 as actuality, 21 Robinson, 122, 180 as potentiality, 21 Rogozhin, 118 intersection, 144, 158 Rosser, 60, 78–79, 87 linearly ordered, 18 Rule locally Euclidean, 183 of construction, 24, 28, 32, 35 normal, 127–129 of inference, 23, 28, 33, 35, 52, 56 property of, 137 Generalization, 33 recognized with oracle, see Oracle-decidable Modus Ponens, 33 set Russell, 18, 24, 39, 42 Russell’s paradoxical R, 18, 20, 24, 42 Russell’s semi-decidable, see Semi-decidable set Paradox, 18, 24, 45 similar, 17, 18 Theory of Types, 24 simple, 272 undecidable, see Undecidable set s-m-n Theorem, 147–149, 158 union, 144, 158 Sacks, 266, 279 well-ordered, 17, 62 Satisfiability (of formulas), 37, 184 Set generation, 127–130, 133–135 Scenario, see Turing program Set recognition, 130–135, 164 Schickard, 6 Set theory Schrodinger,¨ 88 axiomatic Search problem, 162, 204, 210 NBG, 43, 45, 54 Second Incompleteness Theorem, 60–62 ZFC, 42, 44, 54 Self-reference, see Self-reference or Recursion Cantor’s, 13, 42 Semantic Completeness, 63 as a foundation of all mathematics, 15 Semantic Completeness Problem, 52 Shannon, 118 Semantics, 26, 28 Shepherdson, 122 intended, 35 Shortlex order, 91 Index 329

Skolem, 42 of the second order, 46 Soare, 95 paraconsistent, 62 Space, see Abstract computing machine recursively axiomatizable, 56 Spector, Clifford, 266 semantically complete, 52 Statement, 10, 23, 37, 203 semantically incomplete, 57, 63 metamathematical, 59 sound, 52 Stopper, 174 syntactically complete, 50, 56 Storage Thue, 181 capacity system, 181 limited, 70 Tiling unlimited, 70 a path, 185 intuitively, 70 a polygon, 185 Sturgis, 122 Time, see Abstract computing machine Symbol, 24, 27 TM, see Turing machine constant, 32 Topological , 183 equality, 32 Torsion group, 48 function, 32 Total extension, see Function function-variable, 32, 46 TP, see Turing program individual-variable, 32 Transfinite Induction, 54, 61, 62 , 32 Truth-table, 127 mechanical manipulation, 24, 34 Truth-value, 21 proper, 40 dubious, 21 puctuation mark, 32 of a formula, 37, 52 quantifier, 32 Turing, 80–82, 87, 107, 115, 117, 161, 166, relation, 32 202, 220 relation-variable, 32, 46 Turing degree, 240–243, 245–252 Syntactic Completeness Problem, 50 c.e., see c.e. degree Syntactic object, 59 greatest lower bound, 262–263 Syntactic relation, 59 incomparable, 259–261 Syntax, 26 intermediate, 263–264 System least, 262 canonical, 127 least upper bound, 262 normal, 127 minimal, 266–267 of equations, 76, 129 upper and lower cone of, 264–265 Turing jump Term, 32 of a set, 246–252 Terminology (old vs. new), 142 of a Turing degree, 255–267 Theorem Turing machine, x, 97, 101–138, 161 of a theory, 10, 27, 34, 51 accepting a word, 130 Theory as a computer of a function, 126, 138 ω-consistent, 60 as a decider of a set, see Decider axioms of, 10 as a generator of a set, see Generator basic notions of, 10 as a recognizer of a set, see Recognizer cognitive value of, 18, 50, 52 as an acceptor of a set, 131 complete, 39 basic model, 80–82, 101–106, 138 , 44 cell, 81, 102 consistent, 38, 50, 56, 57, 60 computation on, 81, 103, 104 decidable, 51, 56, 127, 204 control unit, 81, 103 development of, 10 empty, 116 in a formal axiomatic system, 27, 33, 35 final state, 103 essentially incomplete, 59 generalization, see Turing machine initial, 10, 33 variants model, 46 halting, 103, 104, 166 of the first order, 40 initial configuration, 104 330 Index

initial state, 103 Turing program, 147 input alphabet Σ, 81, 103, 119 behavior input word, 81, 103 global, 146, 147, 152, 209 instruction, 103, 104 local, 146, 147, 152, 209 internal configuration, 104 coding of TPs, 115, 147 memorizing, 82 enumeration of TPs, 115 non-final state, 103 equivalent, 151–153, 211 starting, 103 index of, 115, 145 state, 81, 103 length, 210, 211 tape, 81, 102 nondeterministic, 108 tape alphabet Γ , 102 decision , 113 tape symbol, 102 indeterminacy of, 112 transition function, 103 instruction indeterminacy, 112 Turing program, 81, 103 scenario of execution, 112 window, 81, 103 property of, 209 Busy Beaver, 174 recursive definition, 153, 158 coding of TMs, 115, 147, 168, 246 recursive execution, 154, 158 enumeration of TMs, 115 activation, 154 equivalence with RAM, 124 activation record, 155 generalized models, 101, 107 actual parameter, 154 external configuration, 107 callee, 155 Finite storage TM, 107 caller, 155 Multi-tape TM, 108 stack, 156 Multi-track TM, 107 shortest equivalent, 210 Multidimensional TM, 108 transformation of, 151, 153 Nondeterministic TM, 108 , 235–237 Two-way unbounded TM, 107 properties of, 237–239 use of, 114 Turing Thesis, 88 index of, 115, 145 Turing-computable function, 81, 87 not recognizing a word, 130 proper function of, 125, 146 Undecidability proving, see Incomputability proper set of, 130, 169, 202 proving reduced model, 113 Undecidable formula, 69, 90 redundant instruction, 145 Undecidable problem, 165, 197–210, 245–252 rejecting a word, 130 examples, see Incomputable problem shrewd, 168, 194 existence of, 166 simulation with the basic model, 109 Halting Problem, 166 of Finite storage TM, 109 Undecidable set, 132, 138, 165, 199, 206, 207 of Multi-tape TM, 110 existence of, 168 of Multi-track TM, 110 Universal language K0, 167, 169 of Multidimensional TM, 111 Universe, 131 ∗ of Nondeterministic TM, 112 standard Σ and N, 136, 170 of Two-way unbounded TM, 110 simulation with the reduced model Validity (of formulas), 37, 184 of basic model, 114 Variable stopper, 174 bound, 32, 77–79 universal, 115–124, 138, 169 free, 32, 37, 77–79 existence of, 117–118 quantified, 24 small, 118 von Neumann, 15, 43, 120 use of a TM, 125–133 von Neumann’s architecture, 120 for function computation, 125 accumulator, 120 for set generation, 127 main memory, 120 for set recognition, 130 memory address, 120 variants, 115, 138 memory location, 120 Index 331

processor, 120 Whitehead, 24, 25, 39 program, 120 Word, 27, 32 program counter, 120 World of infinities, 17 random access to data, 120 registers, 120 Yates, 273, 279 Yes/No problem, see Decision problem Wang, 122 Wave mechanics, 88 Zermelo, 42