Decidability

PRIMITIVE RECURSIVE INCOMPLETENESS DECIDABILITY / INCOMPLETENESS SEP Erik A. Andrejko University of Wisconsin - Madison Summer 2007 ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS PRIMITIVE RECURSIVE FUNCTIONS DEFINITION Primitive recursive functions C contains The successor functions, λ x[x + 1] ∈ C . 1 The constant functions, λ x1 , · · · , xn [m] ∈ C are primitive 2 recursive for 0 ≤ n, m. The identity functions, or projections, λ x1 , · · · , xn [xi ] ∈ C for 3 1 ≤ n and 1 ≤ i ≤ n. (Composition) If g1 , g2 , · · · , gm , h ∈ C then 4 f (x) = h(g1 (x), · · · , gm (x)) ∈ C . ¯ ¯ ¯ (Primitive Recursion) If g, h ∈ C and n ≥ 1 then f ∈ C where 5 f (0, x2 , · · · , xn ) = g(x2 , · · · , xn ) f (x1 + 1, x2 , · · · , xn ) = h(x1 , f (x1 , x2 , · · · , xn ), x2 , · · · , xn ) ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS PRIMITIVE RECURSION FUNCTIONS FACT There exists a ‘computable’ non primitive recursive function. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS PARTIAL RECURSIVE FUNCTIONS DEFINITION The class of partial recursive contains C and (Unbounded Search) If θ (x1 , · · · , xn , y ) is a partial recursive 1 function then ψ(x1 , · · · , xn ) = µy[θ (x1 , · · · , xn , y ) ↓= 0 and ∀z ≤ y [θ (x1 , · · · , xn , z) ↓]] ψ is partial recursive. θ (x1 , · · · , xn , y ) ↓= 0 means that θ (x1 , · · · , xn , y ) deﬁned and equals 0 µyθ (x1 , · · · , xn , y ) ↓= 0 least y such that θ (x1 , · · · , xn , y ) ↓= 0 ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS COMPUTABLE DEFINITION A function is computable ⇐⇒ Turing computable ⇐⇒ partial recursive ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS TURING MACHINES DEFINITION (n) Let ϕe be the partial function of n variables computed by the Turing machine Pe with code e ∈ N. DEFINITION Write ϕe,s (x) = y if w, y , e < s and y is the output of ϕe (x) in < s steps of the Turing machine Pe . ϕe,s (x) converges and we write ϕe,s (x) ↓, ϕe,s (x) diverges which is written as ϕe,s (x) ↑. FACT The set { e, x, s : ϕe,s (x) ↓} is recursive. FACT The set { e, x, y , s : ϕe,s (x) = y } is recursive. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS RECURSIVELY ENUMERABLE SETS DEFINITION A set A is called recursively enumerable if A is the domain of some partial recursive function. We = dom ϕe = {x : ϕe (x) ↓} and We,s = dom ϕe,s Let K = {x : ϕx (x) ↓} = {x : x ∈ Wx }. FACT K is r.e. FACT K is not recursive. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS INCOMPLETENESS N = (N, 0, S, <, +, ·, E) THEOREM Let A ⊆ Th(N) and suppose that the set { ϕ : ϕ ∈ A} is deﬁnable. Then there is a L -sentence σ such that N |= σ and A σ. COROLLARY The set { τ : N |= τ} is not deﬁnable in N. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS PEANO ARITHMETIC ∀x [Sx = 0] ∀x∀y [Sx = Sy =⇒ x = y ] ∀x∀y [x < Sy ⇐⇒ x ≤ y ] ∀x [x < 0] ∀x [x + 0 = x] ∀x∀y [x + Sy = S(x + y )] ∀x [x · 0 = 0] ∀x∀y [x · Sy = x · y + x] ∀x [xE0 = S0] ∀x∀y [xESy = xEy · x] induction axioms for each well formed formula ϕ ϕ(0) ∧ ∀x [ϕ(x) =⇒ ϕ(Sx)] =⇒ ∀xϕ(x) ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS FACT N |= PA. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS INCOMPLETENESS THEOREM (Church’s Thesis) A relation is decidable iff it is deﬁnable in N. COROLLARY Th(N) is not recursive. THEOREM (Gödel Incompleteness Theorem) If A ⊆ Th(N) and A is recursive then A is not a complete theory. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

PRIMITIVE RECURSIVE INCOMPLETENESS UNDECIDABLE THEORIES FACT The theory Th(N, +, ·, S, <) is undecidable. FACT The theory Th(N, +, ·) is undecidable. ERIK A. ANDREJKO DECIDABILITY / INCOMPLETENESS

Decidability

by andrejko on Aug 06, 2007

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