bounded arithmetic Yoriyuki Yamagata At Instite of Mathematics, Czech Academy of Science, 23 Mar, 2020 National Institute of Advanced Industrial Science and Technology (AIST), Japan collaborated work with Arnold Beckmann 1
approach Find theories T1 , T2 such that • S2 Con(T1 ) • S1 2 Con(T2 ) and • T1 ⊆ T2 (as sets of theorems) and reduce the gap between T1 and T2 We concentrate to subsystems T1 and T2 of bounded arithmetic 5
2 ) (P´ udlak, 1990) • S2 BdCon(S−1 2 ) (Takeuti, 1990, Buss and Ignjatovi´ c, 1996) These negative results leaded a conjecture by Takeuti Conjecture (Takeuti, 1991) S2 Con(S−∞ 2 ) where S−∞ 2 is a theory of closed terms of bounded arithmetics with suitable axioms of recursive deﬁnitions 6
was disproved Theorem ([Beckmann, 2002]) Consider an equational theory ET without the substitution rule. If its axioms are “nice”, S1 2 Con(ET) Example (Examples of ET) • S−∞ 2 • PV without induction and substitution (=: PV−−) 7
Con(PV− p +BASICe) where • PV− p = PV −induction + propositional logic • BASICe = BASIC + additional ﬁnitly many axioms PV and S1 2 have the same strength on Πb 1 -formula with a suitable translation 8
algebra A whose constructors of arity ≤ 1. Let C(n) be the set of n-ary constructors and C their union Example • Unary natural numbers 0, S • Binary strings , 1, 0 Convention We omit the parenthesis S(S(· · · 0)) and write S · · · S0. Same for binary strings 11
a sequence of ﬁnite alphabets • xn = [‘x’, n] • c = ‘c’, c ∈ A • f = [ tc1 , . . . , tcm , n #, . . . , #] where A = {c1 , . . . , cm }, n the arity of f and # the “ﬁller” symbol • size(S): the number of primitive symbols in S • nodes(S): the number of nodes in S if S is a sequence, tree or graph Then ar(f ) ≤ size(f ) holds 16
form t, ρ ↓ v where t is term, ρ an environment mapping FV(t) to A∗ and approximate value v ∈ A∗ of t Deﬁne B(ρ) = max x∈dom(ρ) nodes(ρ(x)) B( t, ρ ↓ v) = B(ρ) M( t, ρ ↓ v) = size(t) 19
of an evaluation using computation rules • Conclusions α are sinks of computation as a graph • If α are conclusions of a computation σ, write σ α • M(σ): the maximal M(α) of all evaluations α in σ 20
σ0 of t, ρ ↓ c1 z is obtained and f (c1 x, x) = tc1 (x, f (x, x), x) is an axiom (which is uniquely determined), we can do the computation tc1 (x, y, x), ν ↓ v f (x, x), ξ ↓ w t, ρ ↓ c1 z t1 , ρ ↓ z1 · · · f (t, t), ρ ↓ v∗ where ξ(xi ) = zi , ξ(x) = z, ν = ξ[y → w] and v v∗ 22
∩ dom(ρ2 ), ρ1 (x) ρ2 (x) Lemma (S1 2 ) If ρ1 ρ2 and there are computations for t, ρ1 ↓ v1 and t, ρ2 ↓ v2 , then v1 v2 Corollary (S1 2 ) v, ρ ↓ v is never derived if v = v ∈ A 24
size(rk (#, . . . , #)). Since the claim is Πb 2 , the proof requires Σb 2 − LIND Assume the derivation of r1 (u1 , . . . , un ), ρ ↓ z1 has a form γ s1 (u), ρ ↓ p1 , . . . , sd (u), ρ ↓ pd f (s1 (u), . . . , sd (u)), ρ ↓ z1 where γ does not contain u. Make s1 (u), ρ ↓ p1 , . . . conclusions by duplicating them and obtain κ0 . 27
two condition of induction hypothesis nodes(κ0 ) ≤ nodes(κ) + d ≤ U − d + d a=1 size(sa (u)) + l k=2 size(rk (#, . . . , #)) ≤ U − l k=1 size(rk (#, . . . , #)) Further, κ0 contains the same set of evaluations to κ, wi zi for any ui , ρ ↓ zi in κ0 28
tree-like PET proof of t = u. Then, for any computation σ of an evaluation t, ρ ↓ v, there is a computation τ of the evaluation u, ρ ↓ v s.t. nodes(τ) ≤ nodes(σ) + size(π) Corollary (S2 2 ) PET never proves c1 = c2 if c1 = c2 32
2 ) Fix an integer U. For any 1. A tree-like PET proof χ of t = u, size(χ) ≤ U/2 2. environment ρ and evaluations α: B(ρ) ≤ (U − size(χ))2 3. evaluations α: M(α) ≤ U − size(χ) 4. computation σ t, ρ ↓ v, α: nodes(σ) ≤ U − size(χ) there is a computation τ u, ρ ↓ v, α s.t. nodes(τ) ≤ nodes(σ) + size(χ) 33
S2 2 ) The same as the previous claim but for the computation σ u, ρ ↓ v, α Proof. We prove both claims simultaneously by induction on χ. The proof requires Σb 2 -PIND We consider the cases: • t = u is a recursive axiom • the last rule of χ is the substitution rule 34
ued) Make tc1 (x, y, x), ν ↓ v a conclusion. Since ν = ξ[y → w] and f (x, x), ξ ↓ w is contained in σ, apply the substitution lemma II. τ0 tc1 (x, f (x, x), x), ρ ↓ v, α Can replace ν to ρ because y disappears τ tc1 (x, f (x, x), x), ρ ↓ v, α nodes(τ) ≤ nodes(σ) + size(tc1 (#, #, #)) ≤ nodes(σ) + size(χ) 36
tc1 (x, f (x, x), x), ρ ↓ v, α By the substitution lemma I, there is w s.t. τ0 tc1 (x, y, x), ρ[y → w] ↓ v, f (x, x), ρ ↓ w, α where ξ = ρ[y → w]. Let z = ρ(x), zi = ρ(xi ). τ is obtained by adding τ0 the inference tc1 (x, y, x), ξ ↓ v f (x, x), ρ ↓ w x, ρ ↓ z x1 , ρ ↓ z1 · · · f (c1 x, x), ρ ↓ v∗ Easy to check nodes(τ) ≤ nodes(σ) + size(χ) 37
in bounded arithmetic. Journal of Symbolic Logic, 67(1):279–296. Buss, S. R. and Ignjatovi´ c, A. (1995). Unprovability of consistency statements in fragments of bounded arithmetic. Annals of pure and applied Logic, 74:221–244. 42