Consistency proof of an arithmetic with substitution inside a bounded arithmetic

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1. Consistency proof of an arithmetic with substitution inside a bounded arithmetic January 12, 2017
2. Language Definition (Language L) 1. Constant: 0 2. Function symbols: S, +, ×, · 2 , | · |, # 3. Relation symbols: =, ≤ Remark 1. a 2 is the division by two 2. |a| is the length of bits of a 3. a#b = 2|a|·|b|
3. Bounded arithmetic Definition (Bounded arithmetic over L) A theory of arithmetic of which axioms consist in 1. BASIC axioms 2. induction schema only for bounded formulas Remark All quantifiers of a bounded formula must be bounded by terms of L.
4. Hierarchy of bounded formulas ∀x ≤ |t|, ∃x ≤ |t| : sharply bounded quantifiers ∀x ≤ t, ∃x ≤ t : bounded quantifiers The hierarchy of bounded formulas Πb 0 ⊂ Πb 1 ⊂ · · · ⊂ Πb i ⊂ · · · Σb 0 ⊂ Σb 1 ⊂ · · · ⊂ Σb i ⊂ · · · defined by alternation of bounded quantifiers ignoring sharply bounded quantifiers.
5. Many induction schemata For any term t and A ∈ Σb i 1. Σb i − IND: A(0) ⊃ ∀x(A(x) ⊃ A(Sx)) ⊃ A(t) 2. Σb i − PIND: A(0) ⊃ ∀x(A( x 2 ) ⊃ A(x)) ⊃ A(t) 3. Σb i − LIND: A(0) ⊃ ∀x(A(|x|) ⊃ A(S|x|)) ⊃ A(|t|) etc... Remark 1. Σb i+1 − PIND ⇒ Σb i − IND ⇒ Σb i − PIND 2. Σb i − PIND ⇔ Σb i − LIND
6. Buss’s bounded arithmetics Definition 1. Si 2 : BASIC-axioms + Σb i − PIND 2. Ti 2 : BASIC-axioms + Σb i − IND 3. S2 = i∈N Si 2 = i∈N Ti 2 Remark S1 2 ⊆ T1 2 ⊆ · · · ⊆ Si 2 ⊆ Ti 2 ⊆ · · ·
7. Si 2 and Polynomial Hierarchy (PH) T: a theory of arithmetic ⇒ PT(T): the set of provably total functions in T Fact (Buss 1988) 1. PT(S1 2 ) = {PTIME functions} 2. PT(Si 2 ) = {PTIME functions using a Σp i−1 oracle}
8. Language Definition (LPV) 1. Constant: 0 2. Function symbols: s0 , s1 , S, +, ˙ −, ×, Cond, · 2 , | · |, , # plus all PTIME-functions 3. Predicate: =
9. Axioms 1. s0 0 = 0 2. Defining axioms for primitive function symbols 3. Limited recursion on notations: f (0, x) = g(x) f (s0 x, x) = min{g0 (x, f (x, x), x)), k(s0 x, x)} f (s1 x, x) = min{g1 (x, f (x, x), x)), k(s1 x, x)} 4. Equational axioms 5. Substitution axiom t(x) = u(x) ⇒ t(s) = u(s) 6. Induction axiom
10. PV and related systems 1. PV: Full 2. PV−: PV without induction 3. PV−−: PV without induction and substitution 4. Adding p subscript: addition of propositional logic
11. Is Buss’s hierarchy strict? Conjecture S1 2 S2 2 · · · Si 2 Si+1 2 · · · Remark 1. Major open problem 2. If S1 2 = S2 , P = NP
12. Separation of Si 2 through G¨ odel sentences Fact (Buss, 1988) Incompleteness theorems hold in Si 2 , that is, Si 2 Con(Si 2 ) Question S2 Con(S1 2 )? Answer (Wilkie and Paris, 1980) S2 Con(Q)
13. More unprovability results Fact 1. Si 2 BdCon(Si 2 ) (Buss, 1988) 2. S2 BdCon(S1 2 ) (P´ udlak, 1990) 3. S2 BdCon(S−1 2 ) (Takeuti, 1990, Buss and Ignjatovi´ c, 1996) 4. S1 2 Con(PV− p +BASIC) (Buss and Ignjatovi´ c, 1996) etc... Conjecture (Takeuti, 1991) S1 2 Con(S−∞ 2 ) Answer (Beckmann, 2002) Yes!
14. Provability results Theorem (Beckmann, 2002) S1 2 Con(PV −−) Remark 1. Actually, any rewriting system with good properties is okay 2. Not many provability results on consistency are known (except 2nd order propositional logic)
15. Main result: S2 2 Con(PV −) Theorem (Yamagata, 2016 (preprint)) S2 2 Con(PV −) Remark To prove inside S1 2 could be possible, but it requires explicit construction of witness during induction on Πb 2 -formulas
16. BHK reading of equality Read . . . . π t = u as a “construction” from “computation” of t to u and vice verse, which preserves the value of a computation
17. Proof strategy 1. “computation” = derivation in big-step semantics 2. Bounds a number of steps of computations of u by that of t and vice verse 3. Bounds the size of a computation by its steps
18. Judgment t, ρ ↓ v 1. t : a term of PV 2. ρ : a sequence of substitutions 3. v : the approximated value using ∗
19. Big-step semantics t, ρ2 ↓ v x, ρ1[t/x]ρ2 ↓ v Subst t, ρ ↓ ∗ ∗ , () ↓ , ρ ↓ , ρ ↓ , () ↓ n si v∗, () ↓ si v∗ t, ρ ↓ v si t, ρ ↓ si v∗ si v, () ↓ v si v, () ↓ si v si n , () ↓ ( ti , ρ ↓ vi )i∈X m(t1, . . . , tm), ρ ↓ m v∗ i , () ↓ v∗ i ( tj , ρ ↓ vj )j∈X proji m (t1, . . . , tm), ρ ↓ v∗ i proji m g(w∗), () ↓ z h1(v1), () ↓ w1 · · · hm(vm), () ↓ wm ( ti , ρ ↓ vi )i∈X f (t1, . . . , tn), ρ ↓ z comp g (v1), () ↓ z { t, ρ ↓ } ( ti , ρ ↓ vi )i∈X f (t, t1, . . . , tn), ρ ↓ z rec- gi (v1 0 , w1, v1), () ↓ z { t, ρ ↓ si v0} f (v2 0 , v2), () ↓ w ( tj , ρ ↓ vj )j∈X f (t, t1, . . . , tn), ρ ↓ z rec-si
20. Computation Definition Computation is a DAG of which nodes are judgments and edges are inferences of big-step semantics 1. Judgments which are not used as premises, are called conclusions 2. σ t1 , ρ1 ↓ v1 , . . . computation σ, conclusions t1 , ρ1 ↓ v1 , . . . Notation 1. ||t||: Number of primitive symbols in any object t 2. |||σ|||: Number of nodes in σ
21. Main proposition Proposition Fix a large U. For any tree-like PV−-proof π . . . . π t = u and ||ρ||, ||α||, |||σ||| ≤ U − ||π|| s.t. σ t, ρ ↓ v, α ⇒ ∃τ s.t. 1. |||τ||| ≤ |||σ||| + ||π|| 2. τ u, ρ ↓ v, α
22. Proof of main proposition Induction on π. The induction formula is Πb 2 -formula with U as a parameter
23. Transformation for projection projk m (t1 , . . . , tm ) = tk v∗ i , ρ ↓ v∗ i t1 , ρ ↓ v1 · · · tm , ρ ↓ vm projn i (t1 , . . . , tn ), ρ ↓ v∗ i ⇓ ti , ρ ↓ vi
24. Transformation for projection projk m (t1 , . . . , tm ) = tk vi , ρ ↓ vi t1 , ρ ↓ ∗ · · · ti , ρ ↓ vi · · · tm , ρ ↓ ∗ projn i (t1 , . . . , tn ), ρ ↓ vi ⇑ ti , ρ ↓ vi
25. Transformation for composition f (u) = g(h1 (u), . . . , hm (u)) β g(w∗), () ↓ v γ1 h1 (z1), () ↓ w1 . . . ( ui , ρ ↓ zi )i∈X f (u), ρ ↓ v. β γ1 ( ui , ρ ↓ z1 i )i∈X h1 (u), ρ ↓ w1 . . . g(h(u)), ρ ↓ v
26. Substitution Lemma I 1. U : a large integer. 2. σ : a computation s.t. 2.1 σ t1[u/x], ρ ↓ v1, . . . , tm[u/x], ρ ↓ vm, α 2.2 |||σ||| ≤ U − ||t1[u/x]|| − · · · − ||tm[u/x]|| ⇒ ∃τ s.t. 1. τ t1 , [u/x]ρ ↓ v1 , . . . , tm , [u/x]ρ ↓ vm , α 2. |||τ||| ≤ |||σ||| + ||t1 [u/x]|| + · · · + ||tm [u/x]||
27. Substitution Lemma II 1. U : a large integer. 2. σ : a computation s.t. 2.1 σ t1, [u/x]ρ ↓ v1, . . . , tm, [u/x]ρ ↓ vm, α 2.2 |||σ||| ≤ U − ||t1[u/x]|| − · · · − ||tm[u/x]|| ⇒ ∃τ s.t. 1. τ t1 [u/x], ρ ↓ v1 , . . . , tm [u/x], ρ ↓ vm , α 2. |||τ||| ≤ |||σ||| + ||t1 [u/x]|| + · · · + ||tm [u/x]||
28. Transformation for substitution . . . . π1 t(x) = u(x) t(s) = u(s) σ t(s), ρ ↓ v, α
29. Transformation for substitution . . . . π1 t(x) = u(x) t(s) = u(s) σ0 t(x), [s/x]ρ ↓ v, α |||σ0 ||| ≤ |||σ||| + ||t(s)||
30. Transformation for substitution . . . . π1 t(x) = u(x) t(s) = u(s) τ0 u(x), [s/x]ρ ↓ v, α |||τ0 ||| ≤ |||σ||| + ||π1 || + ||t(s)||
31. Transformation for substitution . . . . π1 t(x) = u(x) t(s) = u(s) τ u(s), ρ ↓ v, α |||τ||| ≤ |||σ||| + ||π1 || + ||t(s)|| + ||u(s)||
32. Conclusion Theorem S2 2 Con(PV −)
33. Future works Question S2 Con(PV − p (d))? Question S2 Con(PV − p (d) + BASIC)? Remark The last statement may imply S1 2 S2