Existence and Consistency in Bounded Arithmetic

Existence and Consistency in Bounded Arithmetic Yoriyuki Yamagata National Institute of Advanced Science and Technology (AIST) Kusatsu, August 30, 2011

Outline Self introduction Summary Theories of PV and EA, EA Ramified recurrence Cook Urquhart s PV EA and EA S 2 2 Con EA Valuation and truth Soundness of EA PV Con EA? EA interprets PV EA interprets EA Open problem Conclusion and future work

Self introduction - Data Name Yoriyuki Yamagata title Researcher Web page http://staff.aist.go.jp/yoriyuki.yamagata/ Personal web page http://sites.google.com/site/yoriyukiy/ email yoriyuki.yamagaga@aist.go.jp Twitter yoriyuki

Self introduction - History 2005 Researcher, AIST Verification of software update system Formal system for LCM 2004 Research staff, AIST Development of software update system 2002 PhD(Mathematical Science), University of Tokyo Computational Interpretation of classical logic

Self introduction - Publications 2008 A sequent calculus for limit computable mathematics (with Stefano Berardi, APAL) 2004 Strong normalization of second-order symmetric lambda-mu calculus (Information and Computation) 2002 Strong normalization of a symmetric lambda calculus for second-order classical logic (AML) 2001 Strong normalization of second-order symmetric lambda-mu calculus (TACS2001)

Self introduction - Software Camomile an open source Unicode library for OCaml programming language. Shinji/Mana an HMM-based open-source kana-kanji conversion engine. HatenaTail a web site which gives recommended web pages to the users based on their Hatena Bookmark histories.

Summary - Results Defined EA such that S 2 2 Con EA S 1 2 Con EA EA p( n ) Con EA (n) where EA quantifier-free arithmetic with free logic, without induction Con EA consistency of tree-like EA -proof Con EA (n) consistency of tree-like EA -proofs which are smaller than n (as Gödel numbers) p(x) a polynomial bound of the size of dag-like proof of Con EA (n).

Ramified recurrence Let N 0, N 1 be two copies of N. f : N i0 N in N j is a ramified primitive recursive function if either 1. f is a constant 0 or projection 2. f is obtained from ramified primitive recursive functions by composition 3. f is defined by ramified recurrence f (0, a) = g(a) f (s 0 a, a) = h 0 (a, f (a, a), a) f (s 1 a, a) = h 1 (a, f (a, a), a) where j > i 0 or f (a, a) does not appear in h 0, h 1.

Ramified recurrence Fact (Leivant 1994) The class of ramified primitive recursive functions is exactly same to P. D : defining equations of polynomial-time functions using ramified recurrence.

Cook Urquhart s PV Axioms Equality rules

Cook Urquhart s PV Axioms Defining axioms for all polynomial-time functions

Cook Urquhart s PV Axioms Finite number of valid atomic rules A 1 A n A In particular, under the assumption of P = NP

Cook Urquhart s PV Axioms Finite number of valid atomic rules A 1 A n A In particular, under the assumption of P = NP f (a) t(a) A(f (a), a) A(b, a) where A(x, a) : NP-complete predicate

Cook Urquhart s PV Inference Propositional logic

Cook Urquhart s PV Inference Substitution rule. A(a) A(t) where a does not occur freely in the assumptions of A(a).

Cook Urquhart s PV Inference PIND A(0) [A( a 2 )]. A(a) A(t)

EA and EA - Language

EA and EA - Language Function symbols Function symbols for all polynomial-time functions

EA and EA - Language Function symbols Function symbols for all polynomial-time functions Predicate symbols E, =,

EA and EA - Language Function symbols Function symbols for all polynomial-time functions Predicate symbols E, =, Atomic formula t = u, t u

EA and EA - Language Function symbols Function symbols for all polynomial-time functions Predicate symbols E, =, Atomic formula t = u, t u E-form Et

EA and EA - Language Function symbols Function symbols for all polynomial-time functions Predicate symbols E, =, Atomic formula t = u, t u E-form Et Formula where a is an atomic formula. A ::= a a A A A A Et

EA and EA - Axioms E-axioms E0 Et Es j t j = 0, 1 Ef (t 1,..., t n ) Et j j = 1,..., n p(t 1, t 2 ) Ea j j = 1, 2; p {=, } p(t 1, t 2 ) Et j j = 1, 2; p {=, }

EA and EA - Axioms Equality axioms Et t = t t = s s = t t = s, s = r t = r t = s s i t = s i s i = 0, 1 Et(t), t = s t(t) = t(s) t 1 = s 1, t 2 = s 2, p(t 1, t 2 ) p(s 1, s 2 ) p {=, }

EA and EA - Axioms Defining equations For each f (p(a), a) = t D, we have as an axiom. Et f (p(a), a) = t

EA and EA - Axioms Auxiliary axioms For finite number of valid atomic rules we have A 1 A n A A 1 A n Et A as an axiom, where t are all terms which appear in A, A 1,..., A n.

EA and EA - inference -intro rule [A]. Et A where t are all terms which appear in A

EA and EA - inference excluded middle Et [ A]. A where t are all terms which appear in A

EA and EA - inference substitution. Et A(a) A(t)

EA and EA - inference data elimination [A(a)] [A(a)] Et A(0). A(s 0 a). A(s 1 a) A(t) EA is a system which have the full data elimination rule.

EA and EA - inference degenerated data elimination Et A(0). A(s 0 a). A(s 1 a) A(t) in there is no assumption on A(a). EA is a system which only have the degenerated data elimination rule.

Valuation and truth - Notation e t Code of function a t(a) {e}(u, a) Result of running code e with input a until u clock. If the program is not terminated, it returns undefined Environment ρ Finite map from variables to integer.

Valuation and truth - Definition bounded valuation of term t val( t, u, ρ) := {e t }(u, ρ(a)) Convergence of term t val( t, u, ρ) def val( t, u, ρ(a)) undefined

Valuation and truth - Definition Existence T ( Et, u, ρ) def val( t, u, ρ(a)) Equality T ( t 1 = t 2, u, ρ) def val( t 1, u, ρ(a)) val( t 2, u, ρ(a)) val( t 1, u, ρ(a)) = val( t 2, u, ρ(a))

Valuation and truth - Definition Existence T ( Et, u, ρ) def val( t, u, ρ(a)) Inequality T ( t 1 t 2, u, ρ) def val( t 1, u, a) val( t 2, u, ρ(a)) val( t 1, u, ρ(a)) val( t 2, u, ρ(a))

Soundness of EA - Assumption u Upper bound π tree-like EA -proof π 1 sub-proof of π A 1,..., A n Occurrences of assumptions of π 1 A Conclusion of π 1 Integer assigned to assumption u j ρ Environment

Soundness of EA - Statement Proposition (Soundness of EA ) T ( A 1, u 1, ρ) T ( A n, u n, ρ) u 1 u n π 1 u T ( A, u 1 u n π 1, ρ)

Soundness of EA - Proof Axiom. π 0 t = s s 0 t = s 0 s

Soundness of EA - Proof Axiom. π 0 t = s s 0 t = s 0 s ũ := u 1 u n π 1 ũ 0 := u 1 u n π 0 C Constant (throughout this presentation)

Soundness of EA - Proof Axiom. π 0 t = s s 0 t = s 0 s 1. Since T ( t = s, ũ 0, ρ), v t = val( t, ũ 0, ρ), v s = val( s, ũ 0, ρ) and v t = v s

Soundness of EA - Proof Axiom. π 0 t = s s 0 t = s 0 s 1. Since T ( t = s, ũ 0, ρ), v t = val( t, ũ 0, ρ), v s = val( s, ũ 0, ρ) and v t = v s 2. Therefore s 0 v t = val( s 0 t, ũ 0 C, ρ)

Soundness of EA - Proof Axiom. π 0 t = s s 0 t = s 0 s 1. Since T ( t = s, ũ 0, ρ), v t = val( t, ũ 0, ρ), v s = val( s, ũ 0, ρ) and v t = v s 2. Therefore s 0 v t = val( s 0 t, ũ 0 C, ρ) 3. Similarly s 0 v s = val( s 0 s, ũ 0 C, ρ) 4. Then s 0 v t = s 0 v s

Soundness of EA - Proof Data elimination [A E ] [A ɛ ] [A s0 ] [A s1 ]. π E Et. π ɛ A(0). π s0 A(s 0 a) A(t). π s1 A(s 1 a)

Soundness of EA - Proof Data elimination [A E ] [A ɛ ] [A s0 ] [A s1 ]. π E Et. π ɛ A(0). π s0 A(s 0 a) A(t). π s1 A(s 1 a) ũ E, ũ ɛ, ũ s0, ũ s1 Concatenation of integers assigned to A E, A ɛ, A s0, A s1

Soundness of EA - Proof Data elimination [A E ] [A ɛ ] [A s0 ] [A s1 ]. π E Et. π ɛ A(0). π s0 A(s 0 a) A(t). π s1 A(s 1 a) ũ E, ũ ɛ, ũ s0, ũ s1 Concatenation of integers assigned to A E, A ɛ, A s0, A s1 1. v t := val( t, ũ E π E, ρ) 2. Assume v t = s 0 v

Soundness of EA - Consistency Theorem (Consistency) S 2 2 Con EA

Soundness of EA - Consistency Theorem (Consistency) S 2 2 Con EA Proof. S 2 2 Soundness Proposition

EA interprets PV

EA interprets PV 1. EA is almost equal to PV

EA interprets PV 1. EA is almost equal to PV 2. Only difference is EA requires to prove Et for each t used in the proof

EA interprets PV 1. EA is almost equal to PV 2. Only difference is EA requires to prove Et for each t used in the proof 3. Therefore we prove Et for every term t

EA interprets PV 1. EA is almost equal to PV 2. Only difference is EA requires to prove Et for each t used in the proof 3. Therefore we prove Et for every term t 4. But this fact can be derived by the totality Ea f (a) of every function f.

EA interprets PV - Totality Since f is defined by ramified recurrence, we can assume where i 0,..., i n, j = 0, 1 Claim f : N i0 N in N j EA Ea Ef (a) Moreover, if i k = 0, in the proof Ea k is not used as assumptions of Et in the non-degenerated data elimination. We call i k the tier of a k.

EA interprets PV - Totality Proof. By induction on the definition of f. We only consider of degenerated recurrence

EA interprets PV - Totality Proof. Since f is defined by recurrence, f (0, a) = g(a) f (s 0 a, a) = g 0 (f (a, a), a, a) f (s 1 a, a) = g 1 (f (a, a), a, a) where tier of a is 1 and tier of f is 0.

EA interprets PV - Totality Proof. We have the following axioms. Eg(a) f (0, a) = g(a) Eg 0 (f (a, a), a, a) f (s 0 a, a) = g 0 (f (a, a), a, a) Eg 1 (f (a, a), a, a) f (s 1 a, a) = g 1 (f (a, a), a, a).

EA interprets PV - Totality Proof. Ef (a, a) Ef (a, a). Eg(a). Eg 0 (f (a, a), a, a). Eg 1 (f (a, a), a, a) Ea Ef (0, a) Ef (s 0 a, a) Ef (s 1 a, a) Ef (a, a)

EA interprets PV - Totality Proof. Ef (a, a) Ef (a, a). Eg(a). Eg 0 (f (a, a), a, a). Eg 1 (f (a, a), a, a) Ea Ef (0, a) Ef (s 0 a, a) Ef (s 1 a, a) Ef (a, a) 1. Since the tier of f is 0, Ef (a, a) is not used as an assumption of the first premise Et of data elimination.

EA interprets EA Proposition EA Ea, Γ(a) (a) EA π1 a n, Γ(a) (a) where π 1 is a dag-like proof and π 1 p( n ), p : polynomial.

EA interprets EA - proof Lemma EA π Ea A(a) EA π a n t(a) n π. where t is a term which appears in π and π p( n ).

EA interprets EA - proof Proof of Lemma

EA interprets EA - proof Proof of Lemma. π E Es. π 0 A(0). π s0 A(s 0 a) A(s). π s1 A(s 1 a) and t t(s) is a term of A

EA interprets EA - proof Proof of Lemma. π E Es. π 0 A(0). π s0 A(s 0 a) A(s). π s1 A(s 1 a) and t t(s) is a term of A By induction hypothesis t(0) n 0 π 0

EA interprets EA - proof Proof of Lemma. π E Es. π 0 A(0). π s0 A(s 0 a) A(s). π s1 A(s 1 a) and t t(s) is a term of A By induction hypothesis t(0) n 0 π 0 s n E π E

EA interprets EA - proof Proof of Lemma. π E Es. π 0 A(0). π s0 A(s 0 a) A(s). π s1 A(s 1 a) and t t(s) is a term of A By induction hypothesis t(0) n 0 π 0 s 2 n E π E

EA interprets EA - proof Proof of Lemma. π E Es. π 0 A(0). π s0 A(s 0 a) A(s). π s1 A(s 1 a) and t t(s) is a term of A By induction hypothesis t(0) n 0 π 0 t(s 0 s 2 ) n s0 n E π E π s0

EA interprets EA - proof Proof of Lemma. π E Es. π 0 A(0). π s0 A(s 0 a) A(s). π s1 A(s 1 a) and t t(s) is a term of A By induction hypothesis t(0) n 0 π 0 t(s 0 s 2 ) n s0 n E π E π s0 t(s 1 s 2 ) n s1 n E π E π s1.

EA interprets EA - proof Proof of Lemma By induction hypothesis t(0) n 0 π 0 Combining these proofs with and equality axioms, we obtain t(s 0 s 2 ) n s0 n E π E π s0 t(s 1 s 2 ) n s1 n E π E π s1. s = 0 s = s 0 s 2 s = s 1 s 2

EA interprets EA - proof Proof of Lemma t(s) n π Note that we are constructing a dag-like proof

EA interprets EA - proof Proof of Proposition We construct the EA -proof from the leaf of EA-proof to the conclusion.

EA interprets EA - proof Proof of Proposition Non-degenerate data elimination. σ E Et. σ ɛ A(0) [A(a)]. σ s0 A(s 0 a) A(t) [A(a)]. σ s1 A(s 1 a)

EA interprets EA - proof Proof of Proposition Non-degenerate data elimination. σ E Et. σ ɛ A(0) [A(a)]. σ s0 A(s 0 a) A(t) [A(a)]. σ s1 A(s 1 a) We use predicativity and the previous lemma σ E t n σ E

EA interprets EA - proof Proof of Proposition

EA interprets EA - proof Proof of Proposition If n = 0, we have σ ɛ A(0)

EA interprets EA - proof Proof of Proposition If n = 0, we have σ ɛ A(0) Assume we have a proof for A( n 2 )

EA interprets EA - proof Proof of Proposition If n = 0, we have σ ɛ A(0) Assume we have a proof for A( n 2 ) By combining σ s0 [ n 2 /a] A( n 2 ) A(s 0 n 2 ) σ s1 [ n 2 /a] A( n 2 ) A(s 1 n 2 )

EA interprets EA - proof Proof of Proposition If n = 0, we have σ ɛ A(0) Assume we have a proof for A( n 2 ) By combining σ s0 [ n 2 /a] A( n 2 ) A(s 0 n 2 ) σ s1 [ n 2 /a] A( n 2 ) A(s 1 n 2 ) t = t n 2 t = s 0 n 2 t = s 1 n 2

EA interprets EA - proof Proof of Proposition If n = 0, we have σ ɛ A(0) Assume we have a proof for A( n 2 ) By combining σ s0 [ n 2 /a] A( n 2 ) A(s 0 n 2 ) σ s1 [ n 2 /a] A( n 2 ) A(s 1 n 2 ) and equational axioms t = t n 2 t = s 0 n 2 t = s 1 n 2

Open Problem Conjecture For enough large n N, there is no dag-like EA -proof such that π n Con EA (n)

Open Problem Conjecture For enough large n N, there is no dag-like EA -proof such that π n Con EA (n) Open problem Is the conjecture true?

Open problem - simple Gödel argument

Open problem - simple Gödel argument Definition (φ) EA φ(n) w n. Prf EA (w, φ(n) ) where Prf(w, φ(n) ) means there is a tree-like proof w of φ()

Open problem - simple Gödel argument Definition (φ) EA φ(n) w n. Prf EA (w, φ(n) ) where Prf(w, φ(n) ) means there is a tree-like proof w of φ() Fact There is a k such that k {}}{ Con EA ( n# #n) φ(n)

Open problem - speedup induction Fact (Buss, Ignjatović 1995) There is a fixed polynomial p such that for any φ, n, S i 2 x.φ(x) S 1 2 (Bb i ) p( n ) φ(n)

Open problem - speedup induction Fact (Buss, Ignjatović 1995) There is a fixed polynomial p such that for any φ, n, S i 2 x.φ(x) S 1 2 (Bb i ) p( n ) φ(n) Can be applied this technique to our case?

Open problem - speedup induction Reason

Open problem - speedup induction Reason k k {}}{{}}{ EA x. x# #x = x# #x

Open problem - speedup induction Reason but by soundness of EA k k {}}{{}}{ EA x. x# #x = x# #x π EA with arbitrary large k k k {}}{{}}{ n# #n = n# #n π k {}}{ n# #n

Future work

Future work Consistency of dag-like EA proof Maybe achieved by replacing with max

Future work Consistency of dag-like EA proof Maybe achieved by replacing with max Counting argument on f : n {0, 1}

Future work Consistency of dag-like EA proof Maybe achieved by replacing with max Counting argument on f : n {0, 1} 1. There are 2 n such f

Future work Consistency of dag-like EA proof Maybe achieved by replacing with max Counting argument on f : n {0, 1} 1. There are 2 n such f 2. If Ef has a polynomial (respect to n ) length proof, f has a variation of 2 p( n ) (?)

Conclusion Defined EA such that S 2 2 Con EA S 1 2 Con EA EA p( n ) Con EA (n) where EA quantifier-free arithmetic with free logic, without induction Con EA consistency of tree-like EA -proof Con EA (n) consistency of tree-like EA -proofs which are smaller than n (as Gödel numbers) p(x) a polynomial bound of the size of dag-like proof of Con EA (n).