Some News on the Proof Complexity of Deep Inference

Transcription

1 Some News on the Proof Complexity of Deep Inference Alessio Guglielmi University of Bth nd LORIA & INRIA Nncy-Grnd Est 11 November 2009 This tlk is vilble t

2 Outline Aims of the tlk: Put some of the current deep-inference reserch in the wider context of proof complexity. Stte surprising result on cut elimintion being t most qusipolynomil in deep inference (insted of exponentil). Provide n introduction for the following tlk by Tom, who will get into some detils of qusipolynomil cut elimintion. Contents: Overview of Complexity Clsses Proof Systems Compressing Proofs Deep Inference Atomic Flows Cut Elimintion Concluding Remrks

3 Overview of (Some!) Complexity Clsses SAT /\,/\ VAL co-hp Ge9gT6 PRodt4p5 N P = clss of problems tht re verifible in polynomil time. SAT = Is propositionl formul stisfible? (Yes: here is stisfying ssignment.) co-n P = clss of problems tht re disqulifible in polynomil time. VAL = Is propositionl formul vlid? (No: here is flsifying ssignment.) P = clss of problems tht cn be solved in polynomil time. N P co-n P implies P N P.

4 Proof Systems Proof complexity = proof size. Proof system = lgorithm tht verifies proofs in polynomil time on their size. Importnt question: Wht is the reltion between size of tutologies nd size of miniml proofs?

5 inference rules. There re severl equivlent wys of presenting such system, the 3.2following Let HTis be onethe of forml the simplest, systemgiving whosesyntx xiom to schemes propositionl re: Exmple clssicl logic. of Proof System: Frege 3.2 Let HT (HT 1 be ) the forml A system (B whose A), xiom schemes re: (HT 2 ) (A (B C)) ((A B) (A C)), (HT 1 ) (HT 3 ) A (B A), ( B A) (( B A) B), (HT 2 ) Axioms: (A (B C)) ((A B) (A C)), (HT 3 ) nd whose ( B inference A) rule (( B is modus A) ponens: B), A A B nd whose Modus inference ponens, rule isor modus cut, ponens: rule: mp. B A A B mp. 3.3 HT cn be extended tob first order clssicl logic by dding the xioms Exmple: 3.3 HT cn (HT be 4 ) extended to first order x.a clssicl A[t/x], logic by dding the xioms (HT 5 ) x.(a B) (A x.b), (HT 4 ) x.a A[t/x], (HT 5 ) ;;;il where t is ny x.(a term ir(o,tq nd B) A[t/x] )) (A> x.b), stndsfortheformulobtinedfroma (.:r) by substituting x with t. Thefollowinginferencerule (generliztion) islso where t is dded: )q ny term nd A[t/x] stndsfortheformulobtinedfroma by A substituting x with t. Thefollowinginference gen rule (generliztion). islso dded: x.a A All the reltionsgen between. connectives re derivble from the xiom x.a schemes provided. In sense, they re hrd-coded inside the xioms, nd t(;;) ) ") ( : ({^> ") >q ) ((^>(o>))>(nj^) Robustness: ll Frege systems re polynomilly equivlent. All the reltions between connectives re derivble from the xiom schemes provided. In sense, they re hrd-coded inside the xioms, nd

6 Exmple of Proof System: Gentzen Sequent Clculus One xiom, mny rules. Exmple: This is specil cse of Frege, importnt becuse it dmits complete nd nlytic proof systems (i.e., cut-free proof systems, by which consistency proofs cn be obtined). Frege nd Gentzen systems re polynomilly equivlent.

7 Exmple of Proof System: Deep Inference Proofs cn be composed by the sme opertors s formule. (Atomic) Flows Exmple: s ā m [ t] [t ā] t [ t] ā s ā t f t = s ā ā ā ā f t ā ā f ā b b b m [ b] [ b] This is generlistion of Frege, which dmits complete nd locl proof systems (i.e., where steps cn be verified in constnt time). Frege nd deep-inference systems re polynomilly equivlent. The clculus Below of derivtions, structures their (CoS) (tomic) is now flows completely re shown. developed deepinference Only structurl formlism. informtion is retined in flows.

8 Proof Complexity nd the N P Vs. co-n P Problem Theorem [Cook & Reckhow(1974)]: There exists n efficient proof system iff N P = co-n P where efficient = dmitting proofs tht re verifible in polynomil time over the size of the proved formul. Is there n lwys efficient proof system? Probbly not, nd this is, obviously, hrd. Is there n optiml proof system? (in the sense tht it polynomilly simultes ll others.) We don t know, nd this is perhps fesible.

9 Proof. Consider n xsksg proof s in Definition 5.3. The sttement is n consequence of Theorem 4.11, fter observing tht there is n O(h)-length size xfrege proof Thus, n importntaquestion 1 β 1,...,A is: h β h,...,(a 1 β 1 ) (A h β h ). How cn we mke proofs smller? Compressing Proofs 1 TheseCorollry re known 5.6. mechnisms: Systems xfrege nd xsksg re p-equivlent. 1. UseWe higher now move orders to(for the substitution exmple, second rule. order propositionl, for propositionl Definition 5.7. formule). A substitution Frege (proof ) system is Frege system ugm A 2. Add the substitution: rule sub. We denote by sfrege the proof system where Aσ 3. Add derivtion Tseitin with extension: no premisses, p conclusion A (whereαp k,ndshpe is fresh tom). α i1 α ih 4. Use the sme sub-proof mny α 1,...,α i1 1, {}}{ times, vi the cut α j1 σ 1,α i1 +1,...,α i h 1, {}} rule. { α jh σ h,α ih +1,...,α k, 5. Use the sme sub-proof mny times, in dg-ness, or cocontrction. where ll the conclusions of substitution instnces α i1,...,α ih re singled {α 1,...,α i1 1 },...,α j h {α 1,...,α ih 1 Only 5 is llowed in nlytic proof systems. },ndtherestoftheproofissinfreg 4 is the We most relystudied on the following form of compression, result. nd the min topic of this tlk, Theorem together 5.8. with (Cook-Reckhow 5. nd Krjíček-Pudlák, [CR79, KP89]) Sys nd sfrege re p-equivlent.

10 Compressing Proofs 2 Some fcts: Substitution nd extension re equivlent when dded to Frege nd to deep inference (not trivil result). Any of these systems is usully clled EF (for Extended Frege) nd is considered the most interesting cndidte s optiml proof system. Deep inference hs the best representtion for EF (the equivlence between extension nd substitution becomes lmost trivil). The EF compression in deep inference leds to bureucrcy-free formlism (but this is topic for nother tlk).

11 ism in deep inference [Gug07b], nd in prticulr for its proof system SKS, introduced by Brünnler in [Brü04] nd then extensively studied [Brü03, Brü03b, Brü06, Brü06d, Proof BG04, Complexity BT01]. nd Deep Inference Our contributions fit in the following picture. nlytic CoS 1 nlytic Gentzen Brünnler 04 open Sttmn 78 CoS 2 Frege Gentzen open Cook- Reckhow 74 CoS + extension 3 Frege + extension 4 Krjíček-Pudlák 89 Cook-Reckhow 79 CoS + substitution 5 Frege + substitution The nottion indictes tht formlism polynomilly simultes formlism ; the nottion indictes tht it is known tht this does not hppen. The left side of the picture represents, in prt, the following. Anlytic Gentzen systems, i.e., Gentzen proof systems without the cut rule, cn only prove certin formule, nd which we cll Sttmn tutologies, with proofs tht grow exponentilly in the size of it hs normlistion theory the formule. On the contrry, Gentzen systems with the cut rule cn prove Sttmn tutologies by polynomilly growing proofs. So, Gentzen systems p-simulte nlytic Deep inference hs s smll proofs s the best systems (2,3,4,5,*) nd its nlytic proof systems re more powerful thn Gentzen ones (1) Dte: April 19, This reserch ws prtilly supported by EPSRC grnt EP/E042805/1 Complexity nd Non-determinismin nd Deep Inference. c ACM, This is the uthors version of the work. It is posted here by permission of ACM for your personl use. Not for redistribution. The definitive version ws published in ACM Trnsctions on Computtionl Logic 10 (2:14) 2009, pp. 1 34, cut elimintion is n O(log n), i.e., qusipolynomil (insted of exponentil). 1 (See [Jeřábek(2009), Bruscoli & Guglielmi(2009), Bruscoli et l.(2009)bruscoli, Guglielmi, Gundersen, & Prigot]).

12 System SKS is CoS proof system, defined by the following structurl i t f i t f w c i ā w c identity ā wekening contrction identity wekening contrction, Atomic rules: ā, i ā i f w c w t c f t cut cowekening cocontrction cut cowekening cocontrction nd by the following two logicl inference rules: nd by the following two logicl inference rules: α [β γ] (α β) (γ δ) α s [β γ] (αm β) (γ Liner rules: δ) s (α β) γ m [α γ] [β δ]. (α β) γ [α γ] [β switch medil δ]. switch γmedil In ddition to these rules, there is rule γ = Plus n = liner rule (ssocitivity, commuttivity,, such tht In ddition to these rules, there is rule =, such δ units). γ nd δ re op of the following equtions: tht γ nd δ re oppos Rules of the refollowing pplied nywhere equtions: inside formule. δ Negtion on toms only. α β = β α α f = α α β = α α f = α α β = β α α Cut is tomic. t = α (1) α β = β α t = α. (1) [α β] γ = α [β γ] t SKS is complete nd implictionlly complete for t =. t [α β] γ = α [β γ] t (α β) γ = α (β γ) t f = t propositionl logic. f = f (α β) γ = α (β γ) f f = f (Proof) System SKS [Brünnler & Tiu(2001)]

13 i f t (Atomic) Flows s ( ā) i = f t ā m [ t] [t ā] t [ t] ā s ā t f t = s ā ā ā ā f t ā ā f ā b b b m [ b] [ b] FIGURE 2. Exmples of derivtions in CoS nd Formlism A nottion, nd ssocited tomic flows. Below derivtions, their (tomic) flows re shown. Only structurl informtion is retined in flows. Logicl informtion is lost. the right flow cnnot: Flow size is polynomilly relted to derivtion size. φ ψ ψ

14 Flow Reductions: (Co)Wekening (1) PAOLA BRUSCOLI, ALESSIO GUGLIELMI, TOM GUNDERSEN, AND MICHEL PARIGOT Consider these flow reductions: w -c : 2 1 1,2 c -w : 1 2 1,2 w -i : 1 1 i -w : 1 1 w -w : w -c : c -w : Ech FIGURE of them 6. Wekening corresponds ndtocowekening correct derivtion tomic-flowreduction. reductions. he process termintes in liner time on the size of Π becuse ech trnsformtion elimtes some tom occurrences. The finl proof is in SKS.

15 to identify the tom occurrences in Π ssocited with ech edge of φ, nd 1 Flow Reductions: 1,2 c -w : (Co)Wekening (2) 1,2 them. We repetedly exmine ech cowekening instnce w in Π, for t 2 described in the proof of Theorem 27 re the miniml ones necof in SKS. Π cn operte on flow reductions insted thn on derivtions: it We However, it is possible to further reduce the proof ormtions is much in the esier proof nd of Theorem we get nturl, Π 27, f together syntx-independent with the one induction the proof mesures. ξ of ε Theorem 12, ll belong f to [t the t] ξ clss of wekening s ions studied in [GG08]. In the rest of (f this section, we quickly Φ er trnsformtion t) t ofbecomes the nlytic formε produced by; those reduc- φ, nd we perform one trnsformtion out of the following exhustive some 1 For Π, Φ, exmple, 1 Ψ, ξ { }nd ζ i -w : { }: 1 1 specifies tht w -w : Π Π t ξ ε ā c -w : ξ t f 2 Φ ā becomes ε Φ{ ε /t} ; ekening ζ nd cowekening tomic-flowζ reductions. {t} t Ψ 1 2 n liner time on the size of Π becuse echα trnsformtion elimences. The finl α proof is in SKS. Ψ ε

16 Flow Reductions: (Co)Contrction w -c : Consider these flow reductions: c -w : c -i : 3 3 i -c : 3 3 c -c : Figure 2: Atomic-flow reduction rules. They conserve the number nd length of pths. We would like to use the reductions in Figure 2 s rules for rewriting inside generic tomic flows. To do so, in generl, we should hve mtching upper nd lower edges in the flows It s thtprticipte good thing: thecocontrction reduction, nd the isreductions new compression in the figure clerly do so. However, mechnism we lso hve to (shring?). py ttention to polrities, not to disrupt tomic flows. In fct, consider the following exmple. Note tht they cn blow up derivtion exponentilly. Open problem: does cocontrction provide exponentil compression? Conjecture: yes. Exmple 4.2. The reduction on the left, when used inside lrger tomic flow, might crete sitution s on the right: +

17 Normlistion Overview None of these methods existed before tomic flows, none of them requires permuttions or other syntctic devices. Qusipolynomil procedures re surprising. Conjecture: polynomil normlistion is possible. (1) [Guglielmi & Gundersen(2008)]; (2,4) forthcoming; (3) [Bruscoli et l.(2009)bruscoli, Guglielmi, Gundersen, & Prigot].

18 Cut Elimintion (on Proofs) by Experiments Experiment: We do: Simple, exponentil cut elimintion; proof genertes 2 n experiments. (No use of cocontrction!)

19 Qusipolynomil Cut Elimintion by Threshold Functions 24 PAOLA BRUSCOLI, ALESSIO GUGLIELMI, TOM GUNDERSEN, AND MICHEL PARIGOT φ θ 1. φ 0 θ k φ k φ ψ k TTYJVi., YT.. θ k+1 θ n φ n φ FIGURE Only n + 1 copies of the proof 5. Atomic reflow stitched of proof in cut-free together. form. It s α complicted, Tom will explin, but note locl cocontrction (= better shring, not vilble in Gentzen). where ψ is the union of flows φ 1,...,φ n, nd where we denote by α the edges corresponding to the tom occurrences ppering in the conclusion α of Π. We then hve tht, for 0 < k < n, the flow of Φ k is φ, s in Figure 5, where ψ k k is the flow of the derivtion Ψ k. The flows of Φ 0 nd Φ n re, respectively, φ nd 0 φ. n

20 Some Comments (tht don t ll follow from wht precedes) (Exponentil) normlistion does not depend on logicl rules. It only depends on structurl informtion, i.e., geometry. Normlistion is extremely robust. Deep inference s loclity is key. Complexity-wise, deep inference is s powerful s the best formlisms, nd more powerful if nliticity is requested. Deep inference is the continution of Girrd politics with other mens. In my opinion, much of the future of structurl proof theory is in geometric methods: we hve to free ourselves from the tyrnny of syntx (so, wr to bureucrcy!).

21 References Brünnler, K., & Tiu, A. F. (2001). A locl system for clssicl logic. In R. Nieuwenhuis, & A. Voronkov (Eds.) LPAR 2001, vol of Lecture Notes in Computer Science, (pp ). Springer-Verlg. Bruscoli, P., & Guglielmi, A. (2009). On the proof complexity of deep inference. ACM Trnsctions on Computtionl Logic, 10(2), Article Bruscoli, P., Guglielmi, A., Gundersen, T., & Prigot, M. (2009). Qusipolynomil normlistion in deep inference vi tomic flows nd threshold formule. Submitted. Cook, S., & Reckhow, R. (1974). On the lengths of proofs in the propositionl clculus (preliminry version). In Proceedings of the 6th nnul ACM Symposium on Theory of Computing, (pp ). ACM Press. Guglielmi, A., & Gundersen, T. (2008). Normlistion control in deep inference vi tomic flows. Logicl Methods in Computer Science, 4(1:9), Jeřábek, E. (2009). Proof complexity of the cut-free clculus of structures. Journl of Logic nd Computtion, 19(2),