Propositional Proofs in Frege and Extended Frege Systems (Abstract)

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1 Propositioal Proofs i Frege ad Exteded Frege Systems (Abstract) Sam Buss Departmet of Mathematics Uiversity of Califoria, Sa Diego La Jolla, Califoria , USA sbuss@math.ucsd.edu Abstract. We discuss recet results o the propositioal proof complexity of Frege proof systems, icludig some recetly discovered quasipolyomial size proofs for the pigeohole priciple ad the Keser-Lovász theorem. These are closely related to formalizability i bouded arithmetic. Keywords: Proof complexity, Frege proofs, pigeohole priciple, Keser- Lovász theorem, bouded arithmetic 1 Itroductio The complexity of propositioal proofs has bee studied extesively both because of its coectios to computatioal complexity ad because of the importace of propositioal proof search for propositioal logic ad as a uderpiig for stroger systems such as SMT solvers, modal logics ad first-order logics. Frege systems are arguably the most importat fully expressive, soud ad complete proof system for propositioal proofs: Frege proofs are textbook propositioal proof systems usually formulated with modus poes as the sole rule of iferece. Exteded Frege proofs allow the use of the extesio rule which permits ew variables to be itroduced as abbreviatios for more complex formulas [27]. (This abstract caot do justice to the field of propositioal proof complexity. There are several surveys available, icludig [25, 4, 12, 13, 5, 23].) We will measure proof complexity by coutig the umber of symbols appearig i a proof. We are particularly iterested i polyomial ad quasipolyomial size Frege ad exteded Frege proofs, as these represet proofs of (ear) feasible size. Frege proofs are usually axiomatized with modus poes ad a fiite set of axiom schemes. However, there are a umber of other atural ways to axiomatize Frege proofs, ad they are all polyomially equivalet [15, 24]. Thus, Frege proof Supported i part by NSF grats CCF ad DMS , ad a Simos Foudatio Fellowship

2 2 S. Buss systems are a robust otio for proof complexity. The same holds for exteded Frege proofs. Formulas i a polyomial size Frege proof are polyomial size of course, ad hece express (ouiform) NC 1 properties. By virtue of the expressiveess of extesio variables, formulas i polyomial size exteded Frege proofs represet polyomial size Boolea circuits. 1 Boolea circuits express ouiform polyomial time (P) predicates. It is geerally cojectured NC 1 P ad that Boolea circuits are more expressive tha Boolea formulas, amely that covertig a Boolea circuit to a Boolea formula may cause a expoetial icrease i size. For this reaso, it is geerally cojectured that Frege proofs do ot polyomially or quasipolyomially simulate exteded Frege proofs: Defiitio 1. The size P of a proof P is the umber of occurreces of symbols i P. Frege proofs polyomially simulate exteded Frege proofs provided that there is a polyomial p() such that, for every exteded Frege proof P 1 of a formula ϕ there is a Frege proof P 2 of the same formula ϕ with P 2 p( P 1 ). Frege proofs quasipolyomially simulate exteded Frege proofs if the same holds but with p() = 2 logo(1). However, the coectio betwee the proof complexity of Frege ad exteded Frege systems ad the expressiveess of Boolea formulas ad circuits is oly a aalogy. There is o kow direct coectio. It could be that Frege proofs ca polyomially simulate exteded Frege proofs but Boolea formulas caot polyomially express Boolea circuits. Likewise, it could be that Boolea formulas ca express Boolea circuits with oly a polyomial icrease i size, but Frege proofs caot polyomially simulate exteded Frege proofs. Boet, Buss, ad Pitassi [7] cosidered the questio of what kids of combiatorial tautologies are cadidates for expoetially separatig proof sizes for Frege ad exteded Frege systems, that is for showig Frege systems do ot polyomially or quasipolyomially simulate exteded Frege systems. Surprisigly, oly a small umber of examples were foud. The first type of examples were based o liear algebra, ad icluded the Oddtow Theorem, the Graham Pollack Theorem, the Fisher Iequality, the Ray-Chaudhuri Wilso Theorem, ad the AB = I BA = I tautology (the last was suggested by S. Cook). The remaiig example was Frakl s Theorem o the trace of sets. The five priciples based o liear algebra were kow to have short exteded Frege proofs usig facts about determiats ad eigevalues of 0/1 matrices. Sice there are quasipolyomial size formulas defiig determiats over 0/1 matrices,[7] cojectured that all these priciples have quasipolyomial size Frege proofs. This was oly recetly proved by Hrubeš ad Tzameret [16], who showed that the five liear-algebra-based tautologies have quasipolyomial size Frege proofs by showig that there are quasipolyomial size defiitios of determiats whose properties ca be established by quasipolyomial Frege proofs. 1 See Jeřábek [18] for a alterative formulatio of exteded Frege systems based directly o Boolea circuits.

3 Propositioal Proofs 3 The remaiig priciple, Frakl s Theorem, was show to have polyomial size exteded Frege proofs by [7], but it was ukow whether it had polyomial size Frege proofs. Recetly, Aiseberg, Boet ad Buss[1] showed that it also has quasipolyomial size Frege proofs. Thus, Frakl s theorem does ot provide a example of tautologies which expoetially separate Frege ad exteded Frege proofs. Istrate ad Crãciu [17] recetly proposed the Keser-Lovász Theorem as a family of tautologies that might be hard for (exteded) Frege systems. They showed that the k = 3 versios of these tautologies have polyomial size exteded Frege proofs, but left ope whether they have (quasi)polyomial size Frege proofs. However, as stated i Defiitio 3 ad Theorem 5 below, [2] have ow give polyomial size exteded Frege proofs ad quasipolyomial size Frege proofs for the Keser-Lovász tautologies, for each fixed k. Thus these also do ot give a expoetial separatio of Frege from exteded Frege systems. Other cadidates for expoetially separatig Frege ad exteded Frege systems arose from the work of Ko lodziejczyk, Nguye, ad Thape [19] i the settig of bouded arithmetic [9]. They proposed as cadidates various forms of the local improvemet priciples LI, LI log ad LLI. The results of [19] iclude that the LI priciple is may-oe complete for the NP search problems of V2 1; it follows that LI is equivalet to partial cosistecy statemets for exteded Frege systems. Beckma ad Buss [6] subsequetly proved that LI log is provably equivalet (i S2) 1 to LI ad that the liear local improvemet priciple LLI is provable i U2. 1 The LLI priciple thus has quasipolyomial size Frege proofs. Combiig the results of [6,19] shows that LI log ad LLI are may-oe complete for the NP search problems of V2 1 ad U2, 1 respectively, ad thus equivalet to partial cosistecy statemets for exteded Frege ad Frege systems, respectively. Cook ad Reckhow [14] showed that the partial cosistecy statemets for exteded Frege systems characterize the proof theoretic stregth of exteded Frege systems; Buss [11] showed the same for Frege systems. For this reaso, partial cosistecy statemets do ot provide satisfactory combiatorial priciples for separatig Frege ad exteded Frege systems. The same is true for other statemets equivalet to partial cosistecy statemets. (But, compare to Avigad [3].) This talk will discuss a pair of recetly discovered families of quasipolyomial size Frege proofs. The first is based o the pigeohole priciple; the secod o the Keser-Lovász priciple. Defiitio 2. The propositioal pigeohole priciple PHP +1 is the tautology i=0 1 j=0 p i,j 0 i 1<i 2 (p i1,j p i2,j). 1 j=0 Theorem 1. (Cook-Reckhow [15]) PHP +1 proofs. has polyomial size exteded Frege

4 4 S. Buss Theorem 1 was proved by a iductio proof. Later, the followig was proved by usig a coutig proof: Theorem 2. ([10]) PHP +1 has polyomial size Frege proofs. Sice the proofs of Theorems 1 ad 2 were so differet, this was sometimes take as evidece that Frege proofs caot polyomially simulate exteded Frege proofs. However, recetly the preset author showed that the proof of Theorem 1 ca be carried out with Frege proofs, ad established a weaker result, but with a proof based o the proof of [15]: Theorem 3. ([8]) PHP +1 has quasipolyomial size Frege proofs. This is weaker tha Theorem 2: the poit is that its proof shows that the costructio uderlyig the proof of Theorem 1 ca be carried by quasipolyomial size Frege proofs. We ext state the results about the Keser-Lovász priciple. Defiitio 3. Fix k 1. Let ( deote the set of subsets of [] := {0,..., 1} of cardiality k. The (,k)-kesergraph is the udirected graph (V,E) where the vertex set V is the set ( ( k), ad E is the set of edges {A,B} such that A,B ) k ad A B =. It is ot hard to show that the (,k)-keser graph ca be colored with 2k+2 colors. (That is, so that o two adjacet vertices receive the same color.) This is the optimal umber of colors: Theorem 4. (Lovász [21]) Let k 1 ad 2k. The (,k)-keser graph caot be colored with 2k+1 colors. Note that the k = 1 case of the Theorem 4 is just the usual pigeohole priciple. It is straightforward to traslate the Keser-Lovász priciple as expressed by Theorem 4 ito a family of polyomial size tautologies: Defiitio 4. Let 2k > 1, ad let m = 2k + 1 be the umber of colors. For A ( k) ad i [m], the propositioal variable pa,i has the iteded meaig that vertex A of the Keser graph is assiged the color i. The Keser- Lovász priciple is expressed propositioally by A ( i [m] k) p A,i A,B ( k) A B= i [m] (p A,i p B,i ). Theorem 5. ([2]) For each k 1, the tautologies based o the Keser-Lovász priciple have polyomial size exteded Frege proofs ad quasipolyomial size Frege proofs.

5 Propositioal Proofs 5 The proof of Theorem 5 is based o a simple coutig argumet which avoids the usual topologically-based combiatorial argumets of Matoušek [22] ad others. As already discussed, we ow lack may good combiatorial cadidates for super-quasipolyomially separatig Frege ad exteded Frege systems, apart from partial cosistecy priciples or priciples which are equivalet to partial cosistecy priciples. At the preset momet, we have oly a couple potetial combiatorial cadidates. The first cadidate is the rectagular local improvemet priciples RLI 2 (or more geerally, RLI k for ay costat k 2). For the defiitios of these i the settig of bouded arithmetic, plus characterizatios of the logical stregths of the related priciples RLI 1, RLI log ad RLI, see Beckma-Buss[6]. RLI 1 is provable i U2 1 ad is may-oecomplete for the NP search problems of U2 1, ad thus has quasipolyomial size Frege proofs (for the latter coectio, see Krajíček [20]). RLI log ad RLI are provable i V2 1 ad are may-oe complete for the NP search problems of V2 1 ; hece they are equivalet to partial cosistecy statemets for exteded Frege. The secod cadidate is the trucated Tucker lemma defied by [2]. These are actively uder ivestigatio as this abstract is beig writte; some special cases are kow to have exteded Frege proofs [Aiseberg-Buss, work i progress], but it is still ope whether they has quasipolyomial size Frege proofs. It seems very ulikely however that Frege proofs ca quasipolyomially simulate exteded Frege proofs. We thak Lev Beklemishev ad Vladimir Podolskii for helpful commets. Refereces 1. Aiseberg, J., Boet, M.L., Buss, S.: Quasi-polyomial size Frege proofs of Frakl s theorem o the trace of fiite sets (201?), to appear i Joural of Symbolic Logic 2. Aiseberg, J., Boet, M.L., Buss, S., Crãciu, A., Istrate, G.: Short proofs of the Keser-Lovász priciple (2015), to appear i Proc. 42th Iteratioal Colloquium o Automata, Laguages, ad Programmig (ICALP 15) 3. Avigad, J.: Plausibly hard combiatorial tautologies. I: Beame, P., Buss, S.R. (eds.) Proof Complexity ad Feasible Arithmetics, pp America Mathematical Society (1997) 4. Beame, P.: Proof complexity. I: Computatioal Complexity Theory, pp IAS/Park City Mathematical Series, Vol. 10, America Mathematical Society (2004), lecture otes scribed by Ashish Sabharwal 5. Beame, P., Pitassi, T.: Propositioal proof complexity: Past, preset ad future. I: Pau, G., Rozeberg, G., Salomaa, A. (eds.) Curret Treds i Theoretical Computer Sciece Eterig the 21st Cetury, pp World Scietific (2001), earlier versio appeared i Computatioal Complexity Colum, Bulleti of the EATCS, Beckma, A., Buss, S.R.: Improved witessig ad local improvemet priciples for secod-order bouded arithmetic. ACM Trasactios o Computatioal Logic 15(1) (2014), article 2, 35 pages

6 6 S. Buss 7. Boet, M.L., Buss, S.R., Pitassi, T.: Are there hard examples for Frege systems? I: Clote, P., Remmel, J. (eds.) Feasible Mathematics II. pp Birkhäuser, Bosto (1995) 8. Buss, S.: Quasipolyomial size proofs of the propositioal pigeohole priciple (201?), to appear i Theoretical Computer Sciece 9. Buss, S.R.: Bouded Arithmetic. Bibliopolis (1986), revisio of 1985 Priceto Uiversity Ph.D. thesis 10. Buss, S.R.: Polyomial size proofs of the propositioal pigeohole priciple. Joural of Symbolic Logic 52, (1987) 11. Buss, S.R.: Propositioal cosistecy proofs. Aals of Pure ad Applied Logic 52, 3 29 (1991) 12. Buss, S.R.: Propositioal proof complexity: A itroductio. I: Berger, U., Schwichteberg, H. (eds.) Computatioal Logic, pp Spriger-Verlag, Berli (1999) 13. Buss, S.R.: Towards NP-P via proof complexity ad proof search. Aals of Pure ad Applied Logic 163(9), (2012) 14. Cook, S.A., Reckhow, R.A.: O the legths of proofs i the propositioal calculus, prelimiary versio. I: Proceedigs of the Sixth Aual ACM Symposium o the Theory of Computig. pp (1974) 15. Cook, S.A., Reckhow, R.A.: The relative efficiecy of propositioal proof systems. Joural of Symbolic Logic 44, (1979) 16. Hrubeš, P., Tzameret, I.: Short proofs for determiat idetities. SIAM J. Computig 44(2), (2015) 17. Istrate, G., Crãciu, A.: Proof complexity ad the Keser-Lovász theorem. I: Theory ad Applicatios of Satisfiability Testig (SAT). pp Lecture Notes i Computer Sciece 8561, Spriger Verlag (2014) 18. Jeřábek, E.: Dual weak pigeohole priciple, boolea complexity, ad deradomizatio. Aals of Pure ad Applied Logic 124, 1 37 (2004) 19. Ko lodziejczyk, L.A., Nguye, P., Thape, N.: The provably total NP search problems of weak secod-order bouded arithmetic. Aals of Pure ad Applied Logic 162(2), (2011) 20. Krajíček, J.: Bouded Arithmetic, Propositioal Calculus ad Complexity Theory. Cambridge Uiversity Press, Heidelberg (1995) 21. Lovász, L.: Keser s cojecture, chromatic umber, ad homotopy. Joural of Combiatorial Theory, Series A 25(3), (1978) 22. Matoušek, J.: A combiatorial proof of Keser s cojecture. Combiatorica 24(1), (2004) 23. Pudlák, P.: Twelve problems i proof complexity. I: Computer Sciece Theory ad Applicatios. pp Lecture Notes i Computer Sciece #5010, Spriger, Berli, Heidelberg (2008) 24. Reckhow, R.A.: O the Legths of Proofs i the Propositioal Calculus. Ph.D. thesis, Departmet of Computer Sciece, Uiversity of Toroto (1976), techical Report # Segerlid, N.: The complexity of propositioal proofs. Bulleti of Symbolic Logic 13(4), (2007) 26. Siekma, J., Wrightso, G.: Automatio of Reasoig, vol. 1&2. Spriger-Verlag, Berli (1983) 27. Tsejti, G.S.: O the complexity of derivatio i propositioal logic. Studies i Costructive Mathematics ad Mathematical Logic 2, (1968), reprited i: [26, vol 2], pp