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Automatizability and Simple Stochastic Games
 In Proc. of 38th International Colloquium on Automata, Languages and Programming (ICALP), Luca Aceto, Monika Henzinger, Jiri Sgall (Eds
, 2011
"... The complexity of simple stochastic games (SSGs) has been open since they were dened by Condon in 1992. Despite intensive eort, the complexity of this problem is still unresolved. In this paper, building on the results of [4], we establish a connection between the complexity of SSGs and the complexi ..."
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The complexity of simple stochastic games (SSGs) has been open since they were dened by Condon in 1992. Despite intensive eort, the complexity of this problem is still unresolved. In this paper, building on the results of [4], we establish a connection between the complexity of SSGs and the complexity of an important problem in proof complexity{the proof search problem for low depth Frege systems. We prove that if depth3 Frege systems are weakly automatizable, then SSGs are solvable in polynomialtime. Moreover we identify a natural combinatorial principle, which is a version of the wellknown Graph Ordering Principle (GOP), that we call the integervalued GOP (IGOP). This principle states that for any graph G with nonnegative integer weights associated with each node, there exists a locally maximal vertex (a vertex whose weight is at least as large as its neighbors). We prove that if depth2 Frege plus IGOP is weakly automatizable, then SSG is in P. Supported by NSERC. 1
Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem Extended Abstract?
"... Abstract. We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasible interpolation property imply an efficie ..."
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Abstract. We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [12], as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and Ω(n1.4) clauses. Such small size refutations would improve the state of the art (with respect to the clause density) efficient refutation algorithm, which works only for Ω(n1.5) many clauses [13]. We demonstrate polynomialsize refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly automatizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in [25]). This reduces the problem of refuting random 3CNF with n variables and Ω(n1.4) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin). 1
The ProofSearch Problem between BoundedWidth Resolution and BoundedDegree SemiAlgebraic Proofs
, 2013
"... In recent years there has been some progress in our understanding of the proofsearch problem for very lowdepth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNFformulas (i.e. R(k) systems), or polynomial inequalities (i.e. sem ..."
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In recent years there has been some progress in our understanding of the proofsearch problem for very lowdepth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNFformulas (i.e. R(k) systems), or polynomial inequalities (i.e. semialgebraic proof systems). In this talk I will overview this progress. I will start with boundedwidth resolution, whose specialized proofsearch algorithm is as easy as uninteresting, but whose proofsearch problem is unintentionally solved by certain versions of conflictdriven clauselearning algorithms with restarts. I will continue with R(k) systems, whose proofsearch problem turned out to hide the complexity of certain twoplayer games of interest in the area of systems synthesis and verification. And I will close with boundeddegree semialgebraic proof systems, whose proofsearch problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.