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34
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce
Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic
, 2005
"... ..."
A tight KarpLipton collapse result in bounded arithmetic
 In Proc. 17th Annual Conference on Computer Science Logic
"... Abstract. Cook and Krajíček [9] have obtained the following KarpLipton result in bounded arithmetic: if the theory PV proves NP ⊆ P/poly, then PH collapses to BH, and this collapse is provable in PV. Here we show the converse implication, thus answering an open question from [9]. We obtain this res ..."
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Abstract. Cook and Krajíček [9] have obtained the following KarpLipton result in bounded arithmetic: if the theory PV proves NP ⊆ P/poly, then PH collapses to BH, and this collapse is provable in PV. Here we show the converse implication, thus answering an open question from [9]. We obtain this result by formalizing in PV a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček [9]. In particular, we obtain several optimal and even poptimal proof systems using advice. We further show that these poptimal systems are equivalent to natural extensions of Frege systems.
Declarative Combinatorics: Isomorphisms, Hylomorphisms and Hereditarily Finite Data Types in Haskell – unpublished draft –
, 808
"... This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals etc.) and their extension to hereditarily fin ..."
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This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals etc.) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides anytoany encodings automatically. A few examples of “free algorithms ” obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to succinct data representations and generation of random instances. The paper is part of a larger effort to cover in a declarative programming paradigm some fundamental combinatorial generation algorithms along the lines of Knuth’s recent work (Knuth 2006). In 440 lines of Haskell code we cover 20 data types and, through the use of the embedded combinator language, provide 380 distinct bijective encodings between them. The selfcontained source code of the paper, as generated from a literate Haskell program, is available at
Examining The Fragments of G
"... When restricted to proving \Sigma qi formulas, the quantified propositional proof system G*i is closely related to the \Sigma bitheorems of Buss's theory Si2. Namely, G*i has polynomialsize proofs of the translations of theorems of Si2, and Si2proves that G*i is sound. However, little is known ..."
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When restricted to proving \Sigma qi formulas, the quantified propositional proof system G*i is closely related to the \Sigma bitheorems of Buss's theory Si2. Namely, G*i has polynomialsize proofs of the translations of theorems of Si2, and Si2proves that G*i is sound. However, little is known about G*i when proving more complex formulas. In this paper, weprove a witnessing theorem for G*1 similar in style to theKPT witnessing theorem for T i2. This witnessing theorem is then used to show that Si2 proves G*1 is sound with respectto prenex \Sigma q i+1 formulas. Note that unless the polynomial hierarchy collapses Si 2 is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that G*1 is pequivalent to a quantified version of extendedFrege. This is followed by a proof that Gi psimulates G*i+1. We finish by proving that S2 can be axiomatized by S12 plus axioms stating that the cutfree versionof G * is sound. All together this shows that the connection between G*i and Si2 does not extend to more complex formulas.
Theories and Proof Systems for PSPACE and the EXPTime Hierarchy
, 2005
"... This document is originally a working paper recording our results in progress. It is hoped that with some reorganization, addition of basic definitions, introduction and conclusion, and of course ..."
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This document is originally a working paper recording our results in progress. It is hoped that with some reorganization, addition of basic definitions, introduction and conclusion, and of course
A Propositional Proof System for Log Space
"... Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formulabased proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of ..."
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Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 corresponds to P, but no formulabased proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL ∗. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a nonparameter free variable. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of Σ B 0rec into a family of tautologies that have polynomial size GL ∗ proofs. Σ B 0rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ B 0rec, and put Σ B 0rec proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in Σ B 0rec. This is done by giving a log space algorithm that witnesses GL ∗ proofs. 1
Everything Is Everything” Revisited: Shapeshifting Data Types with Isomorphisms and Hylomorphisms
 Complex Systems
"... This paper is an exploration of isomorphisms between elementary data types (e.g., natural numbers, sets, finite functions, graphs, hypergraphs) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded hi ..."
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This paper is an exploration of isomorphisms between elementary data types (e.g., natural numbers, sets, finite functions, graphs, hypergraphs) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides anytoany encodings automatically. A few examples of free algorithms obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to succinct data representations and the generation of random instances. 1.
The Complexity of Proving Discrete Jordan Curve Theorem
 In Proc. 22nd IEEE Symposium on Logic in Computer Science
, 2007
"... The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded arithmetic that correspond to small complexity classes. The ..."
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The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded arithmetic that correspond to small complexity classes. The theory V 0 (2) (corresponding to AC 0 (2)) proves that any set of edges that form disjoint cycles divides the grid into at least two regions. The theory V 0 (corresponding to AC 0) proves that any sequence of edges that form a simple closed curve divides the grid into exactly two regions. As a consequence, the Hex tautologies and the stConnectivity tautologies have polynomial size AC 0 (2)Fregeproofs, which improves results of Buss which only apply to the stronger proof system TC 0Frege.