ABSTRACT We present methods for removing top-level cuts from a sequent calculus or Tait-style pro... more ABSTRACT We present methods for removing top-level cuts from a sequent calculus or Tait-style proof without significantly increasing the space used for storing the proof. For propositional logic, this requires con-verting a proof from tree-like to dag-like form, but it most doubles the number of lines in the proof. For first-order logic, the proof size can grow exponentially, but the proof has a succinct description and is polynomial-time uniform. We use direct, global constructions that give polynomial time methods for removing all top-level cuts from proofs. By exploiting prenex representations, this extends to removing all cuts, with final proof size bounded superexponentially in the alternation of quantifiers in cut formulas.
ABSTRACT This paper considers theories of bounded arithmetic that are predicative in the sense of... more ABSTRACT This paper considers theories of bounded arithmetic that are predicative in the sense of Nelson, that is, theories that are interpretable in Robinson’s Q.We give a nearly exact characterization of functions that can be total in predicative bounded theories. As an upper bound, any such function has a polynomial growth rate and its bit-graph is in nondeterministic exponential time and in co-nondeterministic exponential time. In fact, any function uniquely defined in a bounded theory of arithmetic lies in this class. Conversely, any function that is in this class (provably in IΔ0+exp) can be uniquely defined and total in a (predicative) bounded theory of arithmetic.
... It is open whether the theory CPV= S\(PV) can prove NPB ... Mi^ M} if and only if i i< j. ... more ... It is open whether the theory CPV= S\(PV) can prove NPB ... Mi^ M} if and only if i i< j. Since it is CPV-normal, IPV* is valid in this Kripke model; it turns ... universal quantifiers and combining like quantifiers we rewrite NPB as where NPBM (x, y, z) is an atomic formula formalizing" y> x ...
ABSTRACT We study the long-standing open problem of giving ∀Σ b 1 separa-tions for fragments of b... more ABSTRACT We study the long-standing open problem of giving ∀Σ b 1 separa-tions for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek&#39;s theories for approximate counting and their subtheories. We show that the ∀Σ b 1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP NP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeon-hole principle for polynomial time functions.
ABSTRACT We prove that the propositional translations of the Kneser-Lov\&#39;asz theorem have... more ABSTRACT We prove that the propositional translations of the Kneser-Lov\&#39;asz theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lov\&#39;asz theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.
ABSTRACT This paper defines a new notion of bounded computable randomness for certain classes of ... more ABSTRACT This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\&quot;of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen&#39;s theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness.
ABSTRACT We present methods for removing top-level cuts from a sequent calculus or Tait-style pro... more ABSTRACT We present methods for removing top-level cuts from a sequent calculus or Tait-style proof without significantly increasing the space used for storing the proof. For propositional logic, this requires con-verting a proof from tree-like to dag-like form, but it most doubles the number of lines in the proof. For first-order logic, the proof size can grow exponentially, but the proof has a succinct description and is polynomial-time uniform. We use direct, global constructions that give polynomial time methods for removing all top-level cuts from proofs. By exploiting prenex representations, this extends to removing all cuts, with final proof size bounded superexponentially in the alternation of quantifiers in cut formulas.
ABSTRACT This paper considers theories of bounded arithmetic that are predicative in the sense of... more ABSTRACT This paper considers theories of bounded arithmetic that are predicative in the sense of Nelson, that is, theories that are interpretable in Robinson’s Q.We give a nearly exact characterization of functions that can be total in predicative bounded theories. As an upper bound, any such function has a polynomial growth rate and its bit-graph is in nondeterministic exponential time and in co-nondeterministic exponential time. In fact, any function uniquely defined in a bounded theory of arithmetic lies in this class. Conversely, any function that is in this class (provably in IΔ0+exp) can be uniquely defined and total in a (predicative) bounded theory of arithmetic.
... It is open whether the theory CPV= S\(PV) can prove NPB ... Mi^ M} if and only if i i< j. ... more ... It is open whether the theory CPV= S\(PV) can prove NPB ... Mi^ M} if and only if i i< j. Since it is CPV-normal, IPV* is valid in this Kripke model; it turns ... universal quantifiers and combining like quantifiers we rewrite NPB as where NPBM (x, y, z) is an atomic formula formalizing" y> x ...
ABSTRACT We study the long-standing open problem of giving ∀Σ b 1 separa-tions for fragments of b... more ABSTRACT We study the long-standing open problem of giving ∀Σ b 1 separa-tions for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek&#39;s theories for approximate counting and their subtheories. We show that the ∀Σ b 1 Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FP NP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of T 1 2 augmented with the surjective weak pigeon-hole principle for polynomial time functions.
ABSTRACT We prove that the propositional translations of the Kneser-Lov\&#39;asz theorem have... more ABSTRACT We prove that the propositional translations of the Kneser-Lov\&#39;asz theorem have polynomial size extended Frege proofs and quasi-polynomial size Frege proofs. We present a new counting-based combinatorial proof of the Kneser-Lov\&#39;asz theorem that avoids the topological arguments of prior proofs for all but finitely many cases for each k. We introduce a miniaturization of the octahedral Tucker lemma, called the truncated Tucker lemma: it is open whether its propositional translations have (quasi-)polynomial size Frege or extended Frege proofs.
ABSTRACT This paper defines a new notion of bounded computable randomness for certain classes of ... more ABSTRACT This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\&quot;of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen&#39;s theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness.
Uploads
Papers