Sam Buss | University of California, San Diego (original) (raw)

Papers by Sam Buss

Research paper thumbnail of Cobham recursive set functions

Annals of Pure and Applied Logic, 2016

Research paper thumbnail of Cut Elimination In Situ

Gentzen's Centenary, 2015

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.

Research paper thumbnail of Some remarks on lengths of propositional proofs

Archive for Mathematical Logic, 1995

ABSTRACT

Research paper thumbnail of Alogtime algorithms for tree isomorphism, comparison, and canonization

Lecture Notes in Computer Science, 1997

ABSTRACT

Research paper thumbnail of The polynomial hierarchy and intuitionistic Bounded Arithmetic

Lecture Notes in Computer Science, 1986

Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the d... more Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the definable functions of IS: a r e precisely the 0: functions of t h e polvnomial hierarchy. This is an extension of earlier work on t h e classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast t o t h e classical theories of Bounded b Arithmetic where Ci-definable functions are of interest, our results for intuitionistic theories concern all the definable functions. The method of proof uses 0;-realizability which is inspired by t h e recursive realizability of S.C. Kleene 131 and D. Nelson 151. I t also involves polynomial hierarchy functionals of finite type which a r e introduced in this paper. * Research supported in part by NSF Grant DMS 85-11465. In general, 0: is P The theories Si a r e most advantageously viewed a s Gentzen-style natural deduction systems. A formal proof in a natural deduction system contains sequents of t h e form where each A. and B. is a formula. The meaning of such a sequent is J J In addition t o t h e usual inference rules for natural deduction. the Z:-PIND inference is b

Research paper thumbnail of First-Order Proof Theory of Arithmetic

Studies in Logic and the Foundations of Mathematics, 1998

Research paper thumbnail of Axiomatizations and conservation results for fragments of bounded arithmetic

Contemporary Mathematics, 1990

ABSTRACT

Research paper thumbnail of The Computational Power of Bounded Arithmetic from the Predicative Viewpoint

New Computational Paradigms, 2008

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.

Research paper thumbnail of On Model Theory for Intuitionistic Bounded Arithmetic with Applications to Independence Results

Feasible Mathematics, 1990

... 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 ...

Research paper thumbnail of SAFE RECURSIVE SET FUNCTIONS

The Journal of Symbolic Logic, 2015

Research paper thumbnail of A note on bootstrapping intuitionistic bounded arithmetic

A selection of papers from the Leeds Proof Theory Programme 1990, 1993

ABSTRACT

Research paper thumbnail of FRAGMENTS OF APPROXIMATE COUNTING

The Journal of Symbolic Logic, 2014

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.

Research paper thumbnail of Short Proofs of the Kneser-Lovász Coloring Principle

Lecture Notes in Computer Science, 2015

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.

Research paper thumbnail of Sub-computable Boundedness Randomness

Logical Methods in Computer Science, 2014

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.

Research paper thumbnail of Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Transactions of the American Mathematical Society, 2015

Research paper thumbnail of Book Review: Matthias Baaz and Alexander Leitsch, Methods of Cut-Elimination

Research paper thumbnail of Limits on Alternation-Trading Proofs for Time-Space Lower Bounds

2012 IEEE 27th Conference on Computational Complexity, 2012

Research paper thumbnail of Resolution and the weak pigeonhole principle

Lecture Notes in Computer Science, 1998

We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give low... more We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for tree-like resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule. n ; in other words,

Research paper thumbnail of THE POLYNOMIAL HIERARCHY AND INTUITIONISTIC BOUNDED ARITHMETIC

Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the d... more Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the definable functions of IS: a r e precisely the 0: functions of t h e polvnomial hierarchy. This is an extension of earlier work on t h e classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast t o t h e classical theories of Bounded b Arithmetic where Ci-definable functions are of interest, our results for intuitionistic theories concern all the definable functions. The method of proof uses 0;-realizability which is inspired by t h e recursive realizability of S.C. Kleene 131 and D. Nelson 151. I t also involves polynomial hierarchy functionals of finite type which a r e introduced in this paper. * Research supported in part by NSF Grant DMS 85-11465. In general, 0: is P The theories Si a r e most advantageously viewed a s Gentzen-style natural deduction systems. A formal proof in a natural deduction system contains sequents of t h e form where each A. and B. is a formula. The meaning of such a sequent is J J In addition t o t h e usual inference rules for natural deduction. the Z:-PIND inference is b

Research paper thumbnail of Immerman Neil. Upper and lower bounds for first order expressibility. Journal of computer and system sciences, vol. 25 (1982), pp. 76–98. Immerman Neil. Relational queries computable in polynomial time. Information and control, vol. 68 (1986), pp. 86–104. Immerman Neil. Languages that capture com...

Research paper thumbnail of Cobham recursive set functions

Annals of Pure and Applied Logic, 2016

Research paper thumbnail of Cut Elimination In Situ

Gentzen's Centenary, 2015

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.

Research paper thumbnail of Some remarks on lengths of propositional proofs

Archive for Mathematical Logic, 1995

ABSTRACT

Research paper thumbnail of Alogtime algorithms for tree isomorphism, comparison, and canonization

Lecture Notes in Computer Science, 1997

ABSTRACT

Research paper thumbnail of The polynomial hierarchy and intuitionistic Bounded Arithmetic

Lecture Notes in Computer Science, 1986

Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the d... more Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the definable functions of IS: a r e precisely the 0: functions of t h e polvnomial hierarchy. This is an extension of earlier work on t h e classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast t o t h e classical theories of Bounded b Arithmetic where Ci-definable functions are of interest, our results for intuitionistic theories concern all the definable functions. The method of proof uses 0;-realizability which is inspired by t h e recursive realizability of S.C. Kleene 131 and D. Nelson 151. I t also involves polynomial hierarchy functionals of finite type which a r e introduced in this paper. * Research supported in part by NSF Grant DMS 85-11465. In general, 0: is P The theories Si a r e most advantageously viewed a s Gentzen-style natural deduction systems. A formal proof in a natural deduction system contains sequents of t h e form where each A. and B. is a formula. The meaning of such a sequent is J J In addition t o t h e usual inference rules for natural deduction. the Z:-PIND inference is b

Research paper thumbnail of First-Order Proof Theory of Arithmetic

Studies in Logic and the Foundations of Mathematics, 1998

Research paper thumbnail of Axiomatizations and conservation results for fragments of bounded arithmetic

Contemporary Mathematics, 1990

ABSTRACT

Research paper thumbnail of The Computational Power of Bounded Arithmetic from the Predicative Viewpoint

New Computational Paradigms, 2008

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.

Research paper thumbnail of On Model Theory for Intuitionistic Bounded Arithmetic with Applications to Independence Results

Feasible Mathematics, 1990

... 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 ...

Research paper thumbnail of SAFE RECURSIVE SET FUNCTIONS

The Journal of Symbolic Logic, 2015

Research paper thumbnail of A note on bootstrapping intuitionistic bounded arithmetic

A selection of papers from the Leeds Proof Theory Programme 1990, 1993

ABSTRACT

Research paper thumbnail of FRAGMENTS OF APPROXIMATE COUNTING

The Journal of Symbolic Logic, 2014

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.

Research paper thumbnail of Short Proofs of the Kneser-Lovász Coloring Principle

Lecture Notes in Computer Science, 2015

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.

Research paper thumbnail of Sub-computable Boundedness Randomness

Logical Methods in Computer Science, 2014

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.

Research paper thumbnail of Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Transactions of the American Mathematical Society, 2015

Research paper thumbnail of Book Review: Matthias Baaz and Alexander Leitsch, Methods of Cut-Elimination

Research paper thumbnail of Limits on Alternation-Trading Proofs for Time-Space Lower Bounds

2012 IEEE 27th Conference on Computational Complexity, 2012

Research paper thumbnail of Resolution and the weak pigeonhole principle

Lecture Notes in Computer Science, 1998

We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give low... more We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for tree-like resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule. n ; in other words,

Research paper thumbnail of THE POLYNOMIAL HIERARCHY AND INTUITIONISTIC BOUNDED ARITHMETIC

Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the d... more Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the definable functions of IS: a r e precisely the 0: functions of t h e polvnomial hierarchy. This is an extension of earlier work on t h e classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast t o t h e classical theories of Bounded b Arithmetic where Ci-definable functions are of interest, our results for intuitionistic theories concern all the definable functions. The method of proof uses 0;-realizability which is inspired by t h e recursive realizability of S.C. Kleene 131 and D. Nelson 151. I t also involves polynomial hierarchy functionals of finite type which a r e introduced in this paper. * Research supported in part by NSF Grant DMS 85-11465. In general, 0: is P The theories Si a r e most advantageously viewed a s Gentzen-style natural deduction systems. A formal proof in a natural deduction system contains sequents of t h e form where each A. and B. is a formula. The meaning of such a sequent is J J In addition t o t h e usual inference rules for natural deduction. the Z:-PIND inference is b

Research paper thumbnail of Immerman Neil. Upper and lower bounds for first order expressibility. Journal of computer and system sciences, vol. 25 (1982), pp. 76–98. Immerman Neil. Relational queries computable in polynomial time. Information and control, vol. 68 (1986), pp. 86–104. Immerman Neil. Languages that capture com...