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The complexity of propositional proofs
 Bulletin of Symbolic Logic
"... Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorit ..."
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Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents
Structure and Definability in General Bounded Arithmetic Theories
, 1999
"... This paper is motivated by the questions: what are the \Sigma ..."
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Cited by 18 (6 self)
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This paper is motivated by the questions: what are the \Sigma
ON THE NUMBER OF STEPS IN PROOFS
, 1989
"... In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. W ..."
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Cited by 17 (2 self)
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In this paper we prove some results about the complexity of proofs. We consider proofs in Hilbertstyle formal systems such as in [17J. Thus a proof is a sequence of formulas satisfying certain conditions. We caD view the formulas as being strings of symbols; hence the whole proof is a string too. We consider the following measures of complexity of proofs: length ( = the number of symbols in the proof), depth ( = the maximal depth of a formula in the proof) and number o! steps ( = the number of formulas in the proof). For a particular formaI system and a given formula A we consider the shortest length of a proof of A, the minimal depth ofa proof of A and the minimal number of steps in a proof of A. The main results are the following: (1) a bound on the depth in terms of the number of steps: Theorem 2.2, (2) a bound on the depth in terms of the length: Theorem 2.3, (3) a bound on the length in terms of the number of steps for restricted systems: Theorem 3.1. These results are applied to obtain several corollaries. In particular we show: (1) a bound on the number of steps in a cutfree proof, (2) some speedup results, (3) bounds on the number of steps in proofs of ParisHarrington sentences. Some paper
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Computational Complexity and Induction for Partial Computable Functions in Type Theory
 In Preprint
, 1999
"... An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in ..."
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An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory. This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computationa...
Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of t ..."
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This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cutfree proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.
On Bounded Set Theory
"... We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation ..."
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We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation between BST, possibly with AntiFoundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory.
Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Cited by 8 (4 self)
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.