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Hybrid Logics: Characterization, Interpolation and Complexity
 Journal of Symbolic Logic
, 1999
"... Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We sho ..."
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Cited by 101 (35 self)
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Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of EhrenfeuchtFrasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rstorder logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...
Polynomial size proofs of the propositional pigeonhole principle
 Journal of Symbolic Logic
, 1987
"... Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems ha ..."
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Cited by 72 (7 self)
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Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in computational complexity such as whether NP = coNP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which
An Exponential Lower Bound to the Size of Bounded Depth Frege . . .
, 1994
"... We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for whic ..."
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Cited by 68 (10 self)
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We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.
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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Cited by 20 (0 self)
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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
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 10 (3 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Spectra with Only Unary Function Symbols
, 1997
"... The spectrum of a firstorder sentence is the set of cardinalities of its finite models. This paper is concerned with spectra of sentences over languages that contain only unary function symbols. In particular, it is shown that a set S of natural numbers is the spectrum of a sentence over the langua ..."
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Cited by 9 (1 self)
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The spectrum of a firstorder sentence is the set of cardinalities of its finite models. This paper is concerned with spectra of sentences over languages that contain only unary function symbols. In particular, it is shown that a set S of natural numbers is the spectrum of a sentence over the language of one unary function symbol precisely if S is an eventually periodic set.
Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms
, 2001
"... We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem ..."
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Cited by 8 (2 self)
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We show that the known boundeddepth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a sizedepth tradeoff upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in I \Delta 0 with a provably weaker `large number assumption' than previously needed.
Arithmetical Definability over Finite Structures
, 2002
"... Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability overfinite structures, motivated by the correspondence between uniform AC and FO(PLUS;TIMES). We prove finite analogs of three classic results in arithmet ..."
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Cited by 3 (0 self)
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Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability overfinite structures, motivated by the correspondence between uniform AC and FO(PLUS;TIMES). We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can firstorder define PLUS, that < and DIVIDES can firstorder define TIMES, and that < and COPRIME can firstorder define TIMES.
Combinatorics in Bounded Arithmetics
, 2004
"... A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabil ..."
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Cited by 3 (0 self)
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A basic aim of logic is to consider what axioms are used in proving various theorems of mathematics. This thesis will be concerned with such issues applied to a particular area of mathematics: combinatorics. We will consider two widely known groups of proof methods in combinatorics, namely, probabilistic methods and methods using linear algebra. We will consider certain applications of such methods, both of which are significant to Ramsey theory. The systems we choose to work in are various theories of bounded arithmetic. For the probabilistic method, the key point is that we use the weak pigeonhole principle to simulate the probabilistic reasoning. We formalize various applications of the ordinary probabilistic method and linearity of expectations, making partial progress on the Local Lemma. In the case of linearity of expectations, we show how to eliminate the weak pigeonhole principle by simulating the derandomization technique of “conditional probabilities.” We consider linear algebra methods applied to various set system theorems. We formalize some theorems using a linear algebra principle as an extra axiom. We also show how weaker results can be attained by giving alternative proofs that avoid linear algebra, and thus also avoid the extra axiom. We formalize upper and lower Ramsey bounds. For the lower bounds, both the probabilistic methods and the linear algebra methods are used. We provide a stratification of the various Ramsey lower bounds, showing that stronger bounds can be proved in stronger theories. A natural question is whether or not the axioms used are necessary. We provide “reversals” in a few cases, showing that the principle used to prove the theorem is in fact a consequence of the theorem (over some base theory). Thus this work can be seen as a (humble) beginning in the direction of developing the Reverse Mathematics of finite combinatorics.
Undecidable extensions of Skolem arithmetic
 Journal of Symbolic Logic
, 1998
"... Let
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Cited by 2 (0 self)
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Let <P 2 be the restriction of usual order relation to integers which are primes or squares of primes, and let ? denote the coprimeness predicate. The elementary theory of hIN; ?;<P 2 i is undecidable. Now denote by < the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are de nable in the structure hIN; ?;<i. Furthermore, the structures hIN; j; <i, hIN; =; ; <i and hIN; =; +; i are interde nable.