1 -formulae on structures (1983)

by Miklos Ajtai
Venue:Annals of Pure and Applied Logic
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Results 1 - 10 of 12

Infinitary Logics and 0-1 Laws

by Phokion G. Kolaitis, Moshe Y. Vardi - Information and Computation , 1992
"... We investigate the in nitary logic L 1! , in which sentences may have arbitrary disjunctions and conjunctions, but they involve only a nite number of distinct variables. We show that various xpoint logics can be viewed as fragments of L 1! , and we describe a game-theoretic characterizat ..."
Abstract - Cited by 42 (4 self) - Add to MetaCart
We investigate the in nitary logic L 1! , in which sentences may have arbitrary disjunctions and conjunctions, but they involve only a nite number of distinct variables. We show that various xpoint logics can be viewed as fragments of L 1! , and we describe a game-theoretic characterization of the expressive power of the logic. Finally, we study asymptotic probabilities of properties 1! on nite structures. We show that the 0-1 law holds for L 1! , i.e., the asymptotic probability of every sentence in this logic exists and is equal to either 0 or 1. This result subsumes earlier work on asymptotic probabilities for various xpoint logics and reveals the boundary of 0-1 laws for in nitary logics.

Pseudorandom Generators in Propositional Proof Complexity

by Michael Alekhnovich, Eli Ben-sasson, Alexander A. Razborov, Avi Wigderson - ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REP. NO.23 , 2000
"... We call a pseudorandom generator Gn : {0, 1}^n → {0, 1}^m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G(x1, ..., xn) ≠ b for any string b ∈ {0, 1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by ..."
Abstract - Cited by 35 (6 self) - Add to MetaCart
We call a pseudorandom generator Gn : {0, 1}^n → {0, 1}^m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G(x1, ..., xn) ≠ b for any string b ∈ {0, 1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by the Nisan-Wigderson generator on the one hand, and by the construction of Tseitin tautologies on the other. We prove that under certain circumstances these generators are hard for such proof systems as Resolution, Polynomial Calculus and Polynomial Calculus with Resolution (PCR).

Theories for Complexity Classes and their Propositional Translations

by Stephen Cook - Complexity of computations and proofs , 2004
"... We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
Abstract - Cited by 25 (6 self) - Add to MetaCart
We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

Quantum Circuits: Fanout, Parity, and Counting

by Cristopher Moore - In Los Alamos Preprint archives , 1999
"... Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC 0 [q], where n-ary MODq gates are also allowed. We show that it is possible to make a ‘cat ’ state on n qubits in constant depth if ..."
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Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC 0 [q], where n-ary MODq gates are also allowed. We show that it is possible to make a ‘cat ’ state on n qubits in constant depth if and only if we can construct a parity or MOD2 gate in constant depth; therefore, any circuit class that can fan out a qubit to n copies in constant depth also includes QACC 0 [2]. In addition, we prove the somewhat surprising result that parity or fanout allows us to construct MODq gates in constant depth for any q, so QACC 0 [2] = QACC 0. Since ACC 0 [p] ̸ = ACC 0 [q] whenever p and q are mutually prime, QACC 0 [2] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. 1

Definable Relations and First-Order Query Languages over Strings

by Michael Benedikt, Leonid Libkin, Thomas Schwentick, Luc Segoufin
"... We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical model-theoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra - a class of n-ary relati ..."
Abstract - Cited by 16 (4 self) - Add to MetaCart
We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical model-theoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra - a class of n-ary relations for every n, closed under projection and Boolean operations. We show that by choosing the string vocabulary carefully, we get string logics that have desirable properties: computable evaluation and normal forms. We identify five distinct models and study the differences in their model-theory and complexity of evaluation. We identify a subset of these models which have additional attractive properties, such as finite VC dimension and quantifier elimination. Once you have a logic,

Counting, Fanout, And The Complexity Of Quantum Acc

by Frederic Green, Steven Homer, Cristopher Moore, Christopher Pollett , 2002
"... q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upp ..."
Abstract - Cited by 15 (2 self) - Add to MetaCart
q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We dene classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomialsize circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth. Keywords: quantum computation, quantum & circuit complexity, threshold circuit Communicated by : R Cleve & J Watrous 1. Introduction Advances in quantum computation

The Strength of Replacement in Weak Arithmetic

by Stephen Cook, Neil Thapen , 2003
"... The replacement (or collection or choice) axiom scheme BB() asserts bounded quanti er exchange as follows: 8i< jaj 9x
Abstract - Cited by 10 (3 self) - Add to MetaCart
The replacement (or collection or choice) axiom scheme BB() asserts bounded quanti er exchange as follows: 8i< jaj 9x<a(i;x) ! 9w 8i< jaj (i; [w] i ) where is in the class of formulas. The theory S 2 proves the scheme BB( 1 ), and thus in S 2 every 1 formula is equivalent to a strict formula (in which all non-sharply-bounded quanti ers are in front).

The Descriptive Complexity Approach to LOGCFL

by Clemens Lautemann , Pierre McKenzie, Thomas Schwentick, Heribert Vollmer , 1998
"... Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory ..."
Abstract - Cited by 9 (4 self) - Add to MetaCart
Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with major...

Programs Over Semigroups of Dot-Depth One

by Alexis Maciel, Pierre Péladeau, Denis Thérien - THEORETICAL COMPUTER SCIENCE , 1996
"... The notion of a p-variety arises in the algebraic approach to Boolean circuit complexity. It has great signi cance, since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p-varieties. In this paper, we prove that semigroups ..."
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The notion of a p-variety arises in the algebraic approach to Boolean circuit complexity. It has great signi cance, since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p-varieties. In this paper, we prove that semigroups of dot-depth one form a pvariety. This example has the following implication: if a Boolean combination of 1 formulas, using arbitrary numerical predicates, de nes a regular language, one can then nd an equivalent 1 formula all of whose numerical predicates are regular.

Lower bounds for bounded depth Frege proofs via Buss-Pudlák games

by Eli Ben-Sasson, Prahladh Harsha - ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY , 2003
"... We present a simple proof of the bounded-depth Frege lower bounds of Pitassi et. al. and Krajicek et. al. for the pigeonhole principle. Our method uses the interpretation of proofs as two player games given by Pudlak and Buss. Our lower bound is conceptually simpler than previous ones, and relies on ..."
Abstract - Cited by 2 (1 self) - Add to MetaCart
We present a simple proof of the bounded-depth Frege lower bounds of Pitassi et. al. and Krajicek et. al. for the pigeonhole principle. Our method uses the interpretation of proofs as two player games given by Pudlak and Buss. Our lower bound is conceptually simpler than previous ones, and relies on tools and intuition that are well-known in the context of computational complexity. This makes the lower bound of Pitassi et. al. and Krajicek et. al. accessible to the general computational complexity audience. We hope this new view will open new directions for research in proof complexity.