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13
Infinitary Logics and 01 Laws
 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 gametheoretic characterizat ..."
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Cited by 43 (4 self)
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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 gametheoretic characterization of the expressive power of the logic. Finally, we study asymptotic probabilities of properties 1! on nite structures. We show that the 01 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 01 laws for in nitary logics.
Pseudorandom Generators in Propositional Proof Complexity
 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 ..."
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Cited by 39 (7 self)
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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 NisanWigderson 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
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted 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. ..."
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Cited by 30 (7 self)
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We present in a uniform manner simple twosorted 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.
Definable Relations and FirstOrder Query Languages over Strings
"... We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary relati ..."
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Cited by 22 (8 self)
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We study analogs of classical relational calculus in the context of strings. We start by studying string logics. Taking a classical modeltheoretic approach, we fix a set of string operations and look at the resulting collection of definable relations. These form an algebra  a class of nary 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 modeltheory 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,
Quantum Circuits: Fanout, Parity, and Counting
 In Los Alamos Preprint archives
, 1999
"... Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constantdepth circuits with AND and OR gates of arbitrary fanin, and QACC 0 [q], where nary 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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Cited by 17 (1 self)
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Abstract. We propose definitions of QAC 0, the quantum analog of the classical class AC 0 of constantdepth circuits with AND and OR gates of arbitrary fanin, and QACC 0 [q], where nary 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
Lower Bounds for Matrix Product, in Bounded Depth Circuits with Arbitrary Gates
 IN PROCEEDINGS OF THE THIRTYTHIRD ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2001
"... We prove superlinear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth Boolean circuits with arbitrary gates. The bou ..."
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Cited by 16 (5 self)
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We prove superlinear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth Boolean circuits with arbitrary gates. The bounds apply for several explicit functions, and, most importantly, for matrix product. In particular, we obtain the following results: 1. We show that the number of edges in any constant depth arithmetic circuit for matrix product (over any eld) is superlinear in m 2 (where m m is the size of each matrix). That is, the lower bound is superlinear in the number of input variables. Moreover, if the circuit is bilinear the result applies also for the case where the circuit gets for free any product of two linear functions.
Counting, Fanout, And The Complexity Of Quantum Acc
, 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 ..."
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Cited by 15 (2 self)
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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 Descriptive Complexity Approach to LOGCFL
, 1998
"... Building upon the known generalizedquantifierbased firstorder 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 ..."
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Cited by 11 (5 self)
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Building upon the known generalizedquantifierbased firstorder 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 contextfree languages, and we obtain the surprising result that a variant of Greibach's "hardest contextfree language" is LOGCFLcomplete under quantifierfree BITfree 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 firstorder logic with majority of pairs is strictly more expressive than firstorder with major...
The strength of replacement in weak arithmetic
 Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science
, 2004
"... The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x
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Cited by 10 (3 self)
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The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x<aφ(i,x) → ∃w ∀i<aφ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a strict Σb1 formula (in which all nonsharplybounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S1 2 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(ΣB 0), where V 0 (essentially IΣ 1,b 0) is the twosorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb 0), assuming
Programs Over Semigroups of DotDepth One
 THEORETICAL COMPUTER SCIENCE
, 1996
"... The notion of a pvariety 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 pvarieties. In this paper, we prove that semigroups ..."
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Cited by 4 (0 self)
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The notion of a pvariety 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 pvarieties. In this paper, we prove that semigroups of dotdepth 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.