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45
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defin ..."
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Cited by 129 (19 self)
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In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].
Relationships Among PL, L, and the Determinant
, 1996
"... Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterizati ..."
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Cited by 35 (8 self)
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Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspacebounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.
A Uniform Circuit Lower Bound for the Permanent
 SIAM Journal on Computing
, 1994
"... We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is ..."
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Cited by 28 (9 self)
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We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is still unknown if there is any set in Ntime #2 n O#1# # that does not have nonuniform ACC circuits.
Logspace and Logtime Leaf Languages
, 1996
"... The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all ..."
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Cited by 25 (2 self)
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The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all x for which the leaf string of M is contained in Y . In this way, in the context of polynomial time computation, leaf languages were shown to capture many complexity classes. In this paper, we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation. We show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages, capturing also many classes within P. In contrast, logtime leaf languages basically behave like logtime reducibilities. Both cases are more subtle to handle than the polynomial time case. We also raise the issue of balanced versus nonbalanced comp...
Are there Hard Examples for Frege Systems?
"... It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. S ..."
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Cited by 25 (2 self)
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It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speedup of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl. It is
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
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Cited by 24 (3 self)
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Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Algorithms for Boolean Formula Evaluation and for Tree Contraction
 Arithmetic, Proof Theory and Computational Complexity
, 1991
"... This paper presents a new, simpler ALOGTIME algorithm for the Boolean sentence value problem (BSVP). Unlike prior work, this algorithm avoids the use of postfixlongeroperandfirst formulas. This paper also shows that treecontraction can be made ALOGTIME uniform. ..."
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Cited by 20 (0 self)
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This paper presents a new, simpler ALOGTIME algorithm for the Boolean sentence value problem (BSVP). Unlike prior work, this algorithm avoids the use of postfixlongeroperandfirst formulas. This paper also shows that treecontraction can be made ALOGTIME uniform.
Alogtime Algorithms for Tree Isomorphism, Comparison, and Canonization
 In Computational Logic and Proof Theory, 5th Kurt Godel Colloquium'97, Lecture Notes in Computer Science #1289
, 1997
"... The tree isomorphism problem is the problem of determining whether two trees are isomorphic. The tree canonization problem is the problem of producing a canonical tree isomorphic to a given tree. The tree comparison problem is the problem of determining whether one tree is less than a second tre ..."
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Cited by 19 (0 self)
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The tree isomorphism problem is the problem of determining whether two trees are isomorphic. The tree canonization problem is the problem of producing a canonical tree isomorphic to a given tree. The tree comparison problem is the problem of determining whether one tree is less than a second tree in a natural ordering on trees. We present alternating logarithmic time algorithms for the tree isomorphism problem, the tree canonization problem and the tree comparison problem. As a consequence, there is a recursive enumeration of the alternating log time tree problems.
Logspace Versions of the Theorems of Bodlaender and Courcelle
, 2010
"... Bodlaender’s Theorem states that for every k there is a lineartime algorithm that decides whether an input graph has tree width k and, if so, computes a widthk tree composition. Courcelle’s Theorem builds on Bodlaender’s Theorem and states that for every monadic secondorder formula φ and for eve ..."
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Cited by 19 (1 self)
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Bodlaender’s Theorem states that for every k there is a lineartime algorithm that decides whether an input graph has tree width k and, if so, computes a widthk tree composition. Courcelle’s Theorem builds on Bodlaender’s Theorem and states that for every monadic secondorder formula φ and for every k there is a lineartime algorithm that decides whether a given logical structure A of tree width at most k satisfies φ. We prove that both theorems still hold when “linear time ” is replaced by “logarithmic space.” The transfer of the powerful theoretical framework of monadic secondorder logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the logspace world.
Rudimentary Reductions Revisited
 Information Processing Letters 40
, 1991
"... We show that logbounded rudimentary reductions #de#ned and studied by Jones in 1975# characterize Dlogtimeuniform AC . ..."
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Cited by 18 (9 self)
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We show that logbounded rudimentary reductions #de#ned and studied by Jones in 1975# characterize Dlogtimeuniform AC .