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32
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 defined by ..."
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Cited by 126 (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 first-order formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level 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 logspace-bounded machine. #This class of functions is known as #L.# By using that characterizati ..."
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Cited by 27 (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 logspace-bounded 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 logspace-bounded #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 23 (10 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 x for whi ..."
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Cited by 21 (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 non-balanced 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 19 (1 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 speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.
Rudimentary Reductions Revisited
- Information Processing Letters 40
, 1991
"... We show that log-bounded rudimentary reductions #de#ned and studied by Jones in 1975# characterize Dlogtime-uniform AC . ..."
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Cited by 18 (9 self)
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We show that log-bounded rudimentary reductions #de#ned and studied by Jones in 1975# characterize Dlogtime-uniform AC .
On the Existence of Hard Sparse Sets under Weak Reductions
, 1996
"... Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more gene ..."
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Cited by 17 (4 self)
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Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more general reducibilities. Our main results are as follows. (1) If there exists a sparse set that is hard for P under bounded truth-table reductions, then P = NC 2 . (2) If there exists a sparse set that is hard for P under randomized logspace reductions with one-sided error, then P = Randomized LOGSPACE. (3) If there exists an NP-hard sparse set under randomized polynomial-time reductions with one-sided error, then NP = RP. (4) If there exists a 2 (log n) O(1) -sparse hard set for P under truth-table reductions, then P ` DSPACE[(logn) O(1) ]. As a by-product of (4), we obtain a uniform O(log 2 n log log n) time parallel algorithm for computing the rank of a 2 log 2 n \Theta n matrix o...
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 postfix-longer-operand-first formulas. This paper also shows that tree-contraction can be made ALOGTIME uniform. ..."
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Cited by 16 (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 postfix-longer-operand-first formulas. This paper also shows that tree-contraction 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 13 (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.
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 10 (4 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

