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26
Counting quantifiers, successor relations, and logarithmic space
 JCSS
, 1997
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The Complexity of Iterated Multiplication
 INFORMATION AND COMPUTATION
, 1995
"... For a monoid G, the iterated multiplication problem is the computation of the product of n elements from G. By refining known completeness arguments, we show that as G varies over a natural series of important groups and monoids, the iterated multiplication problems are complete for most natural, lo ..."
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Cited by 44 (4 self)
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For a monoid G, the iterated multiplication problem is the computation of the product of n elements from G. By refining known completeness arguments, we show that as G varies over a natural series of important groups and monoids, the iterated multiplication problems are complete for most natural, lowlevel complexity classes. The completeness is with respect to "firstorder projections"  lowlevel reductions that do not obscure the algebraic nature of these problems.
The Permanent Requires Large Uniform Threshold Circuits
, 1999
"... We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the unif ..."
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Cited by 36 (9 self)
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We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constantdepth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.
Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem
 Journal of Computer and System Sciences
"... We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions. ..."
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Cited by 28 (12 self)
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We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions.
The Complexity of Satisfiability Problems: Refining Schaefer’s Theorem
 J. COMPUT. SYS. SCI
"... ... problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and ..."
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Cited by 24 (8 self)
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... problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.
On the Structure of Complete Sets
 IN PROCEEDINGS 9TH STRUCTURE IN COMPLEXITY THEORY
, 1994
"... The many types of resource bounded reductions that are both object of study and research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The ..."
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Cited by 20 (1 self)
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The many types of resource bounded reductions that are both object of study and research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The study of its structure can reveal properties that are general in that complexity class, and the study of the structure of complete sets in different classes can reveal secrets about the relation between these classes. The research into all sorts of aspects and properties of complete sets has been and will be a major topic in structural complexity theory. In this expository paper we review the progress that has been made in recent years on selected topics of the study of complete sets.
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completene ..."
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Cited by 14 (1 self)
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In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...
The isomorphism conjecture for constant depth reductions
 Journal of Computer and System Sciences
"... For any class C closed under TC 0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under uuniform AC 0 reductions are isomorphic under uuniform AC 0computable isomorphisms. 1 ..."
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Cited by 4 (0 self)
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For any class C closed under TC 0 reductions, and for any measure u of uniformity containing Dlogtime, it is shown that all sets complete for C under uuniform AC 0 reductions are isomorphic under uuniform AC 0computable isomorphisms. 1
L.: Experiments with reduction finding
 In: Proc. SAT 2013
"... Abstract. Reductions are perhaps the most useful tool in complexity theory and, naturally, it is in general undecidable to determine whether a reduction exists between two given decision problems. However, asking for a reduction on inputs of bounded size is essentially a Σ p 2 problem and can in pri ..."
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Abstract. Reductions are perhaps the most useful tool in complexity theory and, naturally, it is in general undecidable to determine whether a reduction exists between two given decision problems. However, asking for a reduction on inputs of bounded size is essentially a Σ p 2 problem and can in principle be solved by ASP, QBF, or by iterated calls to SAT solvers. We describe our experiences developing and benchmarking automatic reduction finders. We created a dedicated reduction finder that does counterexample guided abstraction refinement by iteratively calling either a SAT solver or BDD package. We benchmark its performance with different SAT solvers and report the tradeoffs between the SAT and BDD approaches. Further, we compare this reduction finder with the direct approach using a number of QBF and ASP solvers. We describe the tradeoffs between the QBF and ASP approaches and show which solvers instances. It turns out that even stateoftheart perform best on our Σ p 2 solvers leave a large room for improvement on problems of this kind. We thus provide our instances as a benchmark for future work on Σ p 2 solvers. 1
For completeness, sublogarithmic space is no space
"... It is shown that for any class C closed under lineartime reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under firstorder reductions. ..."
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It is shown that for any class C closed under lineartime reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under firstorder reductions.