Descriptive Complexity of #P Functions

Saluja, Sanjeev; Subrahmanyam, K. V.; Thakur, M.N.

doi:10.1006/jcss.1995.1039

Cited by 40 publications

(33 citation statements)

References 20 publications

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“…Finally, let us remark that subclasses of #P that contain functions with easy decision version have been defined in [SST95] and [HHKW05]. It is an interesting open question to see how these classes relate to TotP and #PE.…”

Section: Discussionmentioning

confidence: 97%

The Complexity of Counting Functions with Easy Decision Version

Pagourtzis

Zachos

2006

Lecture Notes in Computer Science

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Abstract. We investigate the complexity of counting problems that belong to the complexity class #P and have an easy decision version. These problems constitute the class #PE which has some well-known representatives such as #Perfect Matchings, #DNF-Sat, and NonNegative Permanent. An important property of these problems is that they are all #P-complete, in the Cook sense, while they cannot be #P-complete in the Karp sense unless P = NP.We study these problems in respect to the complexity class TotP, which contains functions that count the number of all paths of a PNTM. We first compare TotP to #P and #PE and show that FP ⊆ TotP ⊆ #PE ⊆ #P and that the inclusions are proper unless P = NP.We then show that several natural #PE problems -including the ones mentioned above -belong to TotP. Moreover, we prove that TotP is exactly the Karp closure of self-reducible functions of #PE. Therefore, all these problems share a remarkable structural property: for each of them there exists a polynomial-time nondeterministic Turing machine which has as many computation paths as the output value.

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Section: Discussionmentioning

confidence: 97%

The Complexity of Counting Functions with Easy Decision Version

Pagourtzis

Zachos

2006

Lecture Notes in Computer Science

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“…Saluja et al [16] have presented a logical characterisation of the class #P (and of some of its subclasses), much in the spirit of Fagin's logical characterisation of NP [6]. In their framework, a counting problem is identified with a sentence ϕ in first-order logic, the objects being counted with models of ϕ.…”

Section: A Logical Characterisation Of #Bis and Its Relativesmentioning

confidence: 99%

“…By placing a syntactic restriction on ϕ, it is possible to identify a subclass #RH 1 of #P whose complete problems include all the ones mentioned in Theorem 5. We follow as closely as possible the notation and terminology of [16], and direct the reader to that article for further information and clarification. A vocabulary is a finite set σ = { R 0 , .…”

Section: A Logical Characterisation Of #Bis and Its Relativesmentioning

confidence: 99%

“…At the other extreme, Saluja et al[16] propose a very demanding notion of approximation-preserving reduction, which is probably not suitable for our purposes.…”

mentioning

confidence: 99%

“…Lemma 22 is due to Mike Paterson. We thank Dominic Welsh for telling us about reference [16] and Marek Karpinski for stimulating discussions on the topic of approximation-preserving reducibility.…”

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confidence: 99%

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The Relative Complexity of Approximate Counting Problems

et al. 2003

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Two natural classes of counting problems that are interreducible under approximation-preserving reductions are: (i) those that admit a particular kind of efficient approximation algorithm known as an "FPRAS", and (ii) those that are complete for #P with respect to approximation-preserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically defined subclass of #P.

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Logical Definability of Counting Functions

Compton

Grädel

1996

Journal of Computer and System Sciences

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The relationship between counting functions and logical expressibility is explored. The most well studied class of counting functions is *P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine. For a logic L, *L is the class of functions on finite structures counting the tuples (T , cÄ ) satisfying a given formula (T , cÄ ) in L. Saluja, Subrahmanyam, and Thakur showed that on classes of ordered structures *FO=*P (where FO denotes first-order logic) and that every function in * 1 has a fully polynomial randomized approximation scheme. We give a probabilistic criterion for membership in * 1 . A consequence is that functions counting the number of cliques, the number of Hamilton cycles, and the number of pairs with distance greater than two in a graph, are not contained in * 1 . It is shown that on ordered structures * 1 1 captures the previously studied class spanP. On unordered structures *FO is a proper subclass of *P and * 1 1 is a proper subclass of spanP; in fact, no class *L contains all polynomial-time computable functions on unordered structures. However, it is shown that on unordered structures every function in *P is identical almost everywhere with some function *FO, and similarly for * 1 1 and spanP. Finally, we discuss the closure properties of *FO under arithmetical operations. ]

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Descriptive Complexity of #P Functions

Cited by 40 publications

References 20 publications

The Complexity of Counting Functions with Easy Decision Version

The Complexity of Counting Functions with Easy Decision Version

The Relative Complexity of Approximate Counting Problems

Logical Definability of Counting Functions

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