D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which are (1) taken from a (complexity) class defined via space or time restrictions for Turing machines, or (2) taken from a (formal language) class of the Chomsky hierarchy. The power of nondeterministic Turing machines whose acceptance is given by a leaf language is well studied, see, e.g. [4, 14, 10, 12]; recently the model has also been applied to Boolean circuits [6] However, in the context of the probably most basic type of computation device, the finite automaton, leaf languages have not been considered so far. The present paper closes this gap. As had to be expected, our results differ ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....sets in other complexity classes. We give in this section a generalization of the previous technique providing a tool to extend automatically this completeness result. This generalization is based on the structure of promise classes. In the classical leaf language or tree structure approach [7, 29, 6], promises are restricted to be predicates on computation trees (respectively leaf strings) of nondeterministic polynomial time Turing machines. We consider a more general approach that is more suited for our purposes, and in some cases allows a more direct characterization of a promise class. In ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....and only consider equivalence relations in computation trees. Under which conditions complexity classes (of sets as well as functions) defined by an arbitrary equivalence relation coincide with classes obtained from the cluster relation Or, what are the connections to the leaf language approach [BCS92, Ver93] In any case this could also be one way in better understanding the world inside non determinism. Acknowledgments. I am very grateful to Gerd Wechsung for supervising my master s thesis, and Harald Hempel for some helpful hints. ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....x) 2 Y g : The class Leaf NC 1 (Y ) is the class of languages recognized by uniform polynomial length programs with leaf language Y . 5. 1 Padding techniques Leaf language classes have been studied in the context of polynomial time, logarithmic space, and logarithmic time computations [8, 12, 13, 14]. The numerous characterizations obtained in those papers make use of padding. Here we observe that in some sense the padding can be done by an automaton. This allows transferring in one blow many known characterizations to the Leaf NC 1 ( Delta) setting. Here is an automaton that is able to ....

D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....which are (1) taken from a (complexity) class defined via space or time restrictions for Turing machines, or (2) taken from a (formal language) class of the Chomsky hierarchy. The power of nondeterministic Turing machines whose acceptance is given by a leaf language is wellstudied, see, e.g. BCS92, Ver93, HLS 93, JMT96] More recently the model has also been applied to Boolean circuits, see [CMTV98] formally, in this latter model so called programs over automata were used as leaf string generators in the case of language decision these programs are known to yield exactly the power ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....which are (1) taken from a (complexity) class defined via space or time restrictions for Turing machines, or (2) taken from a (formal language) class of the Chomsky hierarchy. The power of nondeterministic Turing machines whose acceptance is given by a leaf language is wellstudied, see, e.g. BCS92, Ver93, HLS 93, JMT96] More recently the model has also been applied to Boolean circuits, see [CMTV98] formally, in this latter model so called programs over automata were used as leaf string generators in the case of language decision these programs are known to yield exactly the power ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....1 (Y ) is the class of languages recognized by uniform polynomial length programs 3 with leaf language Y . o o . o o o o Leaf . o , o Leaf o o o o o o . 15 . Leaf language classes have been studied in the context of polynomial time, logarithmic space, and logarithmic time computations [12, 19, 20, 21]. The numerous characterizations obtained in those papers make use of padding. Here we observe that in some sense the padding can be done by an automaton. This allows transferring in one blow many known characterizations to the Leaf NC 1 (1) setting. Consider the following automaton, in which ....

D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....circuits are getting increasingly popular among complexity theorists (see e.g. AH94, Bar92] One of the goals of this paper is to make the above connection precise. We relate quasipolynomial size small depth circuits to polynomial time classes using the concept of the so called leaf languages [BCS92, Ver93] This will allow us to prove an equivalence between (a) a collapse of quasipolynomial size circuit classes, and (b) a collapse of polynomial time classes holding for all oracles. This means that generally a separation of circuit classes is as difficult as constructing a separating oracle ....

....polynomial time Turing machine classes. For this, we will use padding techniques (see e.g. Boo74] in a slightly elaborated form, which has become known under the name leaf language approach to the examination of complexity classes. Leaf languages were first defined and examined independently in [BCS92] and [Ver93] and later considered in a number of papers, e.g. HLS 93, JMT94, HVW96] cf. also the recent textbook [Pap94, 4 pp. 504f] We use the following definition: Let A; B be two sets. We say that A is polynomial time bit reducible to B, A p;bit m B, iff there exist two functions f; ....

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D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....has been examined (see [Lin66, Ste92, Got95] or Chapter 10 of the textbook [EF95] In the field of computational complexity characterizations of complexity classes by so called leaf languages have been studied intensively. This approach was introduced by Bovet, Crescenzi, and Silvestri in 1992 [BCS92] as a unified approach to define complexity classes. Consider a polynomial time nondeterministic Turing machine M that prints a value on every path of its computation on some given input x. By imposing an order upon the machine s nondeterministic choices (e.g. based on the way the Turing program ....

....is evaluated. This evaluation scheme should therefore be given in a way free from computational aspects whatsoever. This goal is achieved using finite model theory to specify leaf languages. 2 2 Preliminaries The definability of complexity classes via leaf languages was introduced in [BCS92, HLS 93] see also the recent textbook [Pap94] Let M be a polynomial time bounded nondeterministic Turing machine. Given an input x, such a machine produces on every path a symbol from some finite alphabet Gamma. Let fi M (x) be the string of the so produced symbols (based on the natural ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....by so called leaf languages, essentially nothing else than conditions on the sequence of leaves in computation trees. This concept was developed by Papadimitriou and Sipser around 1979 when teaching a complexity class at MIT [Pap94b] It was later rediscovered and published independently in [BCS92, Ver93] and has since then been used actively in the study of complexity classes mostly in between NC 1 and PSPACE. In the circuit world, Boolean circuits with gates for multiplication in certain algebraic structures, mostly monoids or groupoids (hence these gates are called monoidal or ....

....terminology of Papadimitriou [Pap94a] while those which cannot are semantic classes. This computation model was introduced, as already mentioned above, by Papadimitriou and Sipser around 1979, and published for the first time by Bovet, Crescenzi, and Silvestri, and independently by Vereshchagin [BCS92, Ver93] see also the textbook [Pap94a, pp. 504f] Let C be a class of languages. The class BLeaf P (C) consists of the union over all B 2 C of the classes BLeaf P (B) In a sequence of papers ( HLS 93, JMT94, CMTV98] and more) the complexity of the classes BLeaf P (C) was studied as ....

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D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

.... computation trees of nondeterministic polynomial time machines (see [18] for a survey) We can roughly distinguish three mechanisms: predicate classes, where the acceptance condition can depend on the complete tree [3] leaf languages, where the acceptance condition only depends on the leaf word [4,19]; and locally definable acceptance types, where the acceptance condition only depends on local, k valued functions in the tree [16,17] Clearly, the second and the third mechanisms are special cases of the first one. In all cases, however, in principle there are two possible agreements with ....

Daniel P. Bovet, Pierluigi Crescenzi, and Riccardo Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104(2):263--283, 1992.

....which are (1) taken from a (complexity) class defined via space or time restrictions for Turing machines, or (2) taken from a (formal language) class of the Chomsky hierarchy. The power of nondeterministic Turing machines whose acceptance is given by a leaf language is well studied, see, e.g. [4, 14, 10, 12]; recently the model has also been applied to Boolean circuits [6] However, in the context of the probably most basic type of computation device, the finite automaton, leaf languages have not been considered so far. The present paper closes this gap. As had to be expected, our results differ ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....characterization easily provides a nondeterministic logspace decision algorithm for L 1 . There is a close connection between concatenation hierarchies and complexity classes, both related via the so called leaf language approach to define complexity classes. This approach was introduced in [BCS92, Ver93] and led to a number of interesting results (cf. HLS 93, JMT94, BV98, CHVW98] In particular in [BV98] it was shown that taking the languages from L k Gamma1=2 as leaf languages yields exactly the k th class of the polynomial time hierarchy. In the last section we state a result of ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....has been examined (see [Lin66, Ste92, Got95] or Chapter 10 of the textbook [EF95] In the field of computational complexity characterizations of complexity classes by so called leaf languages have been studied intensively. This approach was introduced by Bovet, Crescenzi, and Silvestri in 1992 [BCS92] as a unified approach to define complexity classes. Consider a polynomial time nondeterministic Turing machine M that prints a value on every path of its computation on some given input x. By imposing an order upon the machine s nondeterministic choices (e.g. based on the way the Turing ....

....computation is evaluated. This evaluation scheme should therefore be given in a way free from computational aspects whatsoever. This goal is achieved using finite model theory to specify leaf languages. 2 Preliminaries The definability of complexity classes via leaf languages was introduced in [BCS92, HLS 93] see also the recent textbook [Pap94] Let M be a polynomial timebounded nondeterministic Turing machine. Given an input x, such a machine produces on every path a symbol from some finite alphabet Gamma. Let fi M (x) be the string of the so produced symbols (based on the natural ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....PSPACE have been obtained, using nondeterministic polynomial time Turing machines with some kind of global acceptance condition. A very general framework, allowing characterizations of nearly all introduced complexity classes in this area, has been developed by Bovet, Crescenzi, and Silvestri in [BCS91, BCS92]. In [Her92a] see also [Her92b] a polynomial time machine scheme for such characterizations is introduced using a finite set of k valued functions, the so called locally definable acceptance types. A special case of finite acceptance types appears, if we just add up the number of accepting ....

....existence of oracle separations in a wide variety of cases. The organization of the paper is as follows: In Section 2 we give the formal definition of a complexity class with finite acceptance type, and we recall the connection between this kind of complexity and the concept of leaf languages from [BCS91, BCS92] (see also [HLSVW93] Section 3 introduces hypergraph sequences. We prove some easy facts on existence or nonexistence of special types of such sequences. Section 4 connects the existence question for hypergraph sequences with the inclusionship question for certain counting classes. Finally, in ....

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D. P. Bovet, P. Crescenzi, R. Silvestri, A uniform approach to define complexity classes, Theoretical Computer Science 104 (1992), pp. 263--283.

....in the following, for every set A, we have sA 2 N. It should be remarked however that all our results below (except possibly Theorem 5.3) hold also for the above original encoding without neutral element. Leaf Languages The definability of complexity classes via leaf languages was introduced in (Bovet et al. 1992; Vereshchagin, 1993) see also the recent textbook (Papadimitriou, 1994) Let M be a polynomial time bounded nondeterministic Turing machine. Given an input x, such a machine produces on every path a symbol from some finite alphabet Gamma . Let fi M (x) be the string of the so produced symbols ....

Bovet, D. P., Crescenzi, P., and Silvestri, R. (1992). A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283.

....the fact that the leaf language class obtained is a refinement of the leaf language classes defined using log space or polynomial time machines. 5. 1 Padding techniques Leaf language classes have been studied in the context of polynomial time, logarithmic space, and logarithmic time computations [12, 19, 20, 21]. The numerous characterizations obtained in those papers make use of padding. Here we observe that in some sense the padding can be done by an automaton. This allows transferring in one blow many known characterizations to the Leaf NC 1 ( Delta) setting. Consider the following automaton, in ....

D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....Sect. 5 it should be remarked that we know of no result in the literature where an oracle was constructed building on a circuit lower bound, which could not be obtained using our general theorem. Our uniform diagonalization theorem is an application of the general oracle construction method from [BCS92, Ver93] In order to be able to use it we point out some relations between polynomial time classes and constant depth classes which a reader interested in leaf language characterizations [HLS 93, JMT94, BS97] might find interesting in its own. We then turn to the nagging question [For97] if ....

.... PhiP and give equivalent statements in terms of circuits. 2 Preliminaries We assume the reader is familiar with basic complexity theory notions; refer to the standard literature [BC94, Pap94, BDG95] In the early nineties a general method to obtain oracle separations was given independently in [BCS92, Ver93] This became later known as the leaf language approach to the definition of complexity classes [HLS 93, Pap94] In the sequel we use it as our main technical tool. For this, let M be a nondeterministic Turing machine, halting on every path, with some order on the nondeterministic ....

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D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

.... study has been based on idea of defining classes in terms of structural properties of languages corresponding to the recognition of words by non deterministic machines (the so called leaf languages ) Using these languages it was possible to get new results of separation for relativized classes [111, 112]. 1.5 Counting and Decomposition Problems on Languages and Graphs The complexity of counting problems for languages both from the sequential and parallel point of view has been analyzed in [9, 72, 78, 329, 330, 357] In particular , we found a parallel efficient algorithm for computing the ....

D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

....was recently strengthened by Hemaspaandra and Zimand [HZ96] to the strongest result possible: Relative to a random oracle R, NP R contains a P R balanced immune set with probability 1. See these references for the notions not defined here. Many more immunity results are known see, e.g. [HM83,SB84,Bal85,BR88,TvEB89, BJY90,Ko90,Bru92,EHTY92,BCS92,HRW97]. Most important for the present paper are the results and (circuit based) techniques of Ko [Ko90] and Bruschi [Bru92] In particular, both papers provide relativizations in which the levels of the polynomial hierarchy (PH) separate with immunity, Bruschi s results being somewhat stronger and more ....

....reject simultaneously. Gre91,Bei94] Regarding relativized strong separations, however, the only results known are the above mentioned result that for some A, PhiP A contains a PH A immune set [Ko90, Bru92] and that for some B, NP B (and thus PH B ) has a PhiP B immune set [BCS92]. In this paper, we strengthen to (relativized) strong separations all the other simple separations that are possible among pairs of classes chosen from fPH;PP; PhiP; C=Pg. Just as Balc azar and Russo [Bal85,BR88] exhaustively settled (in suitable relativizations) all possible immunity and ....

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D. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104(2):263--283, 1992.

....operator A Delta . In fact we will introduce the more general notion of promise dot operators for which the BP operator is an example. In Section 3 we will study properties of dot operators. We will see for example that dot operators turn out to be a refinement of the leaf language concept (see [5, 31, 10, 12, 14, 19, 4], and the recent survey [32] because the class determined by a leaf language A equals A Delta P. Furthermore we show that dot operators are closed under composition, and that complementary dot operators keep the property of classes to have a many one complete set. In Section 4 we show that for ....

....complete set. In Section 4 we show that for two languages A; B it holds: A Delta C is a subset of B Delta C for all classes C if and only if A is reducible to B by polylog time uniform monotone projections. As a corollary we get the main result (for the complementary case) of Bovet et al. [5] about the relativization of leaf language classes. In Section 5 we show that dot operators are able to represent not only classes but reducibilities. For example, we will construct a language T such that for all classes C the class T Delta C is the polynomial time Turing closure P(C) of C. In ....

[Article contains additional citation context not shown here]

D. P. Bovet, P. Crescenzi, R. Silvestri. A uniform approach to define complexity classes, Theoretical Computer Science 104, 1992, pp. 263--283.

....operator A Delta . In fact we will introduce the more general notion of promise dot operators for which the BP operator is an example. In Section 3 we will study properties of dot operators. We will see for example that dot operators turn out to be a refinement of the leaf language concept (see [5, 31, 10, 12, 14, 19, 4], and the recent survey [32] because the class determined by a leaf language A equals A Delta P. Furthermore we show that dot operators are closed under composition, and that complementary dot operators keep the property of classes to have a many one complete set. In Section 4 we show that for ....

....complete set. In Section 4 we show that for two languages A; B it holds: A Delta C is a subset of B Delta C for all classes C if and only if A is reducible to B by polylog time uniform monotone projections. As a corollary we get the main result (for the complementary case) of Bovet et al. [5] about the relativization of leaf language classes. In Section 5 we show that dot operators are able to represent not only classes but reducibilities. For example, we will construct a language T such that for all classes C the class T Delta C is the polynomial time Turing closure P(C) of C. In ....

[Article contains additional citation context not shown here]

D. P. Bovet, P. Crescenzi, R. Silvestri. A uniform approach to define complexity classes, Theoretical Computer Science 104, 1992, pp. 263--283.

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D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

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D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

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D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263--283, 1992.

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