Leivant 1989 Daniel Leivant, "Descriptive Characterizations of Computational Complexity", Journal of Computer and System Sciences 39 (1989), 51--83.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in NP. In fact, over finite strings, even the class ESO(8 ) expresses all languages in NP; this follows from the more general result that over finite successor structures, i.e. finite structures equipped with a successor predicate, every ESO sentence is equivalent to some ESO(8 ) sentence [33, 14]. Main Problems Studied. Combining and extending the results of Buchi and Fagin, it is natural to ask: What about (nonmonadic) prefix classes ESO(Q) over finite strings We know by Fagin s theorem that all these classes describe languages in NP. But there is a large spectrum of languages ....

.... regular ESO prefix class is over strings either equivalent to full MSO, or is contained in first order logic, in fact, in FO(9 8) Theorem 12.3) We further show that there is a unique minimal ESO prefix class which captures NP, namely ESO(8 ) Proposition 10.6) Our proof uses results in [33, 14] and well known hierarchy theorems. 7) We give a precise characterization of those regular prefix classes of ESO which, over strings, are closed under complementation. In particular, we show that any nontrivial regular class ESO(Q) is closed under complementation iff some quantifier prefix Q 2 Q ....

[Article contains additional citation context not shown here]

D. Leivant. Descriptive Characterizations of Computational Complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....On the other hand, with non monadic predicates allowed ESO has much higher expressivity. In particular, by Fagin s result [9] we have the following. Proposition 2.3 (Fagin s Theorem) ESO captures NP. This theorem can be sharpened to various fragments of ESO. In particular, by Leivant s results [14, 7], in the presence of a successor and constants min and max, the fragment ESO(8 ) captures NP; thus, ESO(8 ) expresses all languages in NP. 3 Recent Results on ESO over Strings Combining and extending the results of Buchi and Fagin, it is natural to ask: What about (nonmonadic) prefix ....

....Moreover, in [8] it is established that any regular ESO prefix class is over strings either equivalent to full MSO, or is contained in first order logic, in fact, in FO(9 8) It is further shown that ESO(8 ) is the unique minimal ESO prefix class which captures NP. The proof uses results in [14, 7] and well known hierarchy theorems. The main results of [8] are summarized in Figure 1. In this figure, the ESO prefix classes are divided into four regions. The upper two contain all classes that express nonregular languages, and thus also NP complete languages. The uppermost region contains ....

D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....graph exactly once. 5 Automata The conditions (3) of Theorems 4.1 and 4.2 give rise to a generalized form of automata. In this section we are going to introduce automata models for FO logic and monadic 1 logic, respectively. Vaguely similar machine models are known for relational databases [Lei89, GPPdB94, AV95]. Informally the FO automaton works as follows. First it nondeterministically pebbles vertices b 1 ; b g of its input structure A, for some g. Then, for every vertex a of A, it inspects in a constant number of steps (alternating between nondeterminism and parallelism) the neighbourhood of ....

D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....the class of sets of nite models of all second order formulas. Finding an elegant descriptive style characterization of P proved more illusive. One such characterization of P uses the rst order logic augmented with the successor relation and the least xed point operator [23, 13] Later Leivant [17, 18] found a second order characterization of P using the notion of controlled computational formula , which is related to Horn formula. The motivation for using Horn formulas comes from the existence of a simple polynomial time algorithm for solving the satis ability problem for propositional Horn ....

Daniel Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51-83, 1989.

....the existence of a complete problem of PH would imply its collapse. The subdiscipline of database theory and finite model theory dealing with the description of the expressive power of query languages and related logical formalisms via complexity classes is called descriptive complexity theory [77, 90, 78]. An early foundational result in this field was Fagin s Theorem [58] stating that existential second order logic captures NP. In the eighties and nineties, descriptive complexity theory has become a flourishing discipline with many deep and useful results. To prove that a query language Q ....

D. Leivant. Descriptive Characterizations of Computational Complexity. J. Computer and System Sciences, 39:51--83, 1989. 19

....class of sets of finite models of all second order formulas. Finding an elegant descriptive style characterization of P proved more illusive. One such characterization of P uses the first order logic augmented with the successor relation and the least fixed point operator [23, 13] Later Leivant [17, 18] found a second order characterization of P using the notion of controlled computational formula , which is related to Horn formula. The motivation for using Horn formulas comes from the existence of a simple polynomial time algorithm for solving the satisfiability problem for propositional ....

Daniel Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

.... the class PSPACE and noninflationary fixpoint logic [Var82] cf. AV89] 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While computational devices work on encodings of problems, logic is applied directly to ....

....input relationally. The discerning power of relational machines is best understood by viewing them as an effective fragment of a certain infinitary 3 A closely related idea, of generalizing Turing machines to operate on general structures, goes back to [Fri71] and was investigated extensively in [Lei89a, Lei89b]. 3 logic, studied recently in [KV90a, KV90b] This view yields a precise characterization of the discerning power of relational machines in terms of certain infinite 2 player pebble games. The characterization can then be used by relational machines in order to determine their own discerning ....

[Article contains additional citation context not shown here]

D. Leivant. Descriptive characterization of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

.... structures are typically much harder than their unordered counterparts; see for example [dR84] We focus in this paper on the relational machine of [AV91, AV91c, AVV92a, AVV92b] and the generic machine of [AV91] Section 5 discusses briefly various ancestors and cousins of these devices, from [CH80, Cha81, CH82, Fri71, Lei89a]. 2 FO, fixpoint, and while We review informally three languages which play a central role in our discussion: FO, fixpoint, and while. To begin, we recall some terminology and notation of relational databases [Ull88, Kan91] This is interchangeable with the closely analogous terminology of logic ....

....such a mapping is a query (by the definition of Chandra and Harel) It is easily verified that any query defined by a fap is in logspace. Furthermore, Hartmanis [Har72] and later Gurevich [Gur88] showed that the set of queries computed by fap on ordered structures is exactly logspace. Leivant [Lei89a] reviews these results in the context of a syntactic variant of the fap called sequential on site acceptor. This also provides insight into the connection with FO; since FO is less than logspace even on ordered structures, fap compute non FO queries. On the other hand, there are simple FO queries ....

[Article contains additional citation context not shown here]

D. Leivant. Descriptive characterization of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....1 k ) formula reduces to a 1 k (respectively 1 k ) formula with only one quanti er alternation in the rst order part. Leivant found a simpler normal form for 1 1 formulas over nite successor structures: every such formula reduces to a 1 1 formula with universal rst order part [8]. In Section 2, we give a shorter, simpler and more direct proof of this normal form theorem. Leivant s theorem fails in the case of all nite structures. Moreover, let 1 k (bool) respectively 1 k (bool) be the collection of 1 k (respectively 1 k ) formulas whose rst order parts ....

D. Leivant. Descriptive Characterizations of Computational Complexity. Journal of Computer and System Sciences, 39:51-83, 1989.

....program schemes remain amenable to logical manipulation. Program schemes (of various sorts) originated in the 70s (for example, see [8, 10, 17, 30] without much regard being paid to an analysis of resources, before a closer complexity analysis was undertaken in, mainly, the 80s (for example, see [23, 27, 29, 36]: though the analysis tends to be on ordered nite structures) In [5, 9, 19, 31, 33, 34, 35] the computational power of di erent classes of program schemes, on the class of all nite structures, is compared with the expressive power of logics previously considered in descriptive complexity ....

D. Leivant, Descriptive characterizations of computational complexity, Journal of Computer and System Sciences 39 (1989) 51-83.

....graph exactly once. 5 Automata The conditions (3) of Theorems 4.1 and 4.2 give rise to a generalized form of automata. In this section we are going to introduce automata models for FO logic and monadic 1 1 logic, respectively. Vaguely similar machine models are known for relational databases [Lei89, GPPdB94, AV95]. Informally the FO automaton works as follows. First it nondeterministically pebbles vertices b 1 ; b g of its input structure A, for some g. Then, for every vertex a of A, it inspects in a constant number of steps (alternating between nondeterminism and parallelism) the neighbourhood of ....

D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....graph exactly once. 5 Automata The conditions (3) of Theorems 4.1 and 4.2 give rise to a generalized form of automata. In this section we are going to introduce automata models for FO logic and monadic # 1 1 logic, respectively. Vaguely similar machine models are known for relational databases [Lei89, GPPdB94, AV95]. Informally the FO automaton works as follows. First it nondeterministically pebbles vertices b 1 , b g of its input structure A, for some g. Then, for every vertex a of A, it inspects in a constant number of steps (alternating between nondeterminism and parallelism) the neighbourhood of ....

D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....exactly once. 5 Automata The conditions (3) of Theorems 4.1 and 4.2 give rise to a generalized form of automata. In this section we are going to introduce automata models for FO logic and monadic Sigma 1 1 logic, respectively. Vaguely similar machine models are known for relational databases [Lei89, GPPdB94, AV95]. Informally the FO automaton works as follows. First it nondeterministically pebbles vertices b 1 ; b g of its input structure A, for some g. Then, for every vertex a of A, it inspects in a constant number of steps (alternating between nondeterminism and parallelism) the neighbourhood of ....

D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....by Fagin [Fag74] who discovered that NP coincides with the class of problems that are definable by existential second order sentences on finite structures. Fagin s work sparked a sequence of related investigations by Immerman [Imm86, Imm87, Imm89] Vardi [Var82] Gurevich [Gur84, Gur88] Leivant [Lei89], and other researchers, in which it was established that most major complexity classes can be characterized in terms of logical definability on finite structures. These investigations contributed to the development of descriptive complexity theory as the area of research whose goal is to unveil ....

D. Leivant. Descriptive characterization of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

.... of results detailing logical characterizations of numerous different complexity classes ranging from AC 0 , the class of problems accepted by constant depth polynomial size Boolean circuits, to PSPACE, the class of problems solvable in deterministic polynomial space, and beyond (the references [1, 4, 12, 20, 28, 29, 33, 37, 41, 42, 44, 45, 49] include a selection of such characterizations) It is all very well logically capturing complexity classes; but what can one do with these characterizations For on the face of it, they simply provide translations of (hard) complexity theoretic questions into finite model theory. However, this ....

D. Leivant. Descriptive characterizations of computational complexity. J. Comput. System Sci., 39:51--83, 1989.

....or marker movements ; those remaining stationary are not charged. One classical difference between fingers and pointers is that there is no fixed limit on the number of pointers a program can create. Rather than define a form of the BM analogous to the pointer machines of Schonhage and others [45, 66, 67, 49, 10], we move straight to a model that uses random access addressing, a mechanism usually considered stronger than pointers (for in depth comparisons, see [9, 10] and also [68] The following BM form is based on a random access Turing machine (RAM TM; cf. RTM in [30] and indexing TM in [14, 64, ....

D. Leivant, Descriptive characterizations of computational complexity, J. Comp. Sys. Sci., 39 (1989), pp. 51--83.

.... The study of computable queries originated in the work of Chandra and Harel [CH80, Cha81, CH82] Since then, the complexity and expressiveness of query languages, and the relationship with logic, have been widely investigated, e.g. Var82, CH85, GS86, Imm86, Imm87a, Imm87b, Cha88, Gur88, KV90, Lei89a, Lei89b, AV91a] Below NP, expressiveness results usually assume an ordered input. Without order, languages typically express queries complete within a class, but are unable to express simple queries. It remains open whether there is a language expressing exactly ptime. The relational machine ....

....is a language expressing exactly ptime. The relational machine GM loose is similar in spirit to several devices previously proposed, which operate directly on structures rather than on encodings of structures. Such machines have been introduced by H. Friedman [Fri71] and later by D. Leivant [Lei89a, Lei89b] The device introduced in [Fri71] the FAP, is reminiscent of program schemes. The emphasis in [Fri71] is on extending recursion theory to FAPs. In [Lei89a, Lei89b] ordered structures are emphasized. These results have a different focus than those presented here. The paper is ....

[Article contains additional citation context not shown here]

D. Leivant. Descriptive characterization of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....in NP. In fact, over finite strings, even the class ESO(8 ) expresses all languages in NP; this follows from the more general result that over finite successor structures, i.e. finite structures equipped with a successor predicate, every ESO sentence is equivalent to some ESO(8 ) sentence [32, 13]. Main Problems Studied. Combining and extending the results of Buchi and Fagin, it is natural to ask: What about (nonmonadic) prefix classes ESO(Q) over finite strings We know by Fagin s theorem that all these classes describe languages in NP. But there is a large spectrum of languages contained ....

....any regular ESO prefix class is over strings either equivalent to full MSO, or is contained in first order logic, in fact, in FO(9 8) Theorem 11.3) We further show that there is a unique minimal ESO prefix class which captures NP, namely ESO(8 ) Proposition 9. 1) Our proof uses results in [32, 13] and well known hierarchy theorems. 7) We give a precise characterization of those regular prefix classes of ESO which, over strings, are closed under complementation. In particular, we show that any nontrivial regular class ESO(Q) is closed under complementation iff some quantifier prefix Q 2 Q ....

[Article contains additional citation context not shown here]

D. Leivant. Descriptive Characterizations of Computational Complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

.... class PSPACE and noninflationary fixpoint logic [Var82] see also [Imm82] 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While computational devices work on encodings of problems, logic is applied directly ....

....standard complexity classes can be translated to questions about containments among relational complexity classes. On the other hand, the expres 3 A closely related idea, of generalizing Turing machines to operate on general structures, goes back to [Fri71] and was investigated extensively in [Lei89a, Lei89b]. sive power of fixpoint logic can be precisely characterized in terms of relational complexity classes. This tight, three way relationship among fixpoint logics, relational complexity and standard complexity yields in a uniform way logical analogs to all containments among the complexity ....

[Article contains additional citation context not shown here]

D. Leivant. Descriptive characterization of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

....to a Sigma 1 k (respectively Pi 1 k ) formula with only one quantifier alternation in the first order part. Leivant found a simpler normal form for Sigma 1 1 formulas over finite successor structures: every such formula reduces to a Sigma 1 1 formula with universal first order part [8]. In Section 2, we give a shorter, simpler and more direct proof of this normal form theorem. Leivant s theorem fails in the case of all finite structures. Moreover, let Sigma 1 k (bool) respectively Pi 1 k (bool) be the collection of Sigma 1 k (respectively Pi 1 k ) formulas whose ....

D. Leivant. Descriptive Characterizations of Computational Complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

.... query languages are based on first order logic over relations (FO) However, FO itself cannot express simple and useful database queries such as connectivity: this observation has created a dynamic field of research at the interface of Database Theory, Logic, and Complexity (see the surveys [Gu84, Im87, Lei89a]) Control capabilities (such as iteration and fixpoint) were added to logic [Ch81, Var82, Pa85, Im86, AV89] and familiar computational paradigms emerged as a result (Fagin s important characteriza1 tion of NP [Fa74] although otherwise motivated, also falls in this framework) The relational ....

D. Leivant. Descriptive characterization of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

No context found.

Leivant 1989 Daniel Leivant, "Descriptive Characterizations of Computational Complexity", Journal of Computer and System Sciences 39 (1989), 51--83.

No context found.

D. Leivant. Descriptive Characterizations of Computational Complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

No context found.

D. Leivant. Descriptive characterizations of computational complexity. Journal of Computer and System Sciences, 39:51--83, 1989.

No context found.

Leivant 1989 Daniel Leivant, "Descriptive Characterizations of Computational Complexity", Journal of Computer and System Sciences 39 (1989), 51--83.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC