E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 1 G 2 holds at S i if both G 1 and G 2 do (and branching) or G holds at S i if either G holds at S i 1 , or G holds at S i 1 (or branching) To abstract away from implementation details, we shall use the paradigm of logic programming, which is related to and or trees in an essential way [18]. With each formula F occurring in Current at record i, associate a propositional variable ok F;i , whose intended meaning is F holds at S i . Deduction chains are maintained through the use of constrained clauses of the form ok F0 ;i 0 ok F1 ;i 1 ; ok Fm ;i m j C meaning S; i 0 j= ....

E. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1(1):19--33, 1984.

.... G 1 G 2 holds at S i if both G 1 and G 2 do (and branching) or G holds at S i if either G holds at S i 1 , or G holds at S i 1 (or branching) To abstract away from implementation details, we shall use the paradigm of logic programming, which is related to and or trees in an essential way [21]. Assume first, for simplicity, that we have no variables, so that all constraints are trivial. Then use propositional logic: with each formula F occurring in Current at record i, associate a propositional variable ok F;i , whose intended meaning is F holds at S i . Then generate clauses at ....

E. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1(1):19--33, 1984.

....The size of the derivation turns out to be exponential. NEXPTIME hardness is proved by reduction from the tiling problem for the square 2 n Theta 2 n . Some other fragments of LP with function symbols are known to be decidable. For example, the following result was established in [120], by using a simulation of alternating Turing machines by logic programs and vice versa. Theorem 3.9 ( 120] LP with function symbols is PSPACE complete, if each rule is restricted as follows: The body contains only one atom, the size of the head is greater than or equal to that of the body, and ....

....from the tiling problem for the square 2 n Theta 2 n . Some other fragments of LP with function symbols are known to be decidable. For example, the following result was established in [120] by using a simulation of alternating Turing machines by logic programs and vice versa. Theorem 3. 9 ([120]) LP with function symbols is PSPACE complete, if each rule is restricted as follows: The body contains only one atom, the size of the head is greater than or equal to that of the body, and the number of occurrences of any variable in the body is less than or equal to the number of its occurrences ....

E. Shapiro. Alternation and the Computational Complexity of Logic Programs. J. Logic Programming, 1:19--33, 1984.

....Advantages of substructural logic The common syntactical restrictions of FOL used in logic programming do not tame the computational complexity of the system. Theorem 3. 1 Given a program P and a query Q, the question whether there is a substitution such that (Q) follows from P is undecidable [15]. This means that provability in the Horn fragment of FOL is undecidable. This causes problems for doing ILP in this fragment, because the search space for good solutions will be unlimited. 6 decidable and better 124 For logic programs with restrictions on input output behaviour of the ....

Ehud Y. Shapiro, Alternation and the Computational Complexity of Logic Programs, Journal of Logic Programming 1, pp. 19-33, 1984.

....single exponential time (nexptime) hard. Membership in nexptime, and hence nexptime completeness of mall1, follows from current work of Lincoln and Shankar [26] see below. The hardness proof in [24] combines E. Shapiro s logic programming simulation of nondeterministic Turing machines [34] with the standard proof of the pspace hardness of quantified boolean formula validity as in, e.g. Stockmeyer [35] and utilizing some of the remarkable plasticity of linear logic. The hardness result is achieved through a direct encoding of Turing machine transitions, which are shared via the ....

E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

.... and their (potential) applications are well treated in the literature (e.g. 47] Of special relevance to us is the fact that any implementation of such languages must understand much about the underlying mathematical domain, especially its computational properties; see some treatment of this in [29,28,74,57,33]. In practice, it seems that implementations of CLP languages use f.p. computation although, evidently, any correct CLP compiler must use exact computation Ultimately, the CLP compiler must try to balance the opposing demands of correctness and efficiency: some mixed mode computational ability ....

Ehud Shapiro. Alternation and the computational complexity of logic programs. J. Logic programming, 1:19--33, 1094.

....time. 3 Alternating machines Alternation was defined by Chandra, Kozen and Stockmeyer [1] as a general model for parallel computation. Because of its close relationship to logic programming, alternation has been suggested as a good paradigm to write and reason about parallel algorithms [5, 3]. It is closely related to many other models of parallel computation [2] The fundamental feature of an alternating machine is that existential and universal quantification are allowed as primitive operations. These operations may be considered for arbitrary machines. A general alternating ....

E. Y. Shapiro, "Alternation and the Computational Complexity of Logic Programs," J. Logic Programming 1 (1984) 19--33. Page 7

....are definable by regular FGSs. For example, regular FGSs generate trees, two terminal series parallel graphs, the homeomorphisms of a given graph, outerplanar graphs, graphs of cyclic bandwidth 2 and k decomposable graphs [7, 8, 13] Thus FGSs have a diversity of expressibility. A refutation tree [12, 16, 19] describes how a term is generated by applying the rules of a logic program. A refutation tree of a graph for an FGS is defined in a similar way. It represents the logical structure of the graph in the FGS and simultaneously describes a decomposition of the graph explicitly. Therefore, it is ....

E. Y. Shapiro. Alternation and computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

....varieties (geometry) as opposed to studying complex polynomial ideals (algebraic geometry) This observation suggests that we must exploit geometric properties in order to reduce complexity. It might be objected that single exponential complexity is still too high. In general, this is unavoidable [27, 7] and a central goal of algorithmic research is to identify the interesting subcases which admit simpler solutions. This worst case complexity can be ameliorated in several effective ways, including the use of randomization. The exponential behavior is often a function of only the dimensionality of ....

....problems, this worst case behavior does not necessarily extend to all problems in the domain: it is often the case that problems of interest admit efficient solutions. Finally, it must be argued that since the language PROLOG is already capable of simulating an alternating Turing machine [27], adding a constraint solver for a particular domain does not increase the asymptotic complexity of evaluating a constraint program. In fact, what we are proposing is the use of special purpose constraint solvers which can exploit the geometric aspects of the problem to speed the computation. One ....

Ehud Shapiro. Alternation and the computational complexity of logic programs. J. Logic programming, 1:19--33, 1094.

....exponential time Turing machines by mall1 formulas is given in Section 3. This encoding is reminiscent of the standard proof of the pspace hardness of quantified boolean formula validity [28, 16] The encoding is also related to the logic programming simulation of Turing machines given in [27]. In Section 4 it is shown that whenever a nondeterministic exponential time Turing machine accepts, then the mall1 formula encoding the machine is provable. The converse is shown in Section 5. We would like to thank Yuri Gurevich for suggesting that the logic programming simulation of Turing ....

....4 it is shown that whenever a nondeterministic exponential time Turing machine accepts, then the mall1 formula encoding the machine is provable. The converse is shown in Section 5. We would like to thank Yuri Gurevich for suggesting that the logic programming simulation of Turing machines given in [27] may be a source of 2 PRELIMINARIES 4 lower bounds for the complexity of mall1. This suggestion eventually led us to formulate the encoding given in Section 3. We would also like to thank Jean Marc Andreoli, Jawahar Chirimar, Jean Gallier, Jean Yves Girard, Carl Gunter, Joshua Hodas, Jim Lipton, ....

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E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

....class of locally hierarchical programs [7] called acyclic programs in Apt and Bezem [1] The perfect model M P of a weakly reducing program P is recursive. Proposition 13. Suppose that P be a weakly reducing program. Then the perfect model M P is recursive. Proof. By Theorem 4. 4 in Shapiro [18], we can conclude that if the maximum size of atoms in a derivation from A 2 M P is bounded by a computable function of jAj then for each P n a segment M n of M P is at most a recursive set. Since sizes of atoms in a derivation cannot exceeds that of an initial ground goal A for a weakly reducing ....

E.Y. Shapiro. Alternation and the computational complexity of logic programs. J. of Logic Programming, Vol. 1, No. 1, pp. 19--33, 1984.

....of L, where [X jXs] denotes the term : X; Xs) A variable L in the second clause of elemsum=3 is local. elemsum( N ; N) elemsum( XjXs] M ; N) sum(X;M ; L) elemsum(Xs;L; N) sum(0; Y ; Y ) sum(s(X) Y ; s(Z) sum(X;Y ; Z) A clause a a 1 ; a n (n 0) is called linear [11, 12] if for any substitution and any i = 1; n, ja j ja i j. While the program shown above is not linear because of the local variable L, it is linearly covering. For instance, the derivation steps in Figure 1 shows that the second clause of elemsum=3 is linearly covering. We show another ....

E. Y. Shapiro. Alternation and the Computational Complexity of Logic Programs. J. Logic Programming, 1(1):19--33, 1984.

.... by developing a direct encoding of nondeterministic exponential time Turing machines [33] This encoding is reminiscent of the standard proof of the pspace hardness of quantified boolean formula validity [43, 21] and is related to the logic programming simulation of Turing machines given in [42]. This encoding in first order MALL formulas is somewhat unique in that the computation is read across the top of a completed cut free proof, rather than bottom up , which is utilized in most of the abovedescribed results. The result is that Turing machine instructions are not copied as one ....

E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

....exponential time Turing machines by mall1 formulas is given in Section 3. This encoding is reminiscent of the standard proof of the pspace hardness of quantified boolean formula validity [19, 13, 5] The encoding is also related to the logic programming simulation of Turing machines given in [18]. In Section 4 it is shown that whenever a nondeterministic exponential time Turing machine accepts, then the mall1 formula encoding the machine is provable. The converse is shown in Section 5. We would like to thank Yuri Gurevich for suggesting that the logic programming simulation of Turing ....

....4 it is shown that whenever a nondeterministic exponential time Turing machine accepts, then the mall1 formula encoding the machine is provable. The converse is shown in Section 5. We would like to thank Yuri Gurevich for suggesting that the logic programming simulation of Turing machines given in [18] may be a source of lower bounds for the complexity of mall1. This suggestion eventually led us to formulate the encoding given in Section 3. We would also like to thank Jean Yves Girard, Jean Marc Andreoli, Jawahar Chirimar, Jean Gallier, Carl Gunter, Joshua Hodas, Jim Lipton, Narciso Mart ....

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E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

.... path abstraction in [YS91a] and by path projection in [Fru89] can be represented by proper unary predicate programs. We use an analogy between proper unary predicate logic programs and 2 way automata as well as the natural correspondence between logic programs and alternating algorithms (cf. [Sha84]) to study the complexity of type inference and type checking for types described by proper unary predicate programs. The restriction imposed on such programs enables us to use unfolding 1 techniques, inspired by classical techniques in the theory of 2 way automata, to reduce proper ....

....rules in P T are limited in their ability to manipulate terms. We call unary predicate programs that obey the restrictions in Proposition 3.3 proper. Proper unary predicate programs cannot have rules such as p(g(f(X) Y ) q(g(X; f(Y ) With such rules one can easily simulate Turing machines [Sha84]. Thus, improper unary predicate programs can define all recursively enumerable sets of terms. In contrast, as we shall see, proper unary predicate programs define regular sets of terms. Such sets are often used to describe types, cf. HJ90a, Mis84, YS87, Zob87] though often only a proper ....

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E. Shapiro. Alternation and the computational complexity of logic programs. J. Logic Programming, 1:19--34, 1984.

....single possible computational path. The vertical arrows represent inclusion mappings. We prove that they preserve the fullyabstract congruence relation using the notion of testing sub structures [27] The horizontal arrows represent the known translations between logic programs and Turing machines [21, 25] and trivial embeddings of finite automata into Turing machines. Note that the relations between logic programs and Turing machines are obtained although they have different syntactic structure and employ different computation methods. In addition, the inclusion mappings of finite automata are ....

....C 0 = 2 L(Q [ R 00 ) since the prefix C 0 cannot be obtained in P [ R. Hence, by the induction hypothesis there is a deterministic automaton and a separating word as required. 2 5. 2 Embedding between Logic Programs and Turing Machines We adopt the key ideas of the simulations presented in [21, 25] to conclude that there exist OF homomorphisms between the structures associated with logic programs and those associated with Turing machines. Based on the simulations presented in [21, 25] there exists an embedding of alternating Turing machines into logic programs. The restriction of this ....

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Shapiro, E., Alternation and the Computational Complexity of Logic Programs, Journal of Logic Programming, Vol. 1, No. 1, pp. 19--33, 1984.

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E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

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E.Y. Shapiro. Alternation and the computational complexity of logic programs. Journal of Logic Programming, 1:19--33, 1984.

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