Jos e Luis Balc azar, Josep D az, and Joaquim Gabarr o. Structural Complexity, volume 1. Springer-Verlag, Berlin, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and where circuit size is the number gates in a circuit) circuit families can decide all languages L # # # . It is possible to encode a circuit by a finite symbol sequence, and a circuit family by an infinite symbol sequence. For a more thorough introduction to circuits we refer the reader to [4]. ARNNs are finite size feedback first order neural networks with real weights [18, 17] The state of each neuron at time t 1 is given by an update equation of the form x i (t 1) # # N # j=1 a ij x j (t) M # j=1 b ij u j (t) c i # , i = 1, N (5) where N is the ....

Jos e Luis Balc azar, Josep D az, and Joaquim Gabarr o. Structural Complexity, volume 1. Springer-Verlag, Berlin, 1988.

....in n. FP denotes the class of (partial) functions that can be computed in polynomial time. We use h Delta; Delta Delta Delta ; Deltai to denote a standard polynomial time computable tupling function. The definitions of standard complexity classes like P , NP , etc. can be found in books like [1, 17]. For a class C of sets we call a pair (A; B) of disjoint sets A; B 2 C a disjoint C pair. If for a class D, and some D 2 D it holds A D, and B D = we call the pair (A; B) D separable. A nondeterministic polynomial time Turing machine (NPTM, for short) is a Turing machine N such that for ....

Jos'e Luis Balc'azar, Josep D'iaz, and Joaquim Gabarr'o. Structural Complexity I. Springer-Verlag, 2nd edition, 1995.

....expressive with respect to plan existence iff both X GPP p Y GPP and Y GPP p X GPP hold. The corresponding definitions for bounded plan existence are analogous. 7 The reader not being familiar with the concepts of problems and polynomial transformations may wish to consult the literature [8, 15, 16]. 8 The motivation for this definition of equal expressiveness is as follows. If we know that formalisms X and Y are equally expressive with respect to plan existence, then a proof that X PE belongs to a certain complexity class immediately allows us to conclude that also Y PE belongs to that ....

....be polynomially transformed into each other, then the corresponding search problems can usually be Turing reduced to each other. Furthermore, the bounded plan generation problem can be solved by employing an oracle (or an algorithm) for the bounded plan existence problem and using prefix search [8, 15]. For most problems, such a prefix search strategy is a Turing reduction. This is not the case for the planning, however, since instances of the planning problems may have even minimal solutions of exponential length in the size of the instance [2, 6] It has been argued that such instances should ....

Jos'e Luis Balc'azar, Josep Diaz, and Joaquim Gabarr'o. Structural Complexity I. Springer, 1988.

....proof systems. An excerpt of a preliminary version of this paper appears as [10] In the meanwhile we were able to simplify the proof of the main theorem. 2 Preliminaries We assume some familiarity with standard notions of computational complexity theory, and refer the reader to books like [2] for notions not defined in this paper. Let Sigma be some fixed finite alphabet containing 0 and 1. The output of a Turing transducer M on input x 2 Sigma is denoted by M(x) we write M(x) if M does not accept or runs forever on input x. Similarly for a partial function f : Sigma ....

....that have a polynomial time bound on Sigma . Let h be a function with range R Sigma ; For a function f (and also for a transducer M computing f) we say that f (resp. M) inverts h on S R if h(f(y) y for y 2 S. If additionally f 2 FP we say that h is FP invertible on S. Following [2] a function h is called 1 invertible (in polynomial time) if there is a function, denoted h Gamma1 , in FP such that h Gamma1 (h(x) x for x 2 Sigma ; h is called length increasing if jh(x)j jxj for any x; we call h linearly length bounded if jh(x)j c Delta jxj for some constant c ....

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Jos'e Luis Balc'azar, Josep D'iaz, and Joaquim Gabarr'o. Structural Complexity I. Springer-Verlag, 2 edition, 1995.

....of crossword puzzle solving. In our context, deciding propositional implication (i.e. A j= is a problem, while finding out whether fp q; pg j= q is an instance of this problem. 7 A strict partial order is a transitive and irreflexive relation. 8 Good text books on this topic are [ Balc azar et al. 1988; Balc azar et al. 1990; Garey and Johnson, 1979; Papadimitriou, 1994; Wagner and Wechsung, 1988 ] 3.1 Polynomial vs. Exponential Runtime Requirements When we denote the size of an instance I by jIj, and if f is some monotonically increasing function from positive integers to positive ....

Jos'e Luis Balc'azar, Josep Diaz, and Joaquim Gabarr'o. Structural Complexity I. Springer-Verlag, Berlin, Heidelberg, New York, 1988.

....from P but if this was possible in the F case, the polynomial time hierarchy would collapse on the first level. Those results are meant to be a first attempt to understand lowness for general complexity classes. 2 Preliminaries Most of the notation used in this article is standard according to [BDG88, BDG90]. All languages are over some finite alphabet Sigma, Sigma is the set of all words over Sigma, and for x 2 Sigma ; jxj denotes the length of x. IN means the set of all natural numbers, ZZ the set of integers. The numbers shall be binary encoded. The set of polynomial time computable ....

Jos'e Luis Balc'azar, Josep D'iaz, and Joaquim Gabarr'o. Structural Complexity I. Springer-Verlag, 1988.

....the input. In this context, the term problem means, contrary to the ordinary meaning, generic question, which has different instances. For example, a particular crossword puzzle is not a problem, but an instance of the problem of crossword puzzle solving. 1 Good text books on this topic are [ Balc azar et al. 1988; Balc azar et al. 1990; Garey and Johnson, 1979; Papadimitriou, 1994 ] 2.1 Polynomial vs. Exponential Runtime Requirements A problem is considered to be efficiently solvable if for all instances of the problem an algorithm can find the correct answer in a number of computation steps that ....

Jos'e Luis Balc'azar, Josep Diaz, and Joaquim Gabarr 'o. Structural Complexity I. Springer-Verlag, Berlin, Heidelberg, New York, 1988.

....be easier than its corresponding existence problem, so the results in the previous section provide lower bounds for the bounded plan search problems. Furthermore, if we can solve the bounded plan existence problem, then we can also solve the bounded plan search problem by using prefix search [ Balc azar et al. 1988, Garey and Johnson, 1979, pp. 116 117 ] It is important to note, however, that this method does not provide a polynomial reduction of the bounded search problem to the bounded existence problem. The reason for this is that some of the problems we consider do not have polynomially bounded ....

Jos'e Luis Balc'azar, Josep Diaz, and Joaquim Gabarr'o. Structural Complexity I. Springer, 1988.

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Jos e Luis Balc azar, Josep D az, and Joaquim Gabarr o. Structural Complexity, volume 1. Springer-Verlag, Berlin, 1988.

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Jose Luis Balcazar, Josep Daz, and Joaquim Gabarro. Structural Complexity 2. SpringerVerlag, New York, NY, 1990.

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Jose Luis Balcazar, Josep Daz, and Joaquim Gabarro. Structural Complexity 1. SpringerVerlag, New York, NY, 1988.

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Jos'e Luis Balc'azar, Josep D'iaz, and Joaquim Gabarr'o. Structural Complexity I. Springer-Verlag, 1988.

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