E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is valid iff f is valid. The validity problem for first order formulas is well known to be undecidable, even if we restrict to formulas that contain a single binary predicate; indeed, undecidability holds even if we further restrict to formulas of the form # , where f # is quantifier free [7]. This means that we cannot determine if a single policy implies a permission when the conditions under which the policy applies must be written in first order logic as a formula of the # where f # has a binary predicate other than Permitted. We can get the same result even without assuming ....

....is logically equivalent to 2 ( Librarian(x 2 ) Permitted(x 1 , enter(stacks) which uses only universal quantification. Note that enter is a function in Example 3.2. Unfortunately, it is well known that the validity problem for existential formulas with functions is undecidable [7]. The following result is almost immediate: 1 be the set of closed formulas of the 1 . xm (f where t and t # are terms of the appropriate sort, and f is a quantifier free formula (possibly containing function symbols) The validity problem for 1 is undecidable. Theorem 3.3 ....

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer-Verlag, Berlin Heidelberg, 1nd edition, 1997.

....calculi based on restricted logics from the first part of the paper, and find that they have low data complexity (e.g, NC ) while remaining quite expressive. Many results in this paper are proved by a combination of two techniques. One is the encoding of S#S into S2S, due to Rabin (cf. [7]) which gives us a coding of unranked trees as ranked trees. To be able to use it, we need several results showing how to restrict quantification over various structures; those are proved by Ehrenfeucht Frasse games. Organization. Section 2 defines the main concepts and presents the main proof ....

....toolbox of two techniques: translating unranked trees into ranked b c e f d e f d Figure 1. A tree T and ) ones, and Ehrenfeucht Frasse games. Below we briefly review them. 2.5. 1 Encoding unranked trees This encoding is basically the same as Rabin s encoding of S#S into S2S, cf. [7]. Given a string n 1 n k of positive integers, R(n n k ) 01 nk # 0, 1 # . Also, R(#) #. Given a tree T = D, f) UTREE(#) we define ) D # , f # ) TREE(## ) as follows: D # is the prefix closure of R(D) R(s) D ; If s D, then f # (R(s) f(s) ....

E. Borger, E. Gradel, Y. Gurevich. The Classical Decision Problem. Springer, 1997.

....and be a map from C to . A triple ( is called a colour scheme, and is called a constant distribution. Note 1. The notion of the colour scheme came, of course, from the well known method of the decidability proof for the monadic class in classical first order logic (see, for example, [BGG97]) In our case we construct quotient structures based only on the predicates and propositions which occur in the temporal part of the problem, because only these symbols are really responsible for the satisfiability of temporal constraints. Besides, we have to consider so called constant ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Springer, 1997.

....k ary DULSs are undecidable. Proof. We show that both the theories are undecidable over binary infinite trees. Since binary infinite trees are embeddable into k ary DULSs, we have the thesis. We show that over binary infinite trees is undecidable by a reduction of the N tiling problem [11]. Since MFOP [ 1 , D] we have that MFOP [ 1 , D] is undecidable too. Recall that the N N tiling problem asks: given a finite set of tile types tile N N For every tile type t , let right(t) left(t) up(t) and down(t) be the colors or the corresponding sides of t. ....

E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer, Berlin, 1997.

....be a map from C to G . A triple (G;q;r) is called a colour scheme, and r is called a constant distribution. Note 1. The notion of the colour scheme came, of course, from the well known method within the decidability proof for the monadic class in classical first order logic (see, for example, [BGG97]) In our case G is the quotient domain (a subset of all possible equivalence classes of predicate values) q is a propositional valuation, and r is a standard interpretation of constants in the domain G. We construct quotient structures based only on the predicates and propositions which occur ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Springer, 1997.

....complex fragments of logic that appears to be deeply related to core issues of decidability and complexity. In fact, one of the most fruitful research programs that kept logicians and computer scientists busy for decades was the exhaustive solution of Hilbert s classical decision problem (cf. [2]) i.e. of the problem of determining those prefix classes of first order logic for which formula satisfiability (resp. finite satisfiability of formulas) is decidable. Quantifier prefixes emerged not only in the context of decidability theory (a common branch of recursion theory and theoretical ....

....prefix classes of ESO are (semantically) included in MSO Observe that we assume MSO allows one to use nullary predicate variables (i.e. propositional variables) along with unary predicate variables. Obviously, Buchi s Theorem survives. 5 Note that by Gurevich s classifiability theorem (cf. [2]) and by elementary closure properties of regular languages, it follows that there is a finite number of maximal regular prefix classes ESO(Q) and similarly, of minimal nonregular prefix classes; the latter are, moreover, standard prefix classes (cf. Section 2) It was the aim of [8] to determine ....

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer, 1997.

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E. Borger, E. Gradel and Y. Gurevich, The Classical Decision Problem, Springer 1997.

.... of second order logic, this is equivalent to whether for a given Q: Instance: formula in ESO(Q) Question: Is (finitely) satisfiable This question has been studied in depth over the past decades, and an exhaustive classification of decidable and undecidable prefix classes is known (see [6]) there are huge complexity gaps between elementarily decidable and undecidable classes. The ESO(Q) classification played an important role in the identification of fragments of ESO which obey the 0 1 law, i.e. the property that over finite structures, a sentence is almost surely true or almost ....

....alphabet) satisfying constitutes a regular language Any fragment fulfilling this condition is called regular. By Buchi s Theorem this question is identical to the following: Which prefix classes of ESO are (semantically) included in MSO Note that by Gurevich s classifiability theorem (cf. [6]) and by elementary closure properties of regular languages, it follows that there is a finite number of maximal regular prefix classes ESO(Q) and similarly, of minimal nonregular prefix classes; the latter are, moreover, standard, i.e. the quantifier prefix class Q is either the set of all ....

E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer, Berlin Heidelberg, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. 1996.

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E. Borger, E. Gradel, and Y. Gurevich, "The classical decision problem", Springer-Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer-Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The classical decision problem. Springer, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer, 1997.

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Borger, E., E. Gradel and Y. Gurevich, "The Classical Decision Problem," Perspectives of Mathematical Logic, Springer-Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer-Verlag, 1997.

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Borger, E., E. Gradel and Y. Gurevich, "The Classical Decision Problem," Springer-Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer, Berlin, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The classical decision problem. Springer, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer, Berlin, 1996.

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E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Springer Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer-Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer-Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Springer Verlag, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The classical decision problem. Springer, 1997.

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E. Borger, E. Gradel, and Y. Gurevich. The classical decision problem. Springer, 1997.

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