E. Borger, E. Graedel, and Y. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... reasoning had its beginnings in the pioneering Logic Theorist system of Newell, Shaw, and Simon [NSS57] The theorems they proved were shown by Hao Wang [Wan60b] to fall within simply decidable fragments like propositional logic and the Bernays Schonfinkel fragment of first order logic [BGG97] Many technical ideas from the Logic Theorist such as subgoaling, substitution, replacement, and forward and backward chaining, have been central to automated reasoning, but the dogma that human oriented heuristics are the key to e#ective theorem proving has not been vindicated. Hao Wang ....

Egon Borger, Erich Gradel, and Yuri Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

...., the following holds: if #(x, y) t and #(x 2 n 1 1, y) t # , then (t, t # ) H if #(x, y) t and #(x, y n 1 1) t # , then (t, t # ) #(i, 0) a i for i n. i denotes addition modulo i. 12 As shown in, e.g. Corollary 4. 15 of [37] it follows from results in [13] that the above variant of the domino problem is NExpTime complete. We now define the concrete domain D 1 which will be used in the reduction of the NExpTime complete domino problem to 1 ) concept satisfiability w.r.t. Boolean key boxes. Definition 13 (Concrete Domain D 1 ) The concrete ....

....k = 0) x k = x # k ) which encodes incrementation modulo 2 , i.e. if t is the number (binarily) encoded by the propositional variables x 0 , x n and t # is the number encoded by the propositional variables x # 0 , x # n , then we have t # = t 1 modulo 2 , c.f. [13]. Taking into account x quantifiers in XSuccOk, it is readily checked that this concept has just the desired e#ect: to ensure that, for every R x successor d of e i,j , we have xpsn(x) xpsn(e (i# 2 n 1 1) j ) i 2 n 1 1. The explanation of YSuccOk and how it enforces the lower line of ....

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

.... by L we denote the monadic fragment of L (i.e. the set of formulas which contain only unary predicates and propositional variables) Both the two variable and the monadic fragments of classical (non temporal) first order logic are known to be decidable and have the finite model property; see [3] and references therein. The computational behavior of the corresponding fragments of first order temporal logics turns out to be quite different. From linear time results (Theorem 2 of [14] we easily obtain: THEOREM 2. For any of the FOBTLs L introduced above, L L L is not recursively ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

....its two variable fragment FM 2 [Q ] consisting of all FM[Q ] formulas with the variables x and y only. All formulas in the example above belong to this fragment. The two variable fragment of classical first order logic is known to be decidable [26, 22, 14] and NExpTime complete [8, 4]. We use this result to show that the satisfiability problem for FM 2 [Q ] formulas is decidable ffl in the class D of arbitrary distance spaces, and ffl in the class D sym of all distance spaces satisfying (3) Unfortunately, this does not hold any more as soon as we add the triangular ....

....5. Let K D tr contain Omega R 2 ; d 2 ff . Then the satisfiability problem for MS i formulas in K is undecidable, for any 1 i 4. Proof. We consider only MS 1 ; the other languages are treated analogously. The proof is by reduction of the undecidable N Theta N tiling problem (see [32, 4] and references therein) We remind the reader that the tiling problem for N Theta N is formulated as follows: given a finite set T = fT 1 ; T l g of tiles (i.e. squares T i with colors left(T i ) right(T i ) up(T i ) and down(T i ) on their edges) determine whether tiles in T can ....

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E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

....a. Note that not admitting constants is not crucial. In fact, in our version of FO 2 constants can be simulated through unary predicates similar to the simulation of nominals in L described in the introduction. To the contrary, function symbols cannot be admitted without loosing decidability [2]. The description logic L is ALC extended with Boolean operators on roles, the inverse operator on roles, and the identity role. Here is the formal definition: 3 Definition 1 Let NR = fR 1 ; R 2 ; g and N C = fA 1 ; A 2 ; g be disjoint sets of role names and concept names, ....

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

.... that the reason is not that mostly modal logics have a fixed number of modal operators (alias accessibility relations interpreting them) but two variable logic allows for arbitrarily many binary relations: Even without relation symbols of arity 1, the two variable fragment is NExpTime hard [8, 3]. There are two possible explanations for this phenomenon: 1. Explanation: any standard modal logic contained in FO 2 has strictly less expressive power than FO 2 itself, or 2. Explanation: although the expressive power of the two variable logic coincides with the expressive power of a ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

....I(0) interprets the predicate symbols in by the same predicates as in M , Q 0 = D, and for every i 2 N, P i = ae fa 0 ; a i g if i n, D if i n, It follows from Lemma 13 that 0 j= 0 , and clearly we have 0 6j= Q . 2 Now recall that by Trakhtenbrot s theorem (see e.g. [1]) the set of first order classical formulas that are valid in finite models is not recursively enumerable. As a consequence we obtain the following: Theorem 15. The set of T L = 1 formulas that are valid in all temporal models based on hN; i is not recursively enumerable, and so not ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

....Theorem 5. Let K be a class of metric spaces containing R 2 . Then the satisfiability problem for MS 3 [f0; 9; 10; 20; 80g] formulas (even for those with the operators 8 0 a and 9a only) in K is undecidable. Proof. To prove this result, we reduce the undecidable N Theta N tiling problem (see [17, 2] and references therein) to the satisfiability problem in K. We remind the reader that the tiling problem for N Theta N is formulated as follows: given a finite set T = fT 1 ; T l g of tiles (i.e. squares T i with colors left(T i ) right(T i ) up(T i ) and down(T i ) on their edges) ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

....even recursively enumerable. But in contrast to classical first order logic, where the early undecidability results of Turing and Church stimulated research and led to a rich and profound theory concerned with classifying fragments of first order logic according to their decidability (see, e.g. [9]) there were no serious attempts to convert the negative results in first order temporal logic into a classification problem. Apparently, the extremely weak expressive power of the temporal formulas required to prove undecidability left no hope that any useful decidable fragments located ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer-Verlag, 1997.

....formulas of the form 9y (G(x; y) x; y) where the guard G(x; y) is atomic 2 [1] 1 The fragment with binary predicates and three variables is undecidable [30] 2 For a precise definition see Section 5. 1 (The current state of art in this field is presented in the recent monograph [6]. For modal logicians the decision problem in first order modal logics seemed almost hopeless. The following list covers almost all known results and leaves not so much space for a maneuver: ffl the monadic fragment (even with a single unary predicate symbol) of practically all modal predicate ....

....= n, for every t 2 T . 2 Corollary 31. QK ML 2 1 , QT ML 2 1 , QK ML 1 , and QT ML 1 have both the finite frame and the finite domain properties. Proof It is well known that the two variable fragment and the monadic fragment of first order logic have the finite model property (see [6]) 2 One more interesting fragment of ML is the set of monadic formulas, all predicate symbols in which are at most unary. Denote this fragment by ML m , and let ML m 1 = ML 1 ML m . In this case the formula ff T , corresponding to a world candidate T for 2 ML m 1 , is a monadic ....

E. Borger, E. Gradel, and Yu. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

....logic restricted to two variables augmented with counting quantifiers by an extension of the translation given in [4] If we assume unary coding of numbers, this reduction yields a NEXPTIME upper bound [12] for the complexity TBoxconsistency for ALCQI. By reduction from a bounded domino problem [3], we show that NEXPTIME is also a lower bound for the problem. Again, we can only present the ideas of most proofs. Please refer to [15] for details. Definition 10 (Domino System) For n 2 N, let Zn denote the set f0; n Gamma 1g and Phi n denote the addition modulo n. A domino system is ....

.... behaviour of restricted, so called simple, Turing Machines (TM) This restriction is non essential in the following sense: Every language accepted in time T (n) and space S(n) by some one tape TM is accepted within the same time and space bounds by a simple TM, as long as S(n) T (n) 2n [3]. Exploiting the correspondence between computations of resource bounded TMs and tilings of bounded domino systems the following lemma can easily be shown. Lemma 11 There is a domino system D such that the following is a NEXPTIME hard problem. Given an initial condition w = w 0 ; ....

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

.... behaviour of restricted, so called simple, Turing Machines (TM) This restriction is non essential in the following sense: Every language accepted in time T (n) and space S(n) by some one tape TM is accepted within the same time and space bounds by a simple TM, as long as S(n) T (n) 2n [BGG97]. Theorem 1 ( BGG97] Theorem 6.1.2) Let M be a simple TM with input alphabet Sigma . Then there exists a domino system D = D; H;V ) and a linear time reduction which takes any input x 2 Sigma to a word w 2 D with jxj = jwj such that If M accepts x in time t 0 with space s 0 , then ....

.... restricted, so called simple, Turing Machines (TM) This restriction is non essential in the following sense: Every language accepted in time T (n) and space S(n) by some one tape TM is accepted within the same time and space bounds by a simple TM, as long as S(n) T (n) 2n [BGG97] Theorem 1 ([BGG97], Theorem 6.1.2) Let M be a simple TM with input alphabet Sigma . Then there exists a domino system D = D; H;V ) and a linear time reduction which takes any input x 2 Sigma to a word w 2 D with jxj = jwj such that If M accepts x in time t 0 with space s 0 , then D tiles U(s; t) with ....

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

....of quantificational formulas have been studied since the second decade of this century by many logicians including W. Ackermann, P. Bernays, K. Godel, L. Kalm ar, M. Schonfinkel, T. Skolem, H. Wang [1, 2, 5, 11, 12, 24, 33, 34, 35, 37] and many others. In the late twenties and in the thirties (see [7] and [19] for more informations) the study of classification of solvable classes of prenex formulas was one of the most active areas of logic. Now, after the works of Y. Gurevich [18] M. Rabin [30] S. Shelah [32] and W. Goldfarb [14] the classification of prenex classes has been completed. ....

....classes of prenex formulas was one of the most active areas of logic. Now, after the works of Y. Gurevich [18] M. Rabin [30] S. Shelah [32] and W. Goldfarb [14] the classification of prenex classes has been completed. Accounts of the classical results in this area can be found in several books [3, 7, 9, 25]. More The results included in section 4 have been published as a part of [29] recent results have been obtained by H.R. Lewis and W. Goldfarb [13, 14, 26] A short survey of the research in this area can be found in [19] see also the introduction to [15] In 1962, in a short note, D. ....

E. Borger, E. Gradel, and Y. Gurevich. The classical decission problem. Perspectives in Mathematical Logic. Springer-Verlag, 1997.

....been studied since the second decade of this century by many logicians including W. Ackermann, P. Bernays, K. Godel, L. Kalm ar, M. Schonfinkel, T. Skolem, H. Wang [1, 2, 6, 20, 21, 32, 52, 53, 54, 58] and many others. Accounts of the classical results in this area can be found in several books [3, 8, 19, 37] More recent results have been obtained by H.R. Lewis and W. Goldfarb [22, 23, 38] A short survey of the research in this area can be found in [28] Questions concerning decidability of clause implication and similar problems were also studied by computer scientists motivated by problems in the ....

....clause implication problem can be considered as one of the problems that had to be settled to delineate the boundary between decidable and undecidable classes of first order formulas, i.e. classes whose subclass of valid (or satisfiable) formulas are recursive. This was once, in the thirties (see [8]and [28] for more informations) the most active area of logic. Now, after the work Y. Gurevich [29] M. Rabin [47] S. Shelah [51] and W. Goldfarb [23] the classification of prenex classes has been completed. As we have mentioned in the Introduction the question concerning decidability of clause ....

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E. Borger, E. Gradel, and Y. Gurevich. The classical decission problem. Perspectives in Mathematical Logic. Springer-Verlag, 1997.

....of restricted, so called simple, Turing Machines (TM) This restriction is non essential in the following sense: Every language accepted in time T (n) and space S(n) by some one tape TM is accepted within the same time and space bounds by a simple TM, as long as S(n) T (n) 2n. Theorem 3. 2 ([BGG97], Theorem 6.1.2) Let M be a simple TM with input alphabet Sigma. Then there exists a domino system D = D; H;V ) and a linear time reduction which takes any input x 2 Sigma to a word w 2 D with jxj = jwj such that ffl If M accepts x in time t 0 with space s 0 , then D tiles U(s; t) with ....

.... 2 Delta I we define pos(a) by pos(a) xpos(a) ypos(a) n Gamma1 X i=0 x i Delta 2 i ; n Gamma1 X i=0 y i Delta 2 i ; where x i = 0; if a 62 X I i 1; otherwise y i = 0; if a 62 Y I i 1; otherwise : We use a well known characterisation of binary addition [BGG97] to relate the positions of the elements in the torus: Lemma 3.4 Let x; x 0 be a natural numbers with binary representations x = n Gamma1 X i=0 x i Delta 2 i and x 0 = n Gamma1 X i=0 x 0 i Delta 2 i : Then it holds that: x 0 j x 1 (mod 2 n ) iff n Gamma1 k=0 ( ....

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

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Egon Borger, Erich Gradel, and Yuri Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer-Verlag, 1997.

....advantages. First, the decidability and complexity of logical implication for DLF and its variants are closely related to reasoning in Datalog # nS , a logic programming language of monadic predicates and functions [9, 10] and in turn to the classical Ackermann case of the Decision Problem [2, 4]. This connection allows us to draw on powerful techniques developed for the classical decision problems. Second, the use of Datalog # nS provides an uniform framework in which we can study various extensions of DLF , in particular how roles can be simulated by attributes or how dependencies can ....

....Ackermann formulae allow arbitrary arity relations in their matrix, they still require the use of a single universal (#) quantifier in their prefix. This prevents a direct formulation of the #R. D concept since two # s are needed (then adding unary function symbols leads to undecidable theories [4]) We therefore use a less direct formulation by modeling roles in DLFR via attributes. The essential problem is that #R.D i concepts can force a single object to be related via a role R to multiple objects satisfying di#erent constraints D i , that may be disjoint in general. However, as all ....

Egon Borger, Erich Gradel, and Yuri Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer-Verlag, 1997.

.... In the context of discussing the functional programming language Godel used to exhibit undecidable propositions in Principia Mathematica, as opposed to the imperative programming language developed by Turing and used in his proof of the unsolvability of the Entscheidungsproblem (see [7]) Martin Davis [12] states: The programming languages that are mainly in use in the software industry (like C and FORTRAN) are usually described as being imperative. This is because the successive lines of programs written in these languages can be thought of as commands to be executed by ....

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

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E. Borger, E. Graedel, and Y. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, 1997.

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

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

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