Borger, E., E. Gradel, and Y. Gurevich, The classical decision problem, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1997. Zbl 0865.03004. MR 99b:03004. 19, 23, 26  Home/Search   Document Not in Database   Summary   Related Articles   Check

....not even recursively enumerable. But in contrast to classical rst 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 rst order logic according to their decidability (see, e.g. [13]) for a long time there were few if any serious attempts to convert the negative results in rst order temporal logic into a classi cation problem. Apparently, the extremely weak expressive power of the temporal formulas required to prove undecidability left no hope that any useful decidable ....

.... by T L we denote the monadic fragment of T 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) rst order logic are known to be decidable and have the nite model property; see [13] and references therein. The computational behavior of the corresponding fragments of rst order temporal logics turns out to be quite di erent. Theorem 1 ( 35] Let F be either fhN; ig or fhZ; ig. Then QTL(F) is not recursively enumerable. Theorem 2 ( 35] Let F be one of the ....

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

....not even recursively enumerable. But in contrast to classical rst 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 rst order logic according to their decidability (see, e.g. [13]) for a long time there were few if any serious attempts to convert the negative results in rst order temporal logic into a classi cation problem. Apparently, the extremely weak expressive power of the temporal formulas required to prove undecidability left no hope that any useful decidable ....

.... T L mo we denote the monadic fragment of T 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) rst order logic are known to be decidable and have the nite model property; see [13] and references therein. The computational behavior of the corresponding fragments of rst order temporal logics turns out to be quite di erent. Theorem 1 ( 35] Let F be either fhN; ig or fhZ; ig. Then QT L 2 QT L mo QTL(F) is not recursively enumerable. Theorem 2 ( 35] Let F be ....

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 = 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 rst order classical formulas that are valid in nite 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 recursively ....

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

.... is not due to the fact that modal logics have a xed number of modal operators (alias accessibility relations interpreting them) whereas the two variable fragments 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.

.... (x k = x 0 k ) which encodes incrementation modulo 2 n , i.e. if t is the number (binarly) encoded by the propositional variables x 0 ; xn 1 and t 0 is the number encoded by the propositional variables x 0 0 ; x 0 n 1 , then we have t 0 = t 1 modulo 2 n , c.f. [4]. Assume a 2 (Edge[g ; p] I (where p is either = or conc i ) and let b be the leaf with pos(b) pos(a) 1, x be the g successor of a, and y be the g successor of b. The Edge concept ensures that, for each S[g ; p] successor c of a, we have pos(c) 6= pos(a) 1, i.e. there ....

E. Borger, E. Gradel, and Y. Gurevich. The Classical Decision Problem. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 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 de nition: 3 De nition 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, respectively. ....

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

.... is not due to the fact that modal logics have a xed number of modal operators (alias accessibility relations interpreting them) whereas the two variable fragments 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.

....allowed) In the NExpTimecomplete variant of the domino problem that we use, the task is not to tile the whole plane, but to tile a 2 n 1 2 n 1 torus, i.e. a 2 n 1 2 n 1 rectangle whose edges are glued together. See, e.g. 4, 19] for undecidable versions of the domino problem and [6] for bounded variants. We now formally introduce bounded domino systems. De nition 15 Let D = D; H;V ) be a domino system, where D is a nite set of domino types and H;V D D represent the horizontal and vertical matching conditions. For s; t 2 N, let U(s; t) be the torus Z s Z t , where ....

.... behaviour of restricted, so called simple, Turing Machines (TMs) 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, provided that S(n) T (n) 2n [6]. Theorem 16 [ 6] Theorem 6.1.2] Let M be a simple TM with input alphabet . Then there exists a domino system D = D; H;V ) and a linear time reduction which takes any input x 2 to an n tuple a of dominoe with jxj = n such that If M accepts x in time t 0 with space s 0 , then D tiles ....

[Article contains additional citation context not shown here]

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

.... monadic predicate symbols [27, 5] the fragment with only two individual variables [36, 31] 1 the guarded fragment with quanti cation of the form 9y(G(x; y) x; y) where the guard G(x; y) is atomic 2 [1] The current state of art in this eld is presented in the recent monograph [6]; see also [1, 43, 44, 13, 14, 18, 29, 32] For modal logicians the decision problem in rst order modal logics seemed almost hopeless. The following list covers basically all known results and leaves not too much space for maneuver: the monadic fragment of practically all modal predicate ....

....D of the form (1) in which jD t j = n, for every t 2 T . a Corollary 5.7. ML 2 1 and ML 1 have both the nite frame and the nite domain properties with respect to QK and QT. Proof. It is well known that the two variable fragment of rst order logic has the nite model property (see [6]) a 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 mon , and let ML mon 1 = ML 1 ML mon . In this case the formula T , corresponding to a world candidate T for 2 ML m 1 , is a monadic ....

[Article contains additional citation context not shown here]

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

....The Description Logic ALCRP(D) was introduced in [ 11 ] and extends ALC(D) with a role forming concrete domain constructor, i.e. it allows the de nition of roles with reference to the concrete domain. In this section, we extend the logic ALCI(D) with this role forming constructor. De nition 6 (Predicate Roles) A predicate role is an expression of the form 9(u 1 ; u n ) v 1 ; v n ) P where P is an n m ary predicate. The semantics is given as follows: 9( u 1 ; u n ) v 1 ; v m ) P ) I : f(a; b) 2 I I j u I i (a) x i for 1 i n; v ....

.... 0 k ) which encodes incrementation modulo 2 n , i.e. if k is the number binarly encoded by the propositional variables x 0 ; x n 1 and k 0 is the number binarly encoded by the propositional variables x 0 0 ; x 0 n 1 , then we have k 0 = k 1 modulo 2 n (see, e.g. 6 ] Assume a 2 (Edge[g ; p] I (where p is either = or conc l i ) and let b be the fringe node with pos(b) pos(a) 1, x be the g successor of a, and y be the g successor of b. The Edge concept ensures that, for each S[g ; p] successor c of a, we have pos(c) 6= pos(a) 1, i.e. ....

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

....In the NExpTime complete variant of the domino problem that we use, the task is not to tile the whole plane, but to tile a 2 n 1 2 n 1 torus, i.e. a 2 n 1 2 n 1 rectangle whose edges are glued together. See, e.g. 3, 15] for undecidable versions of the domino problem and [5] for bounded variants. De nition 11 Let D = D; H;V ) be a domino system, where D is a nite set of domino types and H;V D D represent the horizontal and vertical matching conditions. For s; t 2 N, let U(s; t) be the torus Z s Z t , where Z n denotes the set f0; n 1g. Let a = a 0 ....

....= d and (x s 1; y) d 0 , then (d; d 0 ) 2 H if (x; y) d and (x; y t 1) d 0 , then (d; d 0 ) 2 V (i; 0) a i for 0 i n. where n denotes addition modulo n. Such a mapping is called a solution for D w.r.t. a. The following is a consequence of Theorem 6.1. 2 in [5] (see also [16] Theorem 12 There exists a domino system D such that the following is a NExpTimehard problem: Given an initial condition a = a 0 a n 1 of length n, does D tile the torus U(2 n 1 ; 2 n 1 ) with initial condition a We reduce the NExpTime complete variant of the domino ....

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

....allowed) In the NExpTimecomplete variant of the domino problem that we use, the task is not to tile the whole plane, but to tile a 2 n 1 2 n 1 torus, i.e. a 2 n 1 2 n 1 rectangle whose edges are glued together. See, e.g. 2, 13] for undecidable versions of the domino problem and [3] for bounded variants. De nition 11 Let D = D; H; V ) be a domino system, where D is a nite set of domino types and H; V D D represent the horizontal and vertical matching conditions. For s; t 2 N , let U(s; t) be the torus Z s Z t , where Z n denotes the set f0; n 1g. Let a = a 0 ; ....

....y) d and (x s 1; y) d 0 , then (d; d 0 ) 2 H if (x; y) d and (x; y t 1) d 0 , then (d; d 0 ) 2 V (i; 0) a i for 0 i n. where n denotes addition modulo n. Such a mapping is called a solution for D w.r.t. a. The following is a consequence of Theorem 6.1. 2 in [3] (see also [14] Theorem 12 There exists a domino system D such that the following is a NExpTime hard problem: Given an initial condition a = a 0 a n 1 of length n, does D tile the torus U(2 n 1 ; 2 n 1 ) with initial condition a We reduce the NExpTime complete variant of the domino ....

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

....allowed) In the NExpTimecomplete variant of the domino problem that we use, the task is not to tile the whole plane, but to tile a 2 n 1 2 n 1 torus, i.e. a 2 n 1 2 n 1 rectangle whose edges are glued together. See, e.g. 3, 15] for undecidable versions of the domino problem and [5] for bounded variants. We now formally introduce bounded domino systems. De nition 15 Let D = D; H;V ) be a domino system, where D is a nite set of domino types and H;V D D represent the horizontal and vertical matching conditions. For s; t 2 N, let U(s; t) be the torus Z s Z t , where ....

.... behaviour of restricted, so called simple, Turing Machines (TMs) 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, provided that S(n) T (n) 2n [5]. Theorem 16 [ 5] Theorem 6.1.2] Let M be a simple TM with input alphabet . Then there exists a domino system D = D; H;V ) and a linear time reduction which takes any input x 2 to an n tuple a of dominoe with jxj = n such that If M accepts x in time t 0 with space s 0 , then D tiles ....

[Article contains additional citation context not shown here]

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

....even recursively enumerable. But in contrast to classical rst order logic, where the 1 early undecidability results of Turing and Church stimulated research and led to a rich and profound theory concerned with classifying fragments of rst order logic according to their decidability (see, e.g. [9]) there were no serious attempts to convert the negative results in rst order temporal logic into a classi cation 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.

....if (x; y) d and (x; y t 1) d 0 then (d; d 0 ) 2 V (i; 0) w i for 0 i n. where n denotes addition modulo n. B orger et al. show that it is NExpTime complete to decide if, for a given domino system D and a given n tuple w, D tiles U(2 n ; 2 n ) with initial condition w [ 6 ] . In the following, we will reduce this domino problem to satis ability of ALCF concepts w.r.t. TBoxes. We will rst give an informal explanation of 5 i.e. a rectangular grid whose edges are glued together Tree 0 : n # n u 9 :Tree 1 u :Tree 1 u n 1 # n 1 Tree 1 : ....

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

....more expressive than SGs. On the other hand, we will identify a tractable fragment of SGs that is larger than the class of trees. Instead of trying to prove new decidability or tractability results for CGs from scratch, our idea was to transfer decidability results from rst order logics [4] and from description logics [9, 10] to CGs. The goal was to obtain natural sub classes of the class of all CGs in the sense that these sub classes are de ned directly by syntactic restrictions on the graphs, and not by conditions on the rst order formulae obtained by translating CGs into FO. ....

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

....m) t(n; m) for all m and all sufficiently large n, and similarly t(n; m q) t(n; m) for all n and all sufficiently large m. iv) A tiling t is recurrent for d 0 2 D if t Gamma1 (d 0 ) N Theta N is infinite. Dominoes are a classical route to undecidability proofs through reduction, see [9]. Theorem 3.5 (Berger [8] Gurevich Koryakov [17] The class of dominoes that admit a periodic tiling is recursively inseparable from the class of dominoes that admit no tiling at all. In particular both the tiling problem and the periodic tiling problem are undecidable, in fact the tiling ....

....) This lemma proves Theorem 3.1. Indeed by virtue of the Gurevich Koryakov Theorem, i) and (ii) show fin sat(L 2 ) to be recursively inseparable from the complement of sat(L 2 ) so that in the terminology of the classical decision problem, L 2 is a conservative reduction class, see [9]. For the proof of the lemma we consider descriptions of valid tilings for D as Kripke structures. Think of the underlying grid N Theta N as a relational structure with horizontal and vertical successor relations H and V . A placement of domino pieces is encoded by unary predicates (basic ....

E. B orger, E. Gr adel, and Y. Gurevich, The Classical Decision Problem, Perspectives in Mathematical Logic, Springer, 1997.

.... In the context of discussing the functional programming language G odel 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 12 languages can be thought of as commands to be executed ....

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

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Borger, E., E. Gradel, and Y. Gurevich, The classical decision problem, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1997. Zbl 0865.03004. MR 99b:03004. 19, 23, 26

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E. B orger, E. Gr adel, and Yu. 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 in Mathematical Logic. Springer, 1997.

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

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

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

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