R. Maddux. The equational theory of ca3 is undecidable. Journal of Symbolic Logic, 45(2):311--316, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....results of the previous section can be explained by the fact that all the undecidable fragments there are in a sense three dimensional , which is often a cause of bad computational properties. The three variable fragment of classical rst order logic is undecidable even without equality [41], and products of three propositional modal logics are usually undecidable [34] In Theorems 1, 2, and 3, the linear time operator U can be applied to formulas with two free variables, and so we can quantify in three dimensions : one temporal and two domain. In Theorem 4 we also have quanti ....

R. Maddux. The equational theory of CA3 is undecidable. Journal of Symbolic Logic, 45:311-316, 1980.

....i.e. the domains of the models are closed under all permutations and substitutions. MLR n for n at least 3 has an undecidable satisfaction problem when interpreted on the class of cubic models with universes of the form U . This follows simply from similar results in first order logic (see [11] for a strong result) When we interpret MLR n on local cubes, the satisfaction problem becomes exptime complete [12] In fact, an expansion of MLR n can be used to decide the guarded fragment, as is shown in [12] Not only does the complexity increase when we go from two to higher dimensions, it ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45(2):311--316, 1980.

....[5] with sole temporal operator now or some time in the future is not decidable. These negative results become less surprising when we realize that first order (bundled) CT L allows a form of quantification in three dimensions, something that is known to be associated with undecidability [17, 13]. A natural way to limit the interaction between the three dimensions is to restrict applications of first order quantifiers to state (i.e. path independent) formulas, and applications of temporal operators and path quantifiers to formulas with at most one free variable. The resulting fragment ....

....The negative results of Section 3 can be explained by the fact that all the undecidable fragments there are in a sense three dimensional , which is often a cause of bad computational properties. The three variable fragment of classical first order logic is undecidable even without equality [17], and products of three propositional modal logics are usually undecidable [13] In Theorem 4 we also have quantification in three dimensions: temporal operators, path quantifiers and the domain quantification. A natural way to reduce the interaction between the dimensions is to restrict ....

R. Maddux. The equational theory of CA 3 is undecidable. J. Symbolic Logic, 45:311--316, 1980.

....every pair of diamonds (since we are in S5, the con uence axiom follows) The formula (1.1) does not follow from S5 for n 3. cf. 5] Construction 3.2.68) Table 1 shows how quickly the satis ability problem becomes hard. n 3 np complete nexptime complete undecidable [7] 9] [8] Table 1. Complexity classes of S5 . We note that the undecidability result states something stronger: namely every axiomatic system extending S5 (n 3) with axioms valid on the class of n dimensional powers is undecidable as well. The study of cylindric algebras from a modal logical ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45(2):311{ 316, 1980.

....that every satis able S5 formula can be satis ed on a model of size 2 O(j j) Gr adel, Kolaitis and Vardi [3] and that both bounds are optimal. As a corollary, both logics are decidable. For n 3, S5 does not have the nite model property and is in fact undecidable, a result of Maddux [6]. To these results we add that in extensions L of S5 , every L satis able formula is satis able in an L model whose size is bounded by a polynomial function in j j. Recall that the language of S5 is the propositional language equipped with two existential modalities 3 1 and 3 2 and two ....

....one can show that the logic of n times products of S5 frames is a subfragment of the n variable FOL and the translation is de ned in the same way as for above. However, unlike S5, in the case of S5 , for n 3, the lattice of extensions of S5 is much more complicated. It is shown by Maddux [6] that already S5 is undecidable. Moreover, every logic in between S5 and the logic of 3 times products of S5 frames is undecidable and there exist in nitely many logics in that interval which do not have the nite model property. For every formula , let Sub( denote the set of all subformulas ....

R. Maddux, The equational theory of CA 3 is undecidable, Journal of Symbolic Logic 45(1980), pp. 311-317.

....every satis able S5 formula can be satis ed on a model of size 2 O(j j) Gr adel, Kolaitis and Vardi [3] and that both bounds are optimal. As a corollary, both logics are decidable. For n 3, S5 does not have the nite model property and is in fact undecidable, a result of Maddux [6]. To these results we add that in extensions L of S5 , every L satis able formula is satis able in an L model whose size is bounded by a polynomial function in j j. Recall that the language of S5 is the propositional language equipped with two existential modalities 3 1 and 3 2 and two ....

....can show that the logic of n times products of S5 frames is a subfragment of the n variable FOL and the translation is de ned in the same way as for above. However, unlike S5, in the case of S5 , for n 3, the lattice of extensions of S5 is much more complicated. It is shown by Maddux [6] that already S5 is undecidable. Moreover, every logic in between S5 and the logic of 3 times products of S5 frames is undecidable and there exist in nitely many logics in that interval which do not have the nite model property. For every formula , let Sub( denote the set of all ....

R. Maddux, The equational theory of CA 3 is undecidable, Journal of Symbolic Logic 45(1980), pp. 311-317.

....results of the previous section can be explained by the fact that all the undecidable fragments there are in a sense three dimensional , which is often a cause of bad computational properties. The three variable fragment of classical rst order logic is undecidable even without equality [41], and products of three propositional modal logics are usually undecidable [34] In Theorems 1, 2, and 3, the linear time operator U can be applied to formulas with two free variables, and so we can quantify in three dimensions : one temporal and two domain. In Theorem 4 we also have quanti ....

R. Maddux. The equational theory of CA3 is undecidable. Journal of Symbolic Logic, 45:311-316, 1980.

....related to product logics were obtained in algebraic logic. This is due to the fact that the modal algebras corresponding to S5 n are well known in this area: the representable diagonal free cylindric algebras of dimension n. Thus the respective algebraic logic results of Johnson [9] and Maddux [12] imply that S5 n is non nitely axiomatizable and undecidable. Given a recursively enumerable set L of n modal formulas, if we can enumerate those formulas which are not in L then we obtain a decision algorithm for L. Obviously, this can be done if (A) L has the nite model property , i.e. ....

R. Maddux. The equational theory of CA3 is undecidable. Journal of Symbolic Logic, 45:311{ 316, 1980.

....axiomatizable) 2D products are the square of the logic of the frame hN; i (see [25] and the compass logic of Venema [26] see [17] J. Logic Computat. Vol. 11 No. 00 34, pp. 1 23 2001 c Oxford University Press 2 On the products of linear modal logics As concerns 3D products, Maddux [16] proved (in the algebraic setting) that any logic between [S5; S5; S5] and S5 S5 S5 is undecidable. 1 And quite recently Hirsch et al. 12] have shown that no logic between K 3 and S5 3 is decidable and finitely axiomatisable: see also [15] For more details consult [6] The main aim of ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45:311--315, 1980.

....and modal logic methods) showed in particular the decidability and the finite model property (f.m.p. of QCL 2[33] 29] 25] and the undecidability of QCL 3, and even of a restricted fragment of the latter, without equality and with the atoms only of the form P i (x 1 ,x 2 ,x 3 )[23]. A more recent development in the field is the discovery of Guarded Fragment (GF) of classical logic [5] 39] It contains formulas in which all quantifiers are relativised ( bounded ) so that they occur only in the subformulas of the form #x (R(x, y) # #(x, y) where x, y are lists of ....

....equivalent to classical first order logic with the three variables x, y, z, the atoms P 3 k (x, y, z) and non relativised quantifiers. The latter is recursively isomorphic to the equational theory of representable diagonal free 3 dimensional cylindric algebras, and thereofore is undecidable by [23]. Example 3.6 If instead of QCL we take the theory of transitive guards, the Guarded Fragment even with two individual variables becomes undecidable. However, if it is also required that the set A should contain only monadic atoms, the decidability is regained [14] Example 3.7 On the other ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45: pp. 311-315, 1980.

....diamonds (since we are in S5, the con uence axiom follows) The formula (1.1) does not follow from S5 n for n 3. cf. 5] Construction 3.2.68) Table 1 shows how quickly the satis ability problem becomes hard. S5 1 S5 2 S5 n , n 3 np complete nexptime complete undecidable [7] 9] [8] Table 1. Complexity classes of S5 n . We note that the undecidability result states something stronger: namely every axiomatic system extending S5 n (n 3) with axioms valid on the class of n dimensional powers is undecidable as well. The study of cylindric algebras from a modal logical ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45(2):311{ 316, 1980.

....the logics of the form L Theta S5 2 are undecidable and non finitely axiomatizable, if K L S5. K 3 is a natural example of an undecidable, recursively enumerable, Kripke complete logic which has the finite model property (see [2] Note that undecidability of S5 3 follows from Maddux [8]. Non finite axiomatizability of the two logics at the end of the above interval was also known: see Johnson [6] for S5 3 and [7] for K 3 . In the proofs we will use the following result of Hirsch Hodkinson [5] It is undecidable whether a finite relation algebra is representable. 1) For ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45:311--316, 1980.

....related to product logics were obtained in algebraic logic. This is due to the fact that the modal algebras corresponding to S5 n are well known in this area: the representable diagonal free cylindric algebras of dimension n. Thus the respective algebraic logic results of Johnson [8] and Maddux [11] imply that S5 n is non nitely axiomatisable and undecidable, whenever n 3. Undecidability and the lack of product nite model property for all product logics between K4 n and S5 n (n 3) was rst proved by Zakharyaschev. Non nite axiomatisability of K n (for n 3) was shown in ....

R. Maddux. The equational theory of CA3 is undecidable. Journal of Symbolic Logic, 45:311{ 316, 1980.

....are decidable, and so on. Starting from 3, there exist non representable dimensional diagonal free cylindric algebras, and the equational theories of the corresponding classes are undecidable. For detailed results in this direction we refer to the above cited [11] as well as to R. Maddux [16], the papers by the members of Dutch, Hungarian and English groups J. van Benthem, Y. Venema, M. Marx, H. Andreika, I. Nemeti, I. Sain, A. Kuruz, R. Hirsch and I. Hodkinson, 1] 2] 12] 4] 20] 25] 26] 13] and many others. The simplest variety of diagonal free cylindric algebras, that ....

R. Maddux, The equational theory of CA 3 is undecidable, 45(1980), pp. 311-317.

....i.e. the domains of the models are closed under all permutations and substitutions. MLR n for n at least 3 has an undecidable satisfaction problem when interpreted on the class of cubic models with universes of the form n U . This follows simply from similar results in first order logic (see [11] for a strong result) When we interpret MLR n on local cubes, the satisfaction problem becomes exptime complete [12] In fact, an expansion of MLR n can be used to decide the guarded fragment, as is shown in [12] Not only does the complexity increase when we go from two to higher dimensions, it ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45(2):311--316, 1980.

....i.e. the domains of the models are closed under all permutations and substitutions. MLR n for n at least 3 has an undecidable satisfaction problem when interpreted on the class of cubic models with universes of the form n U . This follows simply from similar results in first order logic (see [9] for a strong result) When we interpret MLR n on local cubes, the satisfaction problem becomes exptime complete [10] In fact, an expansion of MLR n can be used to decide the guarded fragment, as is shown in [10] The jump in complexity coincides with a much harder to prove finite model ....

R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45(2):311--316, 1980.

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R. Maddux. The equational theory of ca3 is undecidable. Journal of Symbolic Logic, 45(2):311--316, 1980.

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R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45: pp. 311--315, 1980.

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R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45(2):311-- 316, 1980.

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R. Maddux. The equational theory of CA3 is undecidable. Journal of Symbolic Logic, 45:311-315, 1980.

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R. Maddux. The equational theory of CA 3 is undecidable. Journal of Symbolic Logic, 45: pp. 311--315, 1980.

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R. Maddux. The equational theory of CA 3 is undecidable. J. Symbolic Logic, 45(2), (1980), pp. 311-316.

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