Mortimer, M.: 1975, `On languages with two variables'. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 21, 135-140.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equality then the extension of with the identity role is most closely linked with FO . In fact, 19] show expressive equivalence. One extension of FO is the dual of Maslov s class K, denoted by K. The sense in which K extends FO is that the normal forms for FO formulae of Mortimer [22] belong to K. The language over which formulae in the class K are constructed is the language of first order logic without equality and without function symbols. Let # be a closed formula in negation normal form and # be a subformula of #. The # prefix of the formula # is the sequence of ....

M. Mortimer. On languages with two variables. Z. Math. Logik Grundlagen Math., 21:135--140, 1975.

....in NEXP time. Their results also rely on a small model property. They prove that every satisfiable two variable formula over arbitrary structures has a finite model of size at most exponential in the size of the formula, improving on a previous doublyexponential bound obtained by Mortimer [Mor74]. Despite the similarity between the statement of their result and ours, the two are essentially incompatible and neither result implies the other. The reasons for this are two fold. First, our results hold over words, i.e. over a unary vocabulary with built in ordering. In particular, unlike ....

M. Mortimer. On languages with two variables. Z. Math. Logik Grundlag. Math., 21:135-140, 1975.

....in NEXP time. Their results also rely on a small model property. They prove that every satisfiable two variable formula over arbitrary structures has a finite model of size at most exponential in the size of the formula, improving on a previous doubly exponential bound obtained by Mortimer [Mor74]. Despite the similarity between the statement of their result and ours, the two are essentially incompatible and neither result implies the other. The reasons for this are two fold. First, our results hold over words, i.e. over a unary vocabulary with built in ordering. In particular, unlike ....

M. Mortimer. On languages with two variables. Z. Math. Logik Grundlag. Math., 21:135-140, 1975.

....logic. Moreover, only two variables are needed. Then the decidability of the implication problem for word constraints follows from known results about first order logic with two variables (FO 2 ) Indeed, satisfiability of FO 2 sentences (with relational vocabulary and constants) is decidable [25], and the implication problem for word constraints can be reduced to satisfiability of such an FO 2 sentence. However, the complexity of testing FO 2 satisfiability is doubly exponential in the formula [25] and exponential in the model size [16] In contrast, our direct proof provides a ptime ....

....of FO 2 sentences (with relational vocabulary and constants) is decidable [25] and the implication problem for word constraints can be reduced to satisfiability of such an FO 2 sentence. However, the complexity of testing FO 2 satisfiability is doubly exponential in the formula [25] and exponential in the model size [16] In contrast, our direct proof provides a ptime test for word constraint implication (in the size of the words) Furthermore, results about FO 2 and its extensions are no longer of help for implication of full path constraints, where recursion is present ....

M. Mortimer. On languages with two variables. Zeitschr. f. math. Logik u. Grundlagen d. Math, 21:135--140, 1975.

....fragments of rst order logic de ned according to the number of individual variables used. Thus, FO k is the set of rst order sentences with at most k variables. The unsolvability of the pre x class 898 shows that FO 3 is unsolvable. On the other hand, it is known that FO 2 is solvable [Mor75] (in fact, satis ability of FO 2 is NEXPTIME complete [GKV97] The failure of the 0 1 law for the class 1 1 (898) implies its failure for the class 1 1 (FO 3 ) Le Bars, however, showed that his variant of KERNEL can be expressed in 1 1 (FO 2 ) without equality. Thus, FO 2 is ....

Mortimer, M.: On language with two variables, Zeit. fur Math. Logik und Grund. der Math. 21(1975), pp. 135-140.

....fore description logics, used to effectively specify and reason about classifications and properties of objects. Description logics are fragments of FO, some included in FO 2 . It is well known that FO 2 has many nice properties that the full FO lacks, such as decidability of satisfiability [79]. This explains why reasoning with ontologies can be tractable. A survey of description logics is provided in [73] 6. CONCLUSION In order to meaningfully contribute to the formal foundations of the Web, database theory has embarked upon a fascinating journey of rediscovery. In the process, some ....

M. Mortimer. On languages with two variables. Zeitschr. f. math. Logik u. Grundlagen d. Math, 21:135--140, 1975.

....every such sentence is either finite or cofinite. It was for the proof of this result that Ramsey developed his celebrated combinatorial theorems. 56 ERICH GR ADEL, PHOKION G. KOLAITIS AND MOSHE Y. VARDI not cover the case of FO 2 with equality. The full class FO 2 was considered by Mortimer [45]. He proved that this class is decidable by showing that it has the finite model property. An analysis of his proof shows that he actually established a bounded model property for FO 2 : if a FO 2 sentence # is satisfiable, then it is satisfiable in a model whose size is at most doubly ....

....since it is reducible to that of the G odel class. At that time it had not been detected yet that, contrary to G odel s claim, his decidability proof does not persist in the presence of equality. Thus, Scott s proof covers only FO 2 without equality. As mentioned in the introduction, Mortimer [45] established that FO 2 with equality has the finite model property, which implies that the satisfiability problem for FO 2 with equality is decidable. Actually, Mortimer s proof shows that every satisfiable FO 2 sentence with equality has a finite model whose size is doubly exponential in ....

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M. Mortimer, On language with two variables, Zeit. f ur Math. Logik und Grund. der Math., vol. 21 (1975), pp. 135--140.

....we need to agree on what we mean by the latter. Vardi [31] has stressed the importance of the robust decidability of modal fragments: not only is the modal logic K decidable, but it remains decidable when we add temporal constructs, fixed point operators, counting, FO 2 is decidable [25], and one might want to consider FO 2 as a candidate for explaining the good computational behavior of modal logic, but many simple extensions of FO 2 are undecidable or even highly undecidable, and, hence, FO 2 certainly can t explain robustness [17] Moreover, modal logics using operators ....

M. Mortimer. On languages with two variables. Zeitschr. f. math. Logik u. Grundlagen d. Math., 21:135--140, 1975. 13

....in NEXP time. Their results also rely on a small model property. They prove that every satisfiable two variable formula over arbitrary structures has a finite model of size at most exponential in the size of the formula, improving on a previous doublyexponential bound obtained by Mortimer [Mor74]. Despite the similarity between the statement of their result and ours, the two are essentially incompatible and neither result implies the other. The reasons for this are two fold. First, our results hold over words, i.e. over a unary vocabulary with built in ordering. In particular, unlike ....

M. Mortimer. On languages with two variables. Z. Math. Logik Grundlag. Math., 21:135-140, 1975.

....to other first order logics. The most natural candidate is FO 2 , the set of existential second order sentences with at most 2 variables in their first order part. It is well known that this class is decidable, i.e. FO 2 has a solvable satisfiability problem (see Mortimer s paper [Mor75] in 1975) Therefore, in 1994 in Oberwolfach, where open problems in Finite Model Theory were collected, Flum asked the question Does a 0 1 law hold for Sigma 1 1 (FO 2 ) 2 The aim of this paper is to prove that the 0 1 law does not hold for Sigma 1 1 (FO 2 ) and Sigma 1 1 (Minimal ....

M. Mortimer. On languages with two variables. Zeischr.f. math. Logik u. Grundlagen d. Math., 21:135--140, 1975.

....back into (tableaux proofs of) the original source logic. Logical and model theoretic implications are decidability and the nite model property. These results are not all new, for example, Theorems 6.5 and 6.6 and Corollary 7. 4 follow also from results for more expressive logics, for example [16, 35]. However, the signi cance of the results is that the method of obtaining them is new. Moreover, we believe the methods described here have the potential to be used for obtaining new results for less well studied logics. For instance, one could imagine that by studying selection based resolution ....

Mortimer, M.: 1975, `On Languages with two Variables'. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 21, 135-140.

....(with equality) that only has the variable symbols x and y i.e. the closure of atomic formulae involving no variables apart from x and y under Boolean operations and 9x, 9y. Throughout this paper we restrict attention to finite vocabularies consisting of relation symbols and constants. By [7], L 2 has the finite model property, i.e. every satisfiable sentence has a finite model. Consequently, the satisfiability problem for L 2 is decidable. Standard terminology uses the following: ffl sat(X) for the the set of 2 X that are satisfiable; ffl fin sat(X) for the set of 2 X that ....

M. Mortimer, On languages with two variables, Zeitschr. f. math. Logik u. Grundlagen d. Math., 21 (1975), pp. 135--140.

....2 Proof (Sketch) In other words, given a path l in schema S, we can tell whether there exists an instance and a valuation such that the evaluation of l yields a nonempty result. We consider here the fragment of first order logic called first order logic with two variables, and denoted FO2 [Mor75] The syntax is the usual one of first order logic without function symbols, except that only two variable symbols are allowed (instead of countably many) here z and z 0 . We reduce satisfiability of paths queries to satisfiability of FO2. Given a schema S = C; Sigma ) and a path l typable ....

M. Mortimer. On languages with two variables. In Zeitschr. f. mat. Logik und Grunlagen d. Math, Bd. 21, pages 135--140, 1975.

....undecidability result also entails the undecidability of the natural common extension of FO 2 and computation tree logic CTL. 1 Introduction Two variable rst order logic FO 2 stands out as one of the fragments of rst order logic whose satis ability problem is decidable. In fact Mortimer [9] showed that FO 2 has the nite model property and hence is decidable for satis ability. Recently the bound on model size has been improved by Gr adel, Kolaitis and Vardi [4] to locate the complexity of the satis ability problem for FO 2 in non deterministic exponential time. The renewed ....

....consecutive realizations of non royal 1 types. This can be achieved through the introduction of a series of indistinguishable copies of individual realizations. The observation that a corresponding extension of a structure preserves its normal form FO 2 theory is already used by Mortimer in [9]. Lemma 3.3 Let c be a positive constant. For any A 2 O (respectively A 2 WO) there is some B 2 O (respectively B 2 WO) satisfying exactly the same normal form FO 2 sentences as A, and such that for every non royal b 2 B there is an interval I(b) of at least c consecutive elements around b, ....

M. Mortimer, On languages with two variables, Zeitschr. f. math. Logik u. Grundlagen d. Math., 21, 1975, pp. 135-140.

.... er pre x 889 , a fragment that was proved decidable by G odel [1932] G odel claimed without proof that this fragment remains decidable also with equality, which was later refuted by Goldfarb [1984] The decidability and nite model property for the full class FO 2 was rst established by Mortimer [1975]. From Mortimer s [1975] proof follows also that (the satis ability problem for) FO 2 is decidable in nondeterministic doubly exponential time. This upper bound was recently improved by Gr adel, Kolaitis Vardi [1997] to nondeterministic exponential time. The NEXPTIME hardness of FO 2 even ....

.... a fragment that was proved decidable by G odel [1932] G odel claimed without proof that this fragment remains decidable also with equality, which was later refuted by Goldfarb [1984] The decidability and nite model property for the full class FO 2 was rst established by Mortimer [1975] From Mortimer s [1975] proof follows also that (the satis ability problem for) FO 2 is decidable in nondeterministic doubly exponential time. This upper bound was recently improved by Gr adel, Kolaitis Vardi [1997] to nondeterministic exponential time. The NEXPTIME hardness of FO 2 even without equality follows ....

Mortimer, M. (1975), `On languages with two variables', Zeitschr. f. math. Logic u. Grundlagen d. Math. 21, 135{ 140.

....fragments of first order logic defined according to the number of individual variables used. Thus, FO k is the set of first order sentences with at most k variables. The unsolvability of the prefix class 898 shows that FO 3 is unsolvable. On the other hand, it is known that FO 2 is solvable [Mor75] (in fact, satisfiability of FO 2 is NEXPTIME complete [GKV97] The failure of the 0 1 law for the class 1 1 (898) implies its failure for the class 1 1 (FO 3 ) Le Bars, however, showed that his variant of KERNEL can be expressed in 1 1 (FO 2 ) without equality. Thus, FO 2 is ....

Mortimer, M.: On language with two variables, Zeit. fur Math. Logik und Grund. der Math. 21(1975), pp. 135--140.

....of prenex formulas of the form 9 2 8 n , whose validity is decidable by [3] The result is stated with equality in the language, because at that time it was still believed that the 9 2 8 n class of formulas containing equality is decidable for validity. This was however refuted in [4] [11] extends Scott s result by including equality in the language and showing that such sentences cannot force infinite models, obtaining decidability as a corollary. 5] provides a simpler proof and shows that any formula which can be satisfied can be satisfied on a model whose size is single ....

M. Mortimer. On languages with two variables. Zeitsch. f. math. Logik und Grundlagen d. Math., 21:135--140, 1975.

....can be expressed using at most k variables. Note that L k does not limit the number of nested quantifiers in a formula since the same variable may be reused in nested subformulas, as in #x, y.P (x, y) # #x. Q(y, x) Properties of such language families have been studied, among others, in [16, 9, 8]. In our case, since we are dealing with roles and concepts, we will be interested only in those formulas that (i) have one or two free variables (though they may have closed subformulas) and (ii) have only monadic and dyadic predicates. Henceforth, we will use L k to refer to this ....

....of connectives. Corollary 4 [Mortimer] Subsumption is decidable for DL compose, trans, at least, at most . Lewis] Subsumption is undecidable for DL trans,at least,at most . The first result follows from the decidability of validity in the logic L 2 with equality, proven in [16]: the subsumption problem D=#E can be posed as the validity of the formula #x.T x #D# # T x #E#, which by Theorem 1, is in L 2 . The second result follows from the undecidability of validity for the class of formulas with quantifier prefix ###, shown in [14] the formula exhibited in ....

M. Mortimer, "On languages with two variables", Zeitschr. f. Math. Logik und Grundlagen d. Math., 21, pp.135--140, 1975.

....that only two different variable names occur in the formulae obtained by this translation. Thus, decidability of subsumption and other inference problems for these logics follows from the known decidability result for L 2 , i.e. first order logic with two variables and without function symbols [18, 10]. Recently, this decidability result has been extended to C 2 , i.e. first order logic with 2 variables and counting quantifiers [11] Independently, it has been proved in [19] that satisfiability of C 2 formulae can be decided in nondeterministic doubly exponential time. As an immediate ....

M. Mortimer. On languages with two variables. Zeitschr. f. math. Logik u. Grundlagen d. Math., 21:135--140, 1975.

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Mortimer, M.: 1975, `On languages with two variables'. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 21, 135-140.

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Mortimer, M., \On languages with two variables," Z. Math. Logik Grundlagen Math., vol. 21 (

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M. Mortimer. On languages with two variables. Z. Math. Logik Grundlagen Math., 21:135--140, 1975.

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M. Mortimer. On languages with two variables. Zeitschr. f. math. Logik u. Grundlagen d. Math, 21:135--140, 1975.

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M. Mortimer. On languages with two variables. Z. Math. Logik Grundlagen Math., 21:135-140, 1975.

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#22, 2000. #79# M. Mortimer. On languages with twovariables. Zeitschr. f. math. Logik u. Grundlagen d. Math,

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