J. Makowsky and Y. Pnueli. Computable Quantifiers and Logics over Finite Structures. In M. Krynicki, M. Mostowski, and L. Szczerba, editors, Quantifiers: Logics, Models and Computation, Volume I, pages 313--357. Kluwer Academic Publishers, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....oracle) over ordered structures, 53, 54, 22] The relationship of oracles, generalized quantifiers and subprograms was also investigated in [38, 37, 40, 41] See [31, 13] for a comprehensive overview. Notice that all syntactic complexity classes can be captured by generic generalized quantifiers [36, 39, 61]. In the context of databases, generalized quantifiers have been used as a means for increasing the expressive capability of database query languages. In [1] the transitive closure quantifier was applied ad hoc since connectivity cannot be expressed in Codd s relational calculus. A more ....

J. Makowsky and Y. Pnueli. Computable Quantifiers and Logics over Finite Structures. In M. Krynicki, M. Mostowski, and L. Szczerba, editors, Quantifiers: Logics, Models and Computation, Volume I, pages 313--357. Kluwer Academic Publishers, 1995.

.... i s are quantifierfree and do not include BIT then I is a quantifier free interpretation. A first order interpretation that is a projection in the sense of Valiant is called a first order projections, fop. 2 Now, for any problem whatsoever, Theta, we associate an operator of the same name, cf. [MP] where the operator we call Theta would be denoted Q Theta . Definition 2.3 ( IL] Operator Form of a Problem) Let oe and be vocabularies, and let Theta ae STRUC[ be any problem. Let I be any first order interpretation with I : STRUC[oe] STRUC[ Then Theta[I ] is a well formed formula ....

J.A. Makowsky and Y.B. Pnueli, "Computable quantifiers and logics over finite structures," to appear in Quantifiers: Generalizations, Extensions and Variants of Elementary Logic (Kluwer Academic Publishers, 1993).

....classes, subsuming a variety of known separation results. For instance, if (Q ) 2I consists only of the ATC quantifier (or of QMC from Section 3) we have FO(ATC) j ALOGSPACE [14] but PTIME Q FO(ATC;Q) j ALOGSPACE Q (the latter equivalence holds for an appropriate oracle model, see [18]) Furthermore, the result reveals the (well known) difference between Turing and many one reductions. This is, because IFP(Q) captures those problems reducible to Q in the former way, whereas those problems that are many one reducible to Q, are already contained in FO(Q IFP ; Q) this can be ....

J.A. Makowsky and Y.B. Pnueli. Computable quantifiers and logics over finite structures. In M. Krynicki, M. Mostowski, and L.W. Szczerba, editors, Quantifiers: Logics, Models and Computation, volume I, pages 313--357. Kluwer Academic Publishers, 1995.

....extending slightly the common usage of this term (where Q is the quantifier there exists a least ff ) Our final example of a polyadic lift, resumption, amounts to using a monadic quantifier to quantify over k tuples of individuals. This lift has found uses in computer science (cf. 4] and [16]) but can be given a linguistic motivation as well; cf. cases like (10) Most neighbours like each other (11) Most twins never separate 1.4 Definition. Let Q be of type h1; 1i. Define (Res k (Q) ARS ( Q A k RS Similarly for simple unary quantifiers. The route to the lifts here went via ....

J. Makowsky and Y. Pnueli, Computable quantifiers and logics over finite structures, to appear in M. Krynicki, M. Mostowski and L.W. Szczerba (editors), Quantifiers, Kluwer.

....all arities, i.e. all vectorizations of the operators are needed. In some recent papers dealing with generalized quantifiers on finite structures the assumption of being closed under vectorizations has even been taken as a part of the definition of logics with generalized quantifiers (see, e.g. [MP95]) Closing under vectorization also gives a (partial) solution to the problem of weakness of FO(Q) that prevented capturing PTIME by finitely many generalized quantifiers: by a result of Dawar [Daw95] if there is any reasonable logic that captures PTIME on unordered structures, then there is a ....

J. Makowsky and Y. Pnueli. Computable quantifiers and logics over finite structures. In M. Krynicki, M. Mostowski, and L. Szczerba, editors, Quantifiers: Logics, Models and Computation, volume I, pages 313--358. Kluwer Academic Publishers, Dordrecht/Boston/London, 1995.

.... ( 33] ii ) P = SigmaPS ) FO s ] PS [FO s ] 61] iii ) NP = HP [FO s ] 11, 56] iv) L NP = SAT [FO s ] SigmaSAT ) FO s ] SigmaHP ) FO s ] 3COL [FO s ] Sigma3COL) FO s ] 53, 59, 60] v) PSPACE = SigmaHEX ) FO s ] HEX [FO s ] [43, 44]) We have seen logical characterizations of complexity classes both in the presence and absence of the built in successor relation, and in the previous section we have seen how some logical characterizations of complexity classes fail in the absence of the built in successor relation. An immediate ....

J.A. Makowsky and Y.B. Pnueli, Computable quantifiers and logics over finite structures, in: Quantifiers: Generalizations, Extensions and Variants of Elementary Logic (M. Krynicki, M. Mostowski and L.W. Szczerba eds.), Kluwer Academic Publishers, to appear (1996).

....First order reductions give us a mechanism for forming an operator out of any problem Theta. The operator, which we will also call Theta, acts rather like a quantifier (and in fact this construction extends the generalized quantifiers of [BIS] A very similar construction is used in [MP] and in [KV] where the operator we call Theta would be denoted Q Theta . Definition 2.5 ( IL] Operator Form of a Problem) Let oe and be vocabularies, and let Theta STRUC[ be any problem. Let I be any first order reduction with I : STRUC[oe] STRUC[ Then Theta[I ] is a ....

J.A. Makowsky and Y.B. Pnueli, "Computable quantifiers and logics over finite structures," to appear in Quantifiers: Generalizations, Extensions and Variants of Elementary Logic (Kluwer Academic Publishers, 1993).

....by EPSRC Grant GR K 96564. 2 Most of this work was done whilst the author was visiting the University of Leicester. with an operator corresponding to the well known PSPACE complete decision problem Generalized Hex; that is, the logic ( SigmaHEX) FO s ] It was shown by Makowsky and Pnueli [12] (see also [11] that any problem in PSPACE can be defined by a sentence of the logic ( SigmaHEX) FO s ] and, conversely, that any problem definable by a sentence of this logic is in PSPACE. There are numerous other similar logical characterizations of complexity classes (that is, using ....

....a sentence of the form HEX [x; y (x; y) 0; max) where: jxj = jyj = k, for some k; is a quantifier free projection with successor ; and 0 (resp. max) is the constant symbol 0 (resp. max) repeated k times. PROOF. The result that ( SigmaHEX) FO s ] PSPACE is due to Makowsky and Pnueli [12] (see also [11] Like the proof of [10, Theorem 3.3] we proceed by induction on the complexity of a sentence OE 2 HEX [FO s ] The induction step assumes that every wellformed sub formula of OE is logically equivalent to a formula of the desired form and then treats the different ways in ....

J.A. Makowsky and Y.B. Pnueli, Computable quantifiers and logics over finite structures, in: (M. Krynicki, M. Mostowski and L.W. Szczerba eds.) Quantifiers : Generalizations, Extensions and Variants of Elementary Logic, Kluwer Academic Publishers, to appear.

....We leave these last two questions open for further research. 3 The Logics INV (L) and Delta(L) In this section we deal with weakly regular logics L on finite structures and prove some general facts concerning invariant definability. Background on logics and logical reductions can be found in [EFT94, EF95, MP95]. The proofs of all the statements in this section are easy and left to the reader. We first introduce our notion of parametrically definable classes of structures (or global relations) It will serve as a tool to obtain more invariantly definable classes once we have some at our disposal. ....

J.A. Makowsky and Y.B. Pnueli. Computable quantifiers and logics over finite structures. In M. Krynicki, M. Mostowski, and L.W. Szczerba, editors, Quantifiers: Generalizations, extensions and and variants of elementary logic, volume I, pages 313--357. Kluwer Academic Publishers, 1995.

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