J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation 160(1-2): 62--87 (2000).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....quantifiers. Thus, we obtain that local properties of this modulo counting logic are decidable (Theorem 5.14) The decidability of non local properties is not answered in this paper, the missing tool is an analogue of Gaifman s locality theorem for this logic. Libkin [34 36] and Nurmonen [53] proved generalizations of a version of Gaifman s Theorem for this and other counting extensions of first order logic, but we could not make these generalizations serve our purposes. As mentioned above, the first step in our decidability proof is an application of Gaifman s Theorem. Gaifman s ....

....first order theory of does not work for this more expressive logic; but the second step of our proof, i.e. the recognizability of the set of Note that is built using the signature of while uses the larger signature of . Libkin [34 36] and Nurmonen [53] proved locality theorems for counting logics including modulo counting, but not in the form of Theorem 5.1. We could not make them work in our situation. 27 traces satisfying some local formula in FO (cf. Theorems 5.9 and 5.7) extends to the logic FO MOD. Thus, we obtain the decidability of local ....

J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, 160(1--2):62-- 87, 2000.

....radius neighborhoods of points. If the result of this counting satisfies certain criteria, then the structures considered are guaranteed to be elementary equivalent in a certain logic. This technique has been modified for first order logic [17] first order logic with counting modulo quantifiers [39] and first order logic extended by all unary generalized quantifiers [38] for the case of finite structures. Proofs of applicability of Hanf s technique typically are not very difficult [17, 15, 38, 40] We will see some examples in Section 4. The above results have motivated a study of general ....

....U) which satisfies card(B) 3nm but card(U) nm. It is not difficult to show that the duplicator has a winning strategy in the n round counting modulo m Ehrenfeucht Fraiss e game over A and B. But obviously A j= MAJxU(x) and B 6j= MAJxU(x) These results can be extended to the ordered case, see [39]. 7 If we want to give a game theoretic method to prove expressive bounds for first order logic with all unary quantifiers, different techniques must be used. The method we employ here is based on bijective Ehrenfeucht Fraiss e games. The rules of the game are the following. As before, the ....

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J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, to appear. Extended abstract in Proc. 11th IEEE Symp. on Logic in Computer Science (LICS'96), New Brunswick, NJ, July 1996, pages 484--493.

....and characterizations of these notions on structures of bounded degree. We start with a lemma which is one of our main technical tools and we apply it several times in this section. The idea of the proof given below is similar to all the earlier applications of Hanf s technique mentioned before [11, 16, 24, 25], but we believe this proof is simpler. First, we need two obvious facts stated previously in [6] Claim 4.1 Assume that A 2 STRUCT[oe] and h : N A r ( a) N A r ( b) is an isomorphism. Let d r. Then h restricted to S A d ( a) is an isomorphism between N A d ( a) and N A d ( ....

....queries. We say that a Boolean query Psi is strongly Gaifman local on STRUCT k [oe] if in Definition 2.2 STRUCT[oe] is replaced by STRUCT k [oe] i.e. we restrict the consideration to structures where each point has degree at most k. The idea of the proof given below is similar to the one in [25], where a characterization for Boolean queries definable in FO (and in FO with modular counting quantifiers) on structures of bounded degree, was given. Proposition 6.4 Let Psi be a Boolean query and k a natural number. Then Psi is strongly Gaifman local on STRUCT k [oe] iff Psi is definable ....

J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, to appear. Extended abstract in Proc. 11th IEEE Symp. on Logic in Computer Science, New Brunswick, NJ, July 1996, pages 484--493.

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J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation 160(1-2): 62--87 (2000).

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J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, 160:62--87, 2000. LICS 1996, Part I (New Brunswick, NJ).

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