H. Gaifman, On local and non-local properties, in Logic Colloquium '81, North Holland, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by positive boolean combination in the definition . Thomas proved the following theorem. Theorem 2.1. A language is F 1 definable if and only if it is TLT. In fact, this result is a particular instance of the general fact that first order formulas can express only local properties [9, 25, 26]. Theorem 2.1 gave a combinatorial description of the F 1 definable languages but also led to the next question : given a finite deterministic automaton A, is it decidable whether the language accepted by A is F 1 definable The reader is referred to [3] or to [21, p. 47] for an explanation ....

H. Gaifman, On local and non-local properties, in Proc. of the Herbrandt Symposium, Logic Colloquium'81 (J. Stern, ed.), Studies in Logic 107, NorthHolland, Amsterdam, (1982), 105--135.

....bounds, but the logic there does not have aggregate operators. We thus combine the two, which results in a logic L aggr . It defines every arithmetic operation and every aggregate function. We then show that it has very nice behavior: its formulae satisfy analogs of Hanf s [14; 19] and Gaifman s [16] theorems, meaning that it can only express local properties. In particular, properties such as connectivity of graphs cannot be expressed. We then consider a theoretical language RL , similar to those defined in [5; 33] and explain how it models all the features of SQL. Next, we show an ....

....y) R 2 (y; z) z) c 10 6) 4. AGGREGATE LOGIC: EXPRESSIVE POWER In this section we deal with the expressive power of the aggregate logic. Our main goal is to show that it satisfies a very strong locality property. Locality properties were introduced in model theory by Hanf [19] and Gaifman [16], and recently, following [14] they were a subject of renewed attention (see, e.g. 11; 30; 31; 33; 37] and references therein) Intuitively, those properties say that the behavior of logical formulae depends on the structure of small neighborhoods. They imply strong expressivity bounds for ....

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H. Gaifman. On local and non-local properties, Proceedings of the Herbrand Symposium, Logic Colloquium '81, North Holland, 1982.

....applies: all existence and equivalence results in this section are effective. This fact is not stated explicitly in the following lemmas, but it can always be checked easily. 5. 1 Reduction to local properties The main tool in this section is Gaifman s locality theorem for first order logic [25]. We will only need the special case of Gaifman s Theorem where the signature contains only unary and binary relation symbols which makes the definitions a little bit easier. Let us consider a model of the form 5 where is a set of unary relations, and is a set of binary relations. ....

.... has to be renamed into a fresh variable if ) and for atomic formulae. It is allowed that the formula contains the variable free. Moreover, the formula certainly contains free if contains at least one quantifier. Now Gaifman s Theorem states the following [25]. Theorem 5.1. Let be a first order sentence over the signature of the model 5 . Then is logically equivalent to a Boolean combination of sentences of the form C E is a first order formula over the signature of that contains at most free and ....

H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium '81, pages 105--135. North Holland, 1982.

....Pin [9] A recognizable subset of A is definable by a boolean combination of existential formulae of F 1 (S) if and only if it is strongly locally threshold testable. In fact, these results are particular instances of the general fact that first order formulas can express only local properties [26,78,79]. We now give some effective characterizations of the families of sets introduced above. In order to keep a standard notation in subsequent statements, we shall denote by L a recognizable subset of A , by S(L) the syntactic semigroup of L, by : A S(L) the 2 3 b a 1 24 syntactic ....

....is definable by a boolean combination of existential formulae of F 1 (S) if and only if it is strongly locally threshold testable. only if it is left locally threshold testable. Again these results are just variations on the theme that first order formulas can express only local properties [26,78,79]. An effective characterization of this last class has been obtained by Wilke. First Theorem 3.24 can be extended to infinite words as follows Theorem 3.29. Wilke [84,85] A recognizable subset of A is locally threshold testable if and only if its syntactic semigroup is aperiodic and its ....

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H. Gaifman, 1982, On local and non-local properties, in Proc. of the Herbrandt Symposium, Logic Colloquium'81 (J. Stern, ed.), Studies in Logic 107, North-Holland, Amsterdam, 105--135.

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H. Gaifman, On local and non-local properties, in Logic Colloquium '81, North Holland, 1982.

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H. Gaifman. On local and non-local properties. In Proceedings of the Herbrand Symposium, Logic Colloquium '81, pages 105--135. North Holland, 1982.

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H. Gaifman, On local and non-local properties. in "Proceedings of the Herbrand Symposium, Logic Colloquium '81," North Holland, 1982.

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H. Gaifman. On local and non-local properties. In Proceedings of the Herbrand Symposium, Logic Colloquium '81, pages 105--135, North Holland, 1982.

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H. Gaifman, On local and non-local properties, in "Proceedings of the Herbrand Symposium, Logic Colloquium '81," North Holland, 1982.

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H. Gaifman, On local and non-local properties, in "Proceedings of the Herbrand Symposium, Logic Colloquium '81," North Holland, 1982.

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H. Gaifman, On local and non-local properties, in "Proceedings of the Herbrand Symposium, Logic Colloquium '81," North Holland, 1982.

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H. Gaifman. On local and non-local properties. In Proceedings of the Herbrand Symposium, Logic Colloquium '81, pages 105--135. North Holland, 1982.

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H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium '81, pages 105--135. North Holland, 1982.

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H. Gaifman. On local and non-local properties, Proceedings of the Herbrand Symposium, Logic Colloquium '81, North Holland, 1982.

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H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium '81, pages 105--135. North Holland, 1982.

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H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium '81, pages 105-135. North Holland, 1982.

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H. Gaifman, On local and nonlocal properties, Proceedings of the Herbrand Symposium (Marseilles,

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H. Gaifman. On local and non-local properties. In Proceedings of the Herbrand Symposium, Logic Colloquium '81, North Holland, 1982.

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H. Gaifman. On local and non-local properties. Logic Colloquium 1981, North Holland, 1982.

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H. Gaifman. On local and non-local properties. In Proceedings of the Herbrand Symposium, Logic Colloquium '81, North Holland, 1982.

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H. Gaifman. On local and non-local properties. Logic Colloquium 1981,North Holland, 1982.

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H. Gaifman, On local and nonlocal properties, in Logic Colloquium '81, J. Stern, ed., North Holland, 1982, pp. 105-135.

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H. Gaifman, On local and non-local properties, in Proc. of the Herbrandt Symposium, Logic Colloquium'81 (J. Stern, ed.), Studies in Logic 107, NorthHolland, Amsterdam, (1982), 105-135.

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H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium '81, pages 105-135, 1982, North Holland.

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H. Gaifman. On local and non-local properties. In J. Stern, editor, Logic Colloquium '81, pages 105-135. North Holland, 1982.

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