Y. Gurevich and S. Shelah, Fixed Point Extensions of First Order Logic, Annals of Pure and Applied Logic 32 (1986), 265--280.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....operator. For a map f from relations to relations, as above, de ne R i 1 = R i [ f(R i ) and set ifpf = Rn , were n is minimal such that Rn = Rn 1 . IFP is the least logic closed under rst order connectives and in ationary xed points of formulae (which need not be positive) It is shown in [GS1986] that LFP = IFP, i.e. that a query is LFP de nable if, and only if, it is IFP de nable. Every formula of LFP is equivalent to one which is of the form 9y (lfp X; x ) y : y) for some rstorder (this result is implicit in [Imm1986] a proof appears in [EF1999, Lemma 8.2.4] Since every ....

Y. Gurevich and S. Shelah, Fixed-Point Extensions of First-Order Logic, Journal of Pure and Applied Logic, vol. 32, pp. 265-280, 1986.

....of the input database. For each Turing complexity class C there is a corresponding complexity class of queries, which, for simplicity, we also denote by C. It has been shown that order had a strong impact on the expressive power of query languages. In particular it was shown that fixpoint logic [GS86] captures exactly the class of queries computable in polynomial time over ordered relations [Imm86, Var82] This is know to be false over unordered relations, since there are very basic queries that are not expressible in fixpoint logic. The order plays a fundamental role in this ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

....of examples and scenarios in which deflationary fixed points arise in a natural way. It turns out that IFP, the extension of first order logic by inflationary and deflationary fixed points, has precisely the same expressive power as LFP. This has been known for some time for finite structures [9], but has been established only recently for the general case [13, 14] In finite model theory, LFP and IFP have, due to their expressive equivalence, often be used interchangeably. Nevertheless, we argue that least and inflationary fixed points have quite different properties. This becomes ....

....is the class of finite sets of even cardinality, that can of course easily be decided in polynomial time, but is not definable in LFP. Since both logics capture PTIME, IFP and LFP are equivalent on ordered finite structures. What about unordered structures It was shown by Gurevich and Shelah [9] that the equivalence of IFP and LFP holds on all finite structures. Their proof does not work on infinite structures, and indeed, there are some important aspects in which least and inflationary inductions behave differently. For instance, there are first order operators (on arithmetic, for ....

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Y. Gurevich and S. Shelah. Fixed-point extensions of firstorder logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

....inductions have very different properties and provide much more expressive power than least fixed point inductions. To some extent, these results generalise to monadic fixed point logics based on firstorder logic as well. However, for formulae without arity restrictions, Gurevich and Shelah [7] showed that least and inflationary fixedpoint logic have the same expressive power on finite structures. The most basic question concerning LFP and IFP on arbitrary structures is whether IFP is more expressive than LFP, a question that has was left open since the study of inflationary inductions ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

....is decidable in polynomial time. Proposition 3.44. IFP captures PTIME on ordered finite structures. Least versus inflationary fixed points. As both logics capture PTIME, IFP and LFP are equivalent on ordered finite structures. What about unordered structures It was shown by Gurevich and Shelah [58] that the equivalence of IFP and LFP holds on all finite structures. Their proof does not work on infinite structures, and indeed, there are some important aspects in which least and inflationary inductions behave differently. For instance, there ar first order operators (on arithmetic, say) whose ....

....to show is that the stage comparison relation for IFP inductions is in fact LFP definable. Theorem 3.47 (Inflationary Stage Comparison) For any formula 9(R, 3) in FO or LFP, the stage comparison relation inf is definable in LFP. On finite structures it is even definable in positive LFP. See [33, 58] for proofs in the case of finite structures and [73] for the more difficult construction in the general case. From this result, the equivalence of LFP on IFP follows easily. Theorem 3.48 (Gurevich Shelah, Kreutzer) For every IFP formula there is an equivalent LFP formula. inf 3 , 3) Proof. ....

Y. GUREVICH AND S. SHELAH, Fixed-point extensions of first-order logic, Annals of Pure and Applied Logic, 32 (1986), pp. 265-280.

....converges in polynomial time on all input databases. The first order logic augmented with IFP is called inflationary fixpoint logic and is denoted by FO IFP. The queries computed by FO IFP are the so called fixpoint queries, for which there various exist equivalent definitions in the literature [6, 15]. Close connections exist between the fixpoint FO extensions and the Datalog extensions [5] Datalog under inflationary fixpoint semantics expresses exactly the fixpoint queries, i.e. it is equivalent to FO IFP. This implies that Datalog under the inflationary fixpoint semantics is strictly more ....

Y. Gurevich, S. Shelah. Fixed-Point Extensions of First-Order Logic. Annals of Pure and Applied Logic 32 (1986). pp. 265-280.

....inductions have very different properties and provide much more expressive power than least fixed point inductions. To some extent, these results generalise to monadic fixed point logics based on firstorder logic as well. However, for formulae without arity restrictions, Gurevich and Shelah [7] showed that least and inflationary fixedpoint logic have the same expressive power on finite structures. The most basic question concerning LFP and IFP on arbitrary structures is whether IFP is more expressive than LFP, a question that has was left open since the study of inflationary inductions ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

.... would introduce a contradiction (variant of Russell s paradox) Therefore, logical systems must have certain restrictions on the forms of self references (if ever allowed) in order to keep themselves sound; for example, calculus [24, 20] does not allow negative self references (see also [13]) Through the formulae as types notion, this paradox corresponds to the fact that every type of is inhabited by a diverging program which does not produce any information; for example, the term ( x:xx) x:xx) can be typed with every type in . Therefore, even with the model mentioned ....

Y. Gurevich and S. Shelah. Fixed-point extensions of firstorder logic. Annals of Pure and Applied Logic, 32(3):265-- 280, 1986.

.... (or simply inconsistent, when D(U) is understood from context) otherwise we say that r is consistent with respect to D(U) or simply consistent, when D(U) is understood from context) cf. 3, 13, 15, 26] If r is consistent, then CHASE D (U) r ) satisfies D(U) and is an inflationary fixpoint [14] of D(U) on r ; this fixpoint is in fact the least fixpoint of D(U) on r , by Theorem 3 in [14] since the extended chase procedure is monotone for consistent nested relations. Full details can be found in [17] 21 The next theorem shows that an extended chase of a nested relation, r , with ....

.... with respect to D(U) or simply consistent, when D(U) is understood from context) cf. 3, 13, 15, 26] If r is consistent, then CHASE D (U) r ) satisfies D(U) and is an inflationary fixpoint [14] of D(U) on r ; this fixpoint is in fact the least fixpoint of D(U) on r , by Theorem 3 in [14], since the extended chase procedure is monotone for consistent nested relations. Full details can be found in [17] 21 The next theorem shows that an extended chase of a nested relation, r , with respect to D(U) is information wise equivalent to an extended chase of (r ) with respect to D. ....

Y. GUREVICH AND S. SELAH, Fixed-point extensions of first-order logic, Ann. Pure Appl. Logic 32, (1986), 265-280.

....relations) to the relation (or set of) that can be obtained by applying the recipe. Standard work was done by Moschovakis (Moschovakis 1974) and Aczel (Aczel 1977) These treatments study the theoretical expressivity of positive or monotone induction. One spin off of this field is fixpoint logic (Gurevich Shelah 1986), a subarea of databases (Abiteboul, Hull, Vianu 1995) As shown in (Denecker 1998) the abstract positive inductive definition logic defined in (Aczel 1977) is formally isomorphic with the formalism of propositional Horn programs under least model semantics. In the case of Horn programs, the ....

Gurevich, Y., and Shelah, S. 1986. Fixed-point Extensions of First-Order Logic. Annals of Pure and Applied Logic 32:265--280.

.... to a set of NJDs, D, denoted by ORCHASE D (r) or simply ORCHASE(r) when D is understood from context) as being information wise equivalent to the fixpoint of r with respect to T D , namely ORCHASE D (r) i =0 T D i (r) We note that ORCHASE is an inflationary fixpoint operator [8], since it can easily be verified that for any or relation, r, over U, r T D (r) The motivation for using inflationary semantics is that the uniqueness of the or chase follows directly from its parallel semantics rather than by a standard laborious proof as in [13, 16] In what ....

Y. Gurevich and S. Selah, Fixed-point extensions of first-order logic, Annals of Pure and Applied Logic, Vol. 32, pp. 265-280, 1986. 15

....finite model theory. Good examples are the results of Section 5 (the relationship to complexity classes) and of Section 6 (on 0 1 laws) Another example is Immerman s result [Imm86] that for finite structures, fixpoint logic is closed under complement, and the related result by Gurevich and Shelah [GS86] that for finite structures, different natural fixed point logics all have the same expressive power. A recent nice example was 13 This contrasts in an interesting way with results in abstract model theory (cf. Bar85, Mak85] where Robinson consistency is a very strong property, and in fact ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

....### converges in polynomial time on all input databases. First order logic augmented with ### is called in ationary xpoint logic andisdenotedby FO IFP. The queries computed by the language FO IFP are the so called xpoint queries, for whichvarious equivalent de nitions exist in the literature [7, 20]. Close connections exist between the xpoint ## extensions and the Datalog extensions [4] Datalog# under in ationary xpoint semantics expresses exactly the xpoint queries, i.e. it is equivalenttoFO IFP. This implies that Datalog# under the in ationary xpointsemantics is strictly more ....

Y. Gurevich, S. Shelah. Fixed-Point Extensions of First-Order Logic. Annals of Pure and Applied Logic, 32 (1986). pp. 265-280.

.... = IFP x,X #( x, X, b, C ) a ## a # X# . Inflationary fixed point logic has been introduced to finite model theory by Gurevich [34] but has been studied in generalized recursion theory under the title non monotone inductive definability long before (see [61] Gurevich and Shelah [37] proved that it has the same expressive power as the better known least fixed point logic. Example 3.1. The following IFP sentence defines the class of connected graphs: # = #y#x[IFP x,X x = y # #y(Xy # Eyx) x. Note that the variable x is bound by the fixed point operator and then ....

Y. Gurevich and S. Shelah, Fixed point extensions of first-order logic, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 265--280.

....relations to relations defined by the formula . For our purposes we prefer 2 to alternatively view fixpoint logic as a programming language built from first order logic using the programming constructs of relational assignment, composition, and inflationary while change loop. It is well known [8, 1] that our alternative view on fixpoint logic is equivalent to the usual one. We omit formal definitions of syntax and semantics of fixpoint logic programs; instead, we give an example. Example 2.1 Let oe consists of a single binary relation name E. Structures over oe can be viewed as directed ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

.... A k we let A j= IFP x;X (x; X; b; C) a ( a 2 X1 : Inflationary fixed point logic has been introduced to finite model theory by Gurevich [34] but has been studied in generalized recursion theory under the title non monotone inductive definability long before (see [61] Gurevich and Shelah [37] proved that it has the same expressive power as the better known least fixed point logic. Example 3.1. The following IFP sentence defines the class of connected graphs: 8y8x[IFP x;X x = y 9y(Xy Eyx) x: Note that the variable x is bound by the fixed point operator and then re introduced ....

Y. Gurevich and S. Shelah. Fixed point extensions of first--order logic. Annals of pure and applied logic, 32:265--280, 1986.

....y) where 2 8 TC m . Consequently, for all m, TC m ( 8 TC m ( TC m 1 . Also, the Hierarchy Theorem subsumes the Main Theorem of [31] 6 Extensions of Datalog We now discuss a class of languages intermediate between (FO TC) and L 1 . They are fragments of fixpoint logic (see [17, 20, 32]) and were described in game theoretic terms in [31] Here we describe them as extensions of Datalog. Datalog is a now very popular database query language which, in its pure form, consists of function free and negation free Horn clauses. Although Datalog has very nice properties and deals with ....

Y. Gurevich and S. Shelah, Fixed Point Extensions of First Order Logic, Annals of Pure and Applied Logic 32 (1986), 265--280.

....fixpoint operator binds a predicate symbol R that is free and that appears only positively (i.e. under an even number of negations) in the formula. The semantics is given by the least fixpoint of the formula. The same expressive power can be achieved using an inflationary fixpoint logic (FO IFP) [GS86]. The inflationary fixpoint operator (IFP) no longer requires that R occur positively in OE(R) and convergence is instead guaranteed in polynomial time by accumulating iterations of OE(R) up to a fixpoint. Inflationary fixpoint formulas are defined next. Inflationary fixpoint formulas are ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

....At the heart of our framework is the view of fixpoint logic as 1st order logic augmented with iteration. While traditionally fixpoint logic was viewed as the extension of 1st order logic by recursion (cf. Mos74] iteration proved to be a more general extension to 1st order logic than recursion [AV89, GS86, Lei90]. In both inflationary and noninflationary fixpoint logics, iteration is applied in its simplest form: sequential and deterministic. It turns out, however, that in order to express certain problems in fixpoint logic one seems to require more elaborate forms of iteration, such as nondeterministic ....

....fixpoint of . Positive fixpoint logic is 1st order logic augmented with the least fixpoint formation rule for positive formulas. It is easy to see that IFP is at least as expressive as positive fixpoint logic. Gurevich and Shelah showed that in fact the two logics have the same expressive power [GS86] (see also [Lei90] The complexity theoretic aspects of IFP were studied in [CH82, Imm86, Var82] These papers actually focused on positive fixpoint logic, but, as observed above, positive fixpoint logic and IFP have the same expressive power. It is known that IFP captures the complexity class ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

....variables (X; Y; ii) assignment of FO queries to variables, and (iii) a while construct allowing to iterate a program while some first order condition (e.g. X = holds 5 . We will see that the same expressive power as FP can be achieved using an inflationary fixpoint logic (IFP) [GS86], where the iteration of the formula is cumulative, so converges even if the formula is not monotonic in the bound predicate. Also, while is equivalent to noninflationary fixpoint logic (NFP) AV89] The operator NFP iterates an arbitrary FO formula up to a fixpoint, which may or may not exist. ....

Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

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Y. Gurevich and S. Shelah, Fixed Point Extensions of First Order Logic, Annals of Pure and Applied Logic 32 (1986), 265--280.

No context found.

Y. Gurevich and S. Shelah, Fixed-point extensions of first-order logic, Annals of Pure and Applied Logic 32 (1986), 265--280.

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Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32:265--280, 1986.

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Y. Gurevich and S. Shelah, "Fixed-point extensions of first-order logic," Annals of Pure and Applied Logic 32 (1986), 265--280.

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Y. Gurevich and S. Shelah. Fixed point extensions of first--order logic. Annals of pure and applied logic, 32:265--280, 1986.

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