E. Gradel and I. Walukiewicz. Guarded xed point logic. In Logic in Computer Science, pages 45-54, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....axioms is decidable, on the condition that binary predicates occur only in guards. The complexity bound given there is non elementary. In [ST01] the complexity bound for GF with transitive guards is improved to 2EXPTIME and also shown NEXPTIME hard. Another fragment was explored in [GW99], see also [Gr a99a] There it has been shown that GF; the guarded fragment of rst order logic extended with a calculus style xed point operator, is still decidable and in 2EXPTIME. This fragment does contain the simple modal logics S4, but the resulting decision procedure is much more ....

E. Gradel and I. Walukiewicz. Guarded xed point logic. In LICS'99, pages 45-54, 1999.

.... logic [Var97, ANB98] see also [Gab81] Sometime, these fragments are augmented by features that are not present in the standard rst order language allowing more expressive power, often at the cost of losing decidability (see e.g. GOR97] By contrast, the guarded xed point logic LGF [GW99] is a decidable fragment of xed point rst order logic in which can be naturally embedded the modal calculus (see e.g. Koz83] Once an interesting decidable fragment is identi ed, the design of decision procedures that meet the best worst case complexity upper bounds is often the next step ....

E. Gradel and I. Walukiewicz. Guarded xed point logic. In LICS'99, pages 45-54, 1999.

....including decidability of the satis ability problem [1] Various decision procedures for the fragment and its extensions have been developed. They utilise di erent techniques such as ordered resolution, model theoretic constructions, alternating automata or embedding into second order logic [6, 4, 9, 10]. However, the devised decision procedures also have some drawbacks. In particular, they exhibit at least double exponential worst case time and space complexity [4, 9] which is in contrast to the low complexity of the satis ability problem of basic modal logic. Moreover, extensions of the guarded ....

E. Gradel and I. Walukiewicz. Guarded xed point logic. In Proc. LICS'99, pp. 45-54. IEEE Computer Society Press, 1999.

.... Benthem, and N emeti is the following problem: If we extend GF by least and greatest xed points, do we still get a decidable logic If yes, what is its complexity To put it di erently, what is the penalty, in terms of complexity, that we pay for adding xed points to the guarded fragment In [13] we were able to give a positive answer to this question. The modeltheoretic and algorithmic methods that are available for the calculus on one side, and the guarded fragments of rst order logic on the other side, can indeed be combined and generalized to provide positive results for guarded ....

....by the maximal arity of the relation symbols, but there are other variants of guarded logics where the width may be larger. Note that for guarded xed point sentences of bounded width the complexity level is the same as for calculus and for GF without xed points. The proof that we give in [13] relies on alternating two way tree automata 4 (on trees of unbounded branching) on a forgetful determinacy theorem for parity games, and on a notion of tableaux for guarded xed point sentences, which can be viewed as tree representations of structures. We associate with every guarded xed ....

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E. Gr adel and I. Walukiewicz, Guarded xed point logic, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45-54.

....logics (in particular the good algorithmic properties of logics like CTL or the modal calculus that extend propositional modal logic by natural recursion mechanisms) are also re ected in GF. Most notably, not only is GF decidable itself [1] but so is its canonical xed point extension, GF [9]. GF extends GF so as to render de nable least and greatest xed points of guardedly de nable positive operators of arbitrary arities. In particular it extends the modal calculus to the guarded domain. Unlike its modal companion, however, it no longer shares the nite model property, though it ....

....of the tuple x of free variables, i.e. if (x) is logically equivalent to (x) guarded(x) Note that any formula of GF is a Boolean combination of quanti er free formulae and variable guarded GF formulae. 3. 2 Guarded xed point logic GF Guarded xed point logic GF as introduced in [9] is the natural extension of GF by means of least and greatest xed points (or corresponding systems of simultaneous xed points) Syntax of GF. Starting from GF, with second order variables X;Y;Z; that are treated like predicates in but may not be used in guards, we augment the syntax ....

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E. Gr adel and I. Walukiewicz, Guarded xed point logic, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45-54.

....properties of modal logics (and hence also description logics) Guarded logics have received considerable attention in the last years and it seems that the notion of guarded quanti cation is indeed very important for the design of logics that are both expressive and algorithmically manageable. See [1, 2, 3, 5, 6, 10, 9, 8, 12, 13] for background and further results on guarded logics and [11] for an informal discussion. 2 A general criterion for decidability Guarded second order logic, abbreviated GSO, has the same syntax as ordinary second order logic, but second order quanti ers are restricted semantically to range over ....

....logics are decidable, and indeed have good model theoretic and algorithmic properties ( nite model property, generalized tree model property, ecient model checking, automata based decision procedures etc. which make them interesting for a number of applications of logic in computer science. See [1, 3, 5, 7, 10, 9, 11, 8, 12, 13] The guards in GF can be viewed as generalizations of actions in modal and process logics and of the roles in description logics. However, contrary to modal or description logics, there is no strict separation in GF between state formula (or concept de nitions) and actions (or roles) While this ....

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E. Gr adel and I. Walukiewicz, Guarded xed point logic, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45-54.

....v 0 , v 0 ) f g and C(v 0 ) The rules from Figure 1 are successively applied until either a clash occurs, producing output unsatisfiable , or the tree is complete, in which case satisfiable is output. While our notion of tableaux has many similarities to the tableaux appearing in [6], there are two important di erences that make the notion of tableaux here more suitable as basis for a tableau algorithm. We will see that every completion tree generated by the tableau algorithm is nite. Conversely, tableaux in [6] in general, can be in nite. Also, in [6] every node is ....

....of tableaux has many similarities to the tableaux appearing in [6] there are two important di erences that make the notion of tableaux here more suitable as basis for a tableau algorithm. We will see that every completion tree generated by the tableau algorithm is nite. Conversely, tableaux in [6], in general, can be in nite. Also, in [6] every node is labeled with a complete ( C(v) type, i.e. every formula 2 cl( C(v) is explicitly asserted true of false at v. Conversely, a completion tree contains only assertions about relevant formulas. This implies a lower degree of ....

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E. Gradel and I. Walukiewicz. Guarded xed point logic. In Proc. 14th IEEE Symp. on Logic in Computer Science, pages 45-54, 1999.

....Since the tree model property of the modal fragment can be seen as the main reason behind the robustness of the decidability of that fragment (cf. e.g. Var98] this gives hope as to the robustness of GF. And indeed, adding least and greatest xed points to GF yields a decidable expansion [GW99] Interpolation and de nability are other yardsticks by which to measure the modal behavior of GF. They are investigated in this paper. We also study a natural expansion of GF, called the packed fragment (PF) Roughly speaking, the packed fragment allows for quanti cations of the form 9 y( ....

E. Gradel and I Walukiewicz. Guarded xed point logic. In Proc. 14th Symp. on Logic in Computer Science, LICS`99, pages 45-54, 1999.

....all x 2 M H , M H ; x if and only if M;x t . Thus M H , because for all x 2 M H , M;x t . J The theorem follows directly from the claim. For both, Theorem 4.1 and Theorem 4. 2 matching exptime upper bounds can be obtained by embedding into the loosely guarded fragment [GW99] but we omit details. Using ideas from Section 3.3, we can point to a subset of the language with an np complete satisfaction problem. Consider the formulas of the Until=Since language equivalent to formulas of the form Since(i; or Until(i; It is immediate that, in any structure, these ....

E. Gradel and I. Walukiewicz. Guarded xed point logic. In Proc. of 14th IEEE Symposium on Logic in Computer Science LICS'99, Trento, 1999.

....transitive relations are only admitted in guards, but where non transitive relations and equality are allowed to occur everywhere. There are very few known decidable extensions of GF, one exception is the recent decidability result of the extension of GF with least and greatest xed points by Gr adel Walukiewicz [1999]. Recently, Hans de Nivelle showed 5 that S4 reduces to monadic GF 2 . His reduction exploits the fact that, guarded formulas of the form 8xyR(x; y) P (x) P (y) can be used to encode transitivity of R. The idea is similar to the construction of in Lemma 6. Such results are relevant ....

Gradel, E. & Walukiewicz, I. (1999), Guarded xed point logic, in `Proc. IEEE Conference on Logic in Computer Science (LICS)'. In this volume.

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E. Gradel and I. Walukiewicz. Guarded xed point logic. In Logic in Computer Science, pages 45-54, 1999.

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E. Gr adel and I. Walukiewicz, Guarded xed point logic, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, pp. 45-54. 12

....di erent levels of guardedness . In this paper we consider guarded fragments of rst order logic and least xed point logic with two notions of guardedness. While model theoretic properties and satis ability algorithms for guarded logics have already been studied rather extensively (see, e.g. [1, 8, 11, 12, 15]) the model checking problem has not yet received as much attention. In [9] a guarded variant of Datalog, called Datalog LITE, has been introduced which is shown to admit ecient query evaluation (linear time in the query length and the size of the database) Datalog LITE is equivalent, via ecient ....

....variables of . An atom (x; y) that relativizes a quanti er as in the quanti cation rule for GF is the guard of the quanti er. We sometimes use the notation (9y : and (8y : for guarded formulae. Guarded xed point logic. The natural xed point extension of GF is GF and was introduced in [15]. It relates to GF in the same way as the modal calculus relates to propositional modal logic and as least xed point logic LFP (popular in nite model theory) relates to rst order logic FO. De nition 2. GF extends GF by the following rules for constructing xed point formulae: Let T be a ....

E. Gr adel and I. Walukiewicz, Guarded xed point logic, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, 45-54.

....F does not cycle. Q: So this sentence has only in nite models But this means that guarded xed point logic does not have the nite model property. A: Right. Q: But then I would conjecture that the satis ability problem for GF is undecidable. A: Wrong. 10 Theorem (Gr adel, Walukiewicz) [21] The satis ability problem for guarded xed point logic is decidable and complete for 2Exptime. For GF sentences whose relation symbols have bounded arity, the satis ability problem is Exptime complete. The proof is based on alternating tree automata. It is shown that for every sentence 2 GF ....

....to show that if such an automaton accepts a tree then it also accepts one of bounded branching. So the branching property of the trees falls out of general theorems on automata and games, and it is not necessary to prove a bounded branching property before you can apply automata. For details, see [21] and a forthcoming paper by Thomas Wilke. Q: So do you agree with the explanation proposed by Andr eka, van Benthem and N emeti. A: Yes. I think that the positive decidability results on guarded logics con rm that the guarded nature of quanti cation in modal logics is the main responsable for ....

E. Gr adel and I. Walukiewicz, Guarded xed point logic, in Proc. 14th IEEE Symp. on Logic in Computer Science, 1999, 45-54.

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E. Gradel and I. Walukiewicz. Guarded xed point logic. In Proceedings of 14th IEEE Symposium on Logic in Computer Science LICS `99, Trento, 1999.

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