I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... a powerful formalism, first order temporal logic has On leave from Steklov Institute of Mathematics at St.Petersburg generally been avoided due to complexity problems (e.g. there is no finite axiom system for general first order temporal logic) However, recent work by Hodkinson et al. [11] has showed that a particular fragment of first order temporal logic, termed the monodic fragment, has completeness (sometimes even decidability) properties. This breakthrough has led to considerable research activity examining the monodic fragment, in terms of decidable classes, extensions, ....

.... First Order (discrete linear time) Temporal Logic, FOTL, is an extension of classical first order logic with operators that deal with a linear and discrete model of time (isomorphic to N, and the most commonly used model of time) The first order temporal language is constructed in a standard way [9, 11] from: predicate symbols P 0 ; P 1 ; each of which is of some fixed arity (null ary predicate symbols are called propositions) individual variables x 0 ; x 1 ; individual constants c 0 ; c 1 ; Boolean operators , true ( true ) false ( false ) quantifiers 8 and ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....it is easy to show that first order temporal logic is, in general, incomplete (i.e. not recursively enumerable [14] In fact, until recently, it has been difficult to find any non trivial fragment of first order temporal logic that has reasonable properties. A breakthrough by Hodkinson et al. [8] showed that monodic fragments of first order temporal logic could be complete, even decidable. In spite of this, the addition of equality or function symbols leads to the loss of recursive enumerability [15, 3] The definition of the monodic fragment holds great promise for increasing the ....

.... Logic First Order (linear time) Temporal Logic, FOTL, is an extension of classical first order logic with operators that deal with a linear and discrete model of time (isomorphic to N, and the most commonly used model of time) The first order temporal language is constructed in a standard way [6, 8] from: predicate symbols P 0 ; P 1 ; each of which is of some fixed arity (null ary predicate symbols are called propositions) individual variables x 0 ; x 1 ; individual constants c 0 ; c 1 ; booleans operators , true ( true ) false ( false ) together with ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....problem in SL is decidable for all negation free first order formulae without rigid variables. This result is obtained from a reduction to CL satisfiability. Since first order SL is not recursively axiomatizable [1] satisfiability is undecidable for the full language of SL . Recent work [9] and also [11] however, have taken up the task of identifying first order decidable fragments of SL. Our fourth result contributes to this task. The reduction from the standard SL satisfiability to CL satisfiability also contributes to the understanding of the relationship between (timed) ccp ....

....6 by replacing and j= CL with and j= SL , respectively. We say that F is SL satisfiable iff j= SL F for some , and that F is SL valid iff j= SL F for all . In order to prove our decidability result, we assume that state validity (the set of valid state formulae) is decidable. From [9] we know that even under this assumption the SL defined above is undecidable. In contrast, under the assumption, SL satisfiability is decidable for the fragment in which temporal operators are not allowed within the scope of quantifiers as it can be reduced to that of propositional LTL [1] ....

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I. Hodkinson, F. Wolter, and M. Zakharyasche. Decidable fragments of first-order temporal logic. Ann. Pure. Appl. Logic, 106:85--134, 2000.

....of finite axiomatisability beyond the monodic fragment. 1 Introduction First order (discrete) linear temporal logic (FOLTL) is complex. It is known that even small fragments of FOLTL, such as the two variable monadic fragment (all predicates are unary) are not recursively enumerable [15, 13]. However, the set of valid monodic formulae is known to be finitely axiomatisable [16] an FOLTL formula f is called monodic if any subformulae of the forms T y or y 1 T y 2 , where T is a temporal operator, contains at most one free variable) Moreover, a number of decidable subfragments of the ....

.... formulae is known to be finitely axiomatisable [16] an FOLTL formula f is called monodic if any subformulae of the forms T y or y 1 T y 2 , where T is a temporal operator, contains at most one free variable) Moreover, a number of decidable subfragments of the monodic fragment has been defined [13, 16]; this highlights the importance of the monodic class to practical temporal verification. The addition of either equality or function symbols to monodic FOLTL leads to the loss of recursive enumerability [16, 4] The only known decidable monodic fragment with equality (but without functional ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....it is easy to show that first order temporal logic is, in general, incomplete (i.e. not recursively enumerable [14] In fact, until recently, it has been difficult to find any non trivial fragment of first order temporal logic that has reasonable properties. A breakthrough by Hodkinson et al. [9] showed that monodic fragments of first order temporal logic could be complete, even decidable. In spite of this, the addition of equality or function symbols leads to the loss of recursive enumerability [15, 4] In [3] we presented a resolution procedure for the monodic fragment with ....

.... Logic First Order (linear time) Temporal Logic, FOTL, is an extension of classical first order logic with operators that deal with a linear and discrete model of time (isomorphic to N, and the most commonly used model of time) The first order temporal language is constructed in a standard way [7, 9] from: predicate symbols P 0 ; P 1 ; each of which is On leave from Steklov Institute of Mathematics at St.Petersburg of some fixed arity (null ary predicate symbols are called propositions) individual variables x 0 ; x 1 ; individual constants c 0 ; c 1 ; booleans ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

.... and verification of reactive systems [17, 14, 11] Although recognised a powerful formalism, first order temporal logic has generally been avoided due to complexity problems (e.g. there is no finite axiom system for general first order temporal logic) However, recent work by Hodkinson et al. [10] has showed that a particular fragment of first order temporal logic, termed the monodic fragment, has completeness (sometimes even decidability) properties. This breakthrough has led to considerable research activity examining the monodic fragment, in terms of decidable classes, extensions, ....

.... First Order (discrete linear time) Temporal Logic, FOTL, is an extension of classical first order logic with operators that deal with a linear and discrete model of time (isomorphic to N, and the most commonly used model of time) The first order temporal language is constructed in a standard way [7, 10] from: predicate symbols P 0 ; P 1 ; each of which is of some fixed arity (null ary predicate symbols are called propositions) individual variables x 0 ; x 1 ; individual constants c 0 ; c 1 ; booleans operators , true ( true ) false ( false ) together with ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....of S4, S4.2, S4.3, and S5 and its extensions; see [10] for a recent overview. Still further mathematical questions come up in the search for algorithmically well behaved fragments of modal predicate logics; very powerful results were recently obtained by Hodkinson, Wolter, and Zakharyaschev [22]. 5 Further Readings We conclude this chapter with some pointers to the literature on modal logic. First, details on the history of modern modal logic are available, for instance, in [5, 6, 18, 40] Second, there are several survey papers in recent and not so recent handbooks that can serve as ....

I.M. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Anals of Pure and Applied Logic, 106:85--134, 2000.

....in the knowledge base are interpreted as the empty set in every model of , i.e. they are unsatisfiable in . It is easy to prove that T DLR knowledge bases are translatable in a fragment of the first order temporal logic L fsince, untilg , as defined for example in [Chomicki and Toman, 1998; Hodkinson et al. 2000] The next Section will be devoted to convince the reader that the T DLR fragment of first order temporal logic is actually an interesting one for temporal conceptual modelling in databases. 3 Expressive power This Section provides a formal semantic characterisation using T DLR theories ....

....can still be applied to entity expressions. The reasoning problems in T DLR become decidable and EXPSPACEcomplete. This fragment is called monodic since, if the F.O. translation of T DLR theories is considered, sub formulas beginning with a temporal connective have at most one free variable [Hodkinson et al. 2000] . Theorem 2 (Complexity of T DLR ) The problem of checking logical implication in T DLR (the monodic fragment of T DLR) over a linear, unbounded, and discrete temporal structure is an EXPSPACE complete problem. Proof. Sketch) First of all, the EXPSPACE complete language ALCUS [Wolter and ....

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 2000. To appear.

....x 1 (E( x 1 ) # R 1 ( x 1 ) #E l 1 1 ( b 1 ) # x 2 (E 1 ( x 2 ) # R 2 ( x 1 ) #E l 2 2 ( b 2 ) # x n (E n 1 ( x n ) # R n ( x n ) #E n ( b n ) and E n = E. So, T D sequents do not satisfy, in general, the monodic condition from [1]. To define the separation rules (ISIF ) and (GIS) see below) let us define the following operation ( Definition 2 (operation ) Let S = #, # 1 , # # 0 A be a T D sequent, and E( b) be any elementary formula from #. Then (E( b) P n 1 ( b i ) where P n ( b ....

Hodkinson I., Wolter F., Zakharyaschev M.: Decidable fragments of first-order temporal logics. (To appear in: Annals of Pure and Applied Logic).

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106, 85--134, 2000.

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....packed fragment of firstorder temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. We also prove decidability over models with finite domains, over flows of time including the real order. 1 Introduction In [6], a new monodic fragment of predicate temporal logic was introduced. It consists of those formulas whose subformulas beginning with a temporal connective have at most one free variable. It was shown that if the first order part of the language was also restricted to (roughly speaking) a ....

....have at most one free variable. It was shown that if the first order part of the language was also restricted to (roughly speaking) a decidable fragment of first order logic, then satisfiability of monodic sentences was decidable over a wide range of (strict linear) flows of time. The proof in [6] of decidability of a monodic sentence j relied on representing a temporal model of j by a quasimodel . In a quasimodel, each element of the domain at each time is represented only by its type the set of subformulas y(x) of j that are true at the element at that time. The quasimodel records ....

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I Hodkinson, F Wolter, and M Zakharyaschev, Decidable fragments of first-order temporal logics, Ann. Pure Appl. Logic 106 (2000), 85--134.

....in deduction rather than model checking. An obvious solution to this problem would be to look for well behaved fragments of first order temporal logic (see e.g. 17] and references therein) however this way has not been successful the only promising approach we know of is the recent paper [28]. Another idea is to deviate from the first order paradigm and start from computationally more friendly languages such as description logics which have been used in the area of non temporal information management to characterise in a uniform framework both conceptual modelling and queries [34, 12, ....

....The full DLRUS turns out to be undecidable. The main reason for this is the possibility to postulate that a binary relation does not vary in time a very small fragment of DLRUS (say, DLR augmented with a single time invariant binary relation) can encode the undecidable tiling problem (cf. [47, 28]) The fragment DLR US of DLRUS deprived of the ability to talk about temporal persistence of n ary relations, for n 2, is still very expressive, as is illustrated by examples in this paper, but its computational behaviour is much better. We obtain the following non trivial novel complexity ....

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....this result, we first show decidability of the non local version of propositional CTL , where truth values of atoms may depend on the branch of evaluation, and then reduce the weak one variable fragment to this logic. The main technical instrument in both proofs is the method of quasimodels [4]. 2 Decidability of non local PCTL The propositional language PCT L [3, 7] extends propositional logic with temporal connectives U;S ( until, since ) and a path quantifier E ( there exists a branch (or history) The dual path quantifier A ( for all branches (or histories) is defined as ....

....to formulas with at most one free variable and the other temporal operators and path quantifiers only to sentences. 3) Theorem 8 and its generalisation above still holds if we extend Q PCT L w with individual constants; however, functional symbols and equality may lead to undecidability, cf. [4]. We will prove this result in two steps. Firstly, we show that a Q PCT L w formula is satisfiable iff it is satisfiable in certain quasimodels. Then we will reduce satisfiability in quasimodels to nonlocal propositional satisfiability. We begin the proof of Theorem 8 by recalling that the ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....first order temporal logic, monodic fragment, tableau algorithm. 1 Introduction First order temporal logic (FOTL) based on the flow of time # is notorious for its bad computational behaviour: even the two variable monadic fragment of this logic is not recursively enumerable (see e.g. [10] and references therein) A certain breakthrough has recently been achieved in [10] where the so called monodic fragment of FOTL is introduced by restricting applications of temporal operators to formulas with at most one free variable. The full monodic fragment (containing full first order ....

.... First order temporal logic (FOTL) based on the flow of time # is notorious for its bad computational behaviour: even the two variable monadic fragment of this logic is not recursively enumerable (see e.g. 10] and references therein) A certain breakthrough has recently been achieved in [10], where the so called monodic fragment of FOTL is introduced by restricting applications of temporal operators to formulas with at most one free variable. The full monodic fragment (containing full first order logic) turns out to be axiomatisable [21] Moreover, by restricting its first order part ....

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

.... future ) On the other hand, it was shown that by restricting applications of first order quantifiers to state (i.e. path independent) formulas, and applications of temporal operators and path quantifiers to formulas with at most one free variable (similarly to the linear time monodic logic of [11]) decidable fragments can be obtained. This is so even when we include the past operator Since . Here we prove decidability of various expressive fragments of another kind of fragment of firstorder CTL with Since. This fragment is called the weak monodic fragment and consists of formulas ....

....prove decidability, we first show decidability of the non local version of propositional CTL where truth values of atoms may depend on the branch of evaluation. We then reduce the weak monodic fragment to this logic. The main technical instrument in both proofs is the method of quasimodels [11, 13]. For possible applications of decidable fragments of first order temporal logics, the reader may consult, e.g. 6] 2 Decidability of non local PCT L The propositional language PCT L [5, 14] extends propositional logic with temporal connectives U, S ( until, since ) and a path quantifier ....

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

.... are propositional (the only exceptions we know of are [4, 22, 23] The main aim of this paper is to investigate the computational behavior at least on the level of decidability of first order branching temporal logics (FOBTLs) The starting point of our investigation lies in recent work [14] on first order linear temporal logic (FOLTL) which showed that although very weak fragments (say, the monadic two variable fragment) of standard FOLTLs can be highly undecidable, by restricting applications of temporal operators to formulas with at most one free variable, we obtain a fragment ....

....way [25] and becomes decidable if its purely first order part is restricted to a decidable fragment of first order logic. Of course, FOBTLs inherit bad computational properties of FOLTLs. For example, the guarded two variable fragment of most FOBTLs is not even recursively enumerable; cf. [14]. Unfortunately (and to our surprise) it turned out that the situation is much worse. As will be shown in Section 3, the one variable fragment of first order CT L is undecidable. This is a monodic fragment with very simple decidable first order part; it can be reformulated as the product of ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....in deduction rather than model checking. An obvious solution to this problem would be to look for well behaved fragments of first order temporal logic (see e.g. 14] and references therein) however this way has not been successful the only promising approach we know of is the recent paper [19]. Another idea is to deviate from the first order paradigm and start from computationally more friendly languages such as description logics which have been used in the area of non temporal information management to characterise in a uniform framework both conceptual modelling and queries [9, 6, ....

....constraints. The full turns out to be undecidable. The main reason for this is the possibility to postulate that a binary relation does not vary in time a very small fragment of (say, augmented with a single time invariant binary relation) can encode the undecidable tiling problem (cf. [26, 19]) The fragment deprived of the ability to talk about temporal persistence of n ary relations, for n 2, is still very expressive, as is illustrated by examples in this paper, but its computational behaviour is much better. We obtain the following non trivial novel complexity results: 1) ....

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....and Cartesian products in particular are becoming the subject of one of the most important and interesting research fields in pure and applied modal logic. See e.g. applications of results and techniques developed in multi dimensional logic to first order classical, modal and temporal logics in [8, 30, 13]. Unfortunately, as was observed by Gabbay and Shehtman [7] there is no general transfer theorem that could guarantee the preservation of such properties of logics as decidability or axiomatizability under the formation of products. If we consider only 2D products of standard modal systems, ....

.... logics are decidable: Lin S5, where Lin is the temporal logic with the diamonds in the future and the past on arbitrary strict linear orders (cf. 21] Actually, the products of many other temporal logics with S5 turn out to be decidable, in particular, S5 LoghN; i, S5 LoghQ; i; see [13] for more details. Lin K, K Log hN; i, K Log hQ; i (cf. 31] S4 K, K4 K and even CPDL K (cf. 31] On the other hand, it is not known whether such natural logics as K4 K4, S4 S4, K4 Log hN; i, S4 Log hN; i and similar ones are decidable. 2 Log hN; i 2 and Log hN; i ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....US conceptual schemas. The full DLR US turns out to be undecidable. The main reason for this is the possibility to postulate that a binary relation does not vary in time a very small fragment of DLR (say, the basic description logic ALC) can encode then the undecidable tiling problem (cf. [25, 14]) The fragment DLR US of DLR US deprived of this ability to talk about temporal persistence of n ary relations, for n # 2, is still very expressive, as is illustrated by examples provided in [4] but its computational behaviour is much better. We obtain the following hierarchy of ....

....And # = # iff # # ( # #) # # is not satisfiable. Thus, all reasoning tasks connected with the notions introduced above reduce to satisfiability of formulas. The logic DLR US can be regarded as a rather expressive fragment of the firstorder temporal logic L since, until ; cf. [10, 14] and Section 5 below. 1 For instance, we may have #d 1 , d 2 # # (# R) I(t) because #d 1 , d 2 # # R I(t 2) but #d 1 , d 2 # # (# 2 ) I(t 1) 5 3 Temporal queries One more important reasoning task is known as the problem of query containment (see, e.g. 10, 8, 1] for a ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

.... Moreover, even seemingly simple fragments, such as the two variable fragment of FOT L (containing formulas with the variables x; y only) and the monadic fragment of FOT L (containing fomulas with unary predicates only) are undecidable in any natural class of flows of time [ Merz, 1992; Hodkinson et al. 2000 ] These negative results have been a serious obstacle for applying first order temporal logic in computer science and AI. A certain breakthrough has been recently achieved in [ Hodkinson et al. 2000; Wolter and Zakharyaschev, 2001 ] where a so called monodic fragment of FOT L was shown to ....

....unary predicates only) are undecidable in any natural class of flows of time [ Merz, 1992; Hodkinson et al. 2000 ] These negative results have been a serious obstacle for applying first order temporal logic in computer science and AI. A certain breakthrough has been recently achieved in [ Hodkinson et al. 2000; Wolter and Zakharyaschev, 2001 ] where a so called monodic fragment of FOT L was shown to have a much better computation behavior. The monodic fragment consists of those FOT L formulas that do not contain a subformula starting with S or U and having more than one free variable. Unlike the full ....

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

.... the expressive and decidable description logic DLR which allows the logical reconstruction and the extension of representational tools such as object oriented data models (e.g. class diagrams in UML and ODMG) semantic data model 1 The only promising approach we know of is the recent paper [21]. s (e.g. extended entity relationship, EER, and ORM) frame based ontology languages (e.g. OKBC, XOL, and OIL) and semantic networks [8, 9] In this setting, an interesting feature of DLR is the ability to completely define entities and relations as DLR views over other entities and ....

....conceptual schemas. The full DLRUS turns out to be undecidable. The main reason for this is the possibility to postulate that a binary relation does not vary in time a very small fragment of DLR (say, the basic description logic ALC) can encode then the undecidable tiling problem (cf. [38, 21]) The fragment DLR US of DLRUS deprived of this ability to talk about temporal persistence of n ary relations, for n # 2, is still very expressive, as is illustrated by examples in this paper, but its computational behaviour is much better. We obtain the following hierarchy of complexity ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....OKBC, XOL, and OIL) and semantic networks [8, 9] In this setting, an interesting feature of DLR is the ability to completely define entities and relations as DLR views over other entities and relations of the conceptual schema. 1 The only promising approach we know of is the recent paper [21]. Moreover, DLR formulas can express a large class of integrity constraints that are typical in databases, for instance, existence dependencies, exclusion dependencies, typed inclusion dependencies without projection of relations, unary inclusion dependencies, full key dependencies [5, 6] ....

....by DLRUS conceptual schemas. The full DLRUS turns out to be undecidable. The main reason for this is the possibility to postulate that a binary relation does not vary in time a very small fragment of DLR (say, the basic description logic ALC) can encode then the undecidable tiling problem (cf. [38, 21]) The fragment DLR US of DLRUS deprived of this ability to talk about temporal persistence of n ary relations, for n 2, is still very expressive, as is illustrated by examples in this paper, but its computational behaviour is much better. We obtain the following hierarchy of complexity ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

.... modelling and queries [23, 6] The temporal description logic DLRUS we design in this paper is based on the expressive and decidable description logic DLR which allows the logical reconstruction and the extension of representational 1 The only promising approach we know of is the recent paper [17]. tools such as object oriented data models (e.g. class diagrams in UML and ODMG) semantic data models (e.g. extended entity relationship, EER, and ORM) frame based ontology languages (e.g. OKBC, XOL, and OIL) and semantic networks [7, 8] In this setting, an interesting feature of DLR is ....

....by DLRUS conceptual schemas. The full DLRUS turns out to be undecidable. The main reason for this is the possibility to postulate that a binary relation does not vary in time a very small fragment of DLR (say, the basic description logic ALC) can encode then the undecidable tiling problem (cf. [30, 17]) The fragment DLR US of DLRUS deprived of this ability to talk about temporal persistence of n ary relations, for n 2, is still very expressive, as is illustrated by examples in this paper, but its computational behaviour is much better. We obtain the following hierarchy of complexity results: ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....more expressive, provided that the extension is also supported by a reasonable tableau procedure. One idea we are working on now is to extend this component to expressive fragments of first order logic, thereby obtaining tableau procedures for fragments of first order temporal logic (cf. [6]) having potential applications in a growing number of fields such as specification and verification of reactive systems, model checking, query languages for temporal databases, transaction protocols (or business models) in e commerce, etc. Another interesting aspect of this paper is that the ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of firstorder temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000. 29

....fact, finitely axiomatizable or even decidable and yet expressive fragments with interesting interactions between the common knowledge operator and quantifiers. A promising approach to singling out such kind of fragments of firstorder modal and temporal logics has been proposed in [12,25]. The idea is to restrict attention to the class of monodic 1 formulas which allow applications of modal or temporal operators only to formulas with at most one free variable. In the epistemic context, monodicity means, in particular, that: we have the full expressive power of common ....

....in G(x; y) as well, then the formula 8y(G(x; y) x; y) is in G. Note that although the guarded fragment of classical first order logic is decidable (see [1] the guarded fragment of L(F a ) turns out to be undecidable. For more details and an idea of the proof the reader is referred to [12,25]. Actually, no non trivial decidable fragments of epistemic predicate logics have been constructed before. It maybe also of interest to note that these decidability results make it possible to construct various decidable description logics with common knowledge and other epistemic operators ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of firstorder temporal logics. Annals of Pure and Applied Logic, 2000. Holger Sturm et al.

....thereby obtaining tableau procedures 1 Of course, eventualities which are marked also have to be realized. However, the fact that all unmarked eventualities in a tableau are realized implies that all other eventualities are also realized. 14 for fragments of first order temporal logic (cf. [6]) having potential applications in a growing number of fields such as specification and verification of reactive systems, model checking, query languages for temporal databases, transaction protocols (or business models) in e commerce, etc. Another interesting aspect of this paper is that the ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of firstorder temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

....and the employed techniques as clear as possible. The developed methods can be extended to more 3 sophisticated logics, say, S4 or temporal logics based on ALC. Moreover, the approach developed in this paper can be generalized to the monodic fragments of first order modal and temporal logics [16, 39] by combining tableau procedures for their first order and modal components in a modular way; for details visit http: www.dcs.kcl.ac.uk staff mz. It is worth also noting that KALC is closely related to the Cartesian product K Theta S5 (cf. 13, 33] Thus we obtain for free a tableau based ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 2000.

....fact, finitely axiomatizable or even decidable and yet expressive fragments with interesting interactions between the common knowledge operator and quantifiers. A promising approach to singling out such kind of fragments of first order modal and temporal logics has been proposed in [ Hodkinson et al. 2000; Wolter and Za2 kharyaschev, 2001 ] The idea is to restrict attention to the class of monodic 1 formulas which allow applications of modal or temporal operators only to formulas with at most one free variable. In the epistemic context, monodicity means, in particular, that: ffl we have ....

....by Q1 we obtain S (y 1 ; ym ) 22 Note that although the guarded fragment of classical first order logic is decidable (see [ Andr eka et al. 1998 ] the guarded fragment of L(F a ) turns out to be undecidable. For more details and an idea of the proof the reader is referred to [ Hodkinson et al. 2000; Wolter and Zakharyaschev, 2001 ] Actually, no non trivial decidable fragments of epistemic predicate logics have been constructed before. It maybe also of interest to note that these decidability results make it possible to construct various decidable description logics with common knowledge ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 2000.

....OF COROLLARY 3.4 It is immediate from the lemma that any sentence of LGF(L) is logically equivalent to a sentence of PF(L) The corollary now follows from theorem 3.3. 4 Applications We end by outlining some applications of our results. 4. 1 Decidable fragments of predicate temporal logic In [14], certain decidable fragments of predicate temporal logic with Until and Since (the monodic fragments) were introduced. The idea is to restrict the predicate part of the logic to a known decidable fragment of first order logic (such as the loosely guarded fragment) and restrict temporal ....

I Hodkinson, F Wolter, and M Zakharyaschev, Decidable fragments of first-order temporal logics, Ann. Pure Appl. Logic, to appear.

....of using common knowledge predicate logic Still there exist manageable fragments with non trivial interaction between the common knowledge operator and quantifiers. A promising approach to singling out non trivial decidable fragments of first order modal and temporal logics has been proposed in [9, 20]. The idea is to restrict attention to the class of monodic 2 formulas which allow applications of modal or temporal operators only to formulas with at most one free variable. In the epistemic context, monodicity means, in particular, that ffl we have the full expressive power of first order ....

....we can assume that D is the domain of the first order structures I(w) satisfying the ff oe(w) that c w = r c ; 0) and that r(w) f 2 sub x : I(w) j= r; g; 1) for all runs r and . Note that the underlying first order language does not contain equality; for details see [9], Lemma 9. Let us now define the underlying frame F of the model we are constructing. Its set of worlds is W . The accessibility relations R i depend on S. Namely, we define R i to be ffl i if S = K C n or S = KD C n ; 8 ffl i [ fhw; wi : w 2 Wg if S = T C n ; ffl the transitive ....

[Article contains additional citation context not shown here]

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 2000.

....the language with outer quantifiers is much more expressive than the language with inner ones. For instance, the properties (1) 23) from Hodes s list are only expressible in a modal language with outer domain quantifiers. 3 The concept of world monodicity has first been introduced in [16], where it is proved that the temporal predicate language with operators Since and Until is expressively complete for the world monodic fragment of the corresponding two sorted language relative to, say, the flows of time R (the reals) and N (the natural numbers) 3 2 The modal logic S5 A ....

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 2000.

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I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyasche. Decidable fragments of first-order temporal logic. Ann. Pure. Appl. Logic, 106:85--134, 2000.

No context found.

I. Hodkinson, F. Wolter, M. Zakharyaschev, Decidable fragments of first-order temporal logics, Annals of Pure and Applied Logic 106 (2000) 85--134.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85-- 134, 2000.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85--134, 2000.

No context found.

I. Hodkinson, F. Wolter, and M. Zakharyaschev. Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic, 106:85-- 134, 2000.

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