## Towards the Implementation of First-Order Temporal Resolution: the Expanding Domain Case

### Cached

### Download Links

- [www.dcs.kcl.ac.uk]
- [www.csc.liv.ac.uk]
- [www.csc.liv.ac.uk]
- DBLP

### Other Repositories/Bibliography

Citations: | 11 - 7 self |

### BibTeX

@MISC{Konev_towardsthe,

author = {Boris Konev and Anatoli Degtyarev and Clare Dixon and Michael Fisher and Ullrich Hustadt},

title = {Towards the Implementation of First-Order Temporal Resolution: the Expanding Domain Case},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. In this paper, we develop a clausal resolution method for the monodic fragment of first-order temporal logic over expanding domains. We first define a normal form for monodic formulae and show how arbitrary monodic formulae can be translated into the normal form, while preserving satisfiability. We then introduce novel resolution calculi that can be applied to formulae in this normal form and state correctness and completeness results for the method. We illustrate the method on a comprehensive example. The method is based on classical first-order resolution and can, thus, be efficiently implemented.

### Citations

1573 |
The Temporal Logic of Reactive and Concurrent Systems: Specification
- Manna, Pnueli
- 1991
(Show Context)
Citation Context ...can, thus, be efficiently implemented. 1. Introduction In its propositional form, linear, discrete temporal logic has been widely used in the formal specification and verification of reactive systems =-=[18, 15, 12-=-]. Although recognised 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. the... |

1335 | The model checker SPIN
- Holzmann
- 1997
(Show Context)
Citation Context ...can, thus, be efficiently implemented. 1. Introduction In its propositional form, linear, discrete temporal logic has been widely used in the formal specification and verification of reactive systems =-=[18, 15, 12-=-]. Although recognised 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. the... |

1273 |
The Temporal Logic of Programs
- Pnueli
- 1977
(Show Context)
Citation Context ...can, thus, be efficiently implemented. 1. Introduction In its propositional form, linear, discrete temporal logic has been widely used in the formal specification and verification of reactive systems =-=[18, 15, 12-=-]. Although recognised 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. the... |

162 | Resolution theorem proving
- Bachmair, Ganzinger
- 2001
(Show Context)
Citation Context ... tautology if D is a tautology. (Note that, since we do not have negative occurrences to the left-hand side of step clauses, C cannot be false). Tautologies are deleted. We adopt the terminology from =-=[2]-=-. A (linear) proof by fine-grained resolution of a clause C from a set of clauses S is a sequence of clauses C 1 ; : : : ;Cm such that C =Cm and each clause C i is either an element of S or else the c... |

116 | Paramodulation-based theorem proving
- Nieuwenhuis, Rubio
- 2001
(Show Context)
Citation Context ...nstrained clauses with constraint inheritance. It is known that such inference systems are complete and moreover compatible with redundancy elimination rules and many (liftable) refinements (see e.g. =-=[17]-=-, theorems 5.11 and 5.12, subsections 5.4 and 5.5, resp.). Here we take into account that there are no clauses with equality, and therefore all sets are well-constrained in the terminology of [17]. Th... |

97 | Clausal Temporal Resolution - Fisher, Dixon, et al. - 2001 |

88 | Decidable fragments of first-order temporal logics. Annals of Pure and Applied Logic 106:85–134
- Hodkinson, Wolter, et al.
- 2000
(Show Context)
Citation Context ... 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 ... |

56 |
The Resolution Calculus
- Leitsch
- 1997
(Show Context)
Citation Context ...g fp i ) g l i j i 2 Jg (and universal clauses) of a final clause C ) g false, where C = V (i; j)2I P j (c i ) ^ V j2J p i . We assume, for simplicity of the proof, that the lifting theorem (cf. e.g. =-=[14-=-]) holds for D, that is, there exists a non-ground (first-order) refutation D 0 from fM j (x j ) j (i; j) 2 Ig[fl i j i 2 Jg, such that D s D 0 in the terminology of [14]: Every clause C 0 i of D 0 i... |

48 | Monodic fragments of first-order temporal logics: 2000–2001 A.D
- Hodkinson, Wolter, et al.
- 2001
(Show Context)
Citation Context ...ariable monadic fragment (all predicates are unary), are not recursively enumerable [16, 11]. However, the set of valid monodic formulae (see Definition 1 below) is known to be finitely axiomatisable =-=[20-=-]. Definition 1 An FOTL-formula f is called monodic if any subformulae of the form T y, where T is one of g , , (or y 1 T y 2 , where T is one of U , W ), contains at most one free variable. 3. Divid... |

46 |
Resolution Decision Procedures
- Fermüller, Leitsch, et al.
- 2001
(Show Context)
Citation Context ...rns non-false value, its output is a loop formula in L(x). Note 3 Termination of calls by step resolution can be achieved for the cases when there exists a (first-order) resolution decision procedure =-=[8]-=- for formulae in the universal part, see also [4]. Theorem 9 The calculus consisting of the rules of finegrained step resolution, together with the (both ground and non-ground) eventuality resolution ... |

45 | A Normal Form for Temporal Logic and its Application in TheoremProving and Execution
- Fisher
(Show Context)
Citation Context ...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 ;... |

32 | Qualitative spatio-temporal representation and reasoning: a computational perspective
- Wolter, Zakharyaschev
- 2002
(Show Context)
Citation Context ...quires the maximal combination of all temporal clauses. In this paper, we focus on an important subclass of temporal models, having a wide range of applications, for example in spatio-temporal logics =-=[21, 10]-=- and temporal description logics [1], namely those models that have expanding domains. In such models, the domains over which first-order terms range can increase at each temporal step. The focus on t... |

28 | Decidable fragments of first-order modal logics
- Wolter, Zakharyaschev
(Show Context)
Citation Context ...terpreted, is restricted to structures M = hD n ; I n i, n 2 N, such that D i = D j for all i; j 2 N. The notions of truth and validity are defined similarly to the expanding domain case. It is known =-=[19-=-] that satisfiability over expanding domains can be reduced to satisfiability over constant domains. Example 1 The formula 8xP(x) ^ (8xP(x) ) g 8xQ(x)) ^ :Q(c) is unsatisfiable over both expanding and... |

28 | A normal form for first-order temporal formulae - FISHER - 1992 |

27 | Towards firstorder temporal resolution
- Degtyarev, Fisher
- 2001
(Show Context)
Citation Context ...idable classes, extensions, applications and mechanisation, etc. Concerning the mechanisation of monodic temporal logics, general tableau and resolution calculi have already been defined, in [13] and =-=[5, 3]-=-, respectively. However, neither of these is particularly practical: the tableau method requires representation of all possible first-order models, while the resolution method requires the maximal com... |

27 | Monodic temporal resolution
- DEGTYAREV, FISHER, et al.
(Show Context)
Citation Context ...idable classes, extensions, applications and mechanisation, etc. Concerning the mechanisation of monodic temporal logics, general tableau and resolution calculi have already been defined, in [13] and =-=[5, 3]-=-, respectively. However, neither of these is particularly practical: the tableau method requires representation of all possible first-order models, while the resolution method requires the maximal com... |

23 | A temporal description logic for reasoning over conceptual schemas and queries
- Artale, Franconi, et al.
- 2002
(Show Context)
Citation Context ...poral clauses. In this paper, we focus on an important subclass of temporal models, having a wide range of applications, for example in spatio-temporal logics [21, 10] and temporal description logics =-=[1]-=-, namely those models that have expanding domains. In such models, the domains over which first-order terms range can increase at each temporal step. The focus on this class of models allows us to pro... |

23 | Temporal resolution using a breadth-first search algorithm
- Dixon
- 1998
(Show Context)
Citation Context ...m constants but c l . 2 Then the loop search algorithm from [5] can be reformulated as shown in Fig. 1. (This algorithm is essentially based on the BFS algorithm for propositional temporal resolution =-=[7]-=-.) Lemma 8 The BFS algorithm terminates provided that all calls of saturation by step resolution terminate. If BFS returns non-false value, its output is a loop formula in L(x). Note 3 Termination of ... |

21 | On the computational complexity of spatio-temporal logics
- Gabelaia, Kontchakov, et al.
- 2003
(Show Context)
Citation Context ...quires the maximal combination of all temporal clauses. In this paper, we focus on an important subclass of temporal models, having a wide range of applications, for example in spatio-temporal logics =-=[21, 10]-=- and temporal description logics [1], namely those models that have expanding domains. In such models, the domains over which first-order terms range can increase at each temporal step. The focus on t... |

20 |
Decidability and incompleteness results for first-order temporal logics of linear time
- Merz
- 1992
(Show Context)
Citation Context ... with an expanding domain. This logic is complex. It is known that even "small" fragments of FOTL, such as the two-variable monadic fragment (all predicates are unary), are not recursively e=-=numerable [16, 11]-=-. However, the set of valid monodic formulae (see Definition 1 below) is known to be finitely axiomatisable [20]. Definition 1 An FOTL-formula f is called monodic if any subformulae of the form T y, w... |

17 | Temporalising tableaux
- KONTCHAKOV, LUTZ, et al.
- 2004
(Show Context)
Citation Context ...ms of decidable classes, extensions, applications and mechanisation, etc. Concerning the mechanisation of monodic temporal logics, general tableau and resolution calculi have already been defined, in =-=[13]-=- and [5, 3], respectively. However, neither of these is particularly practical: the tableau method requires representation of all possible first-order models, while the resolution method requires the ... |

3 | Monodic temporal resolution: the expanding domain case
- Degtyarev, Fisher, et al.
- 2003
(Show Context)
Citation Context ...xpanding domains. Soundness of the fine-grained inference steps is easy to prove and completeness is shown relative to the completeness proof for the expanding domain for the non-fine grained version =-=[6]-=-. While the implementation based on the general calculus would involve generating all subsets of the step clauses with which to apply the step and eventuality resolution rules, the fine-grained resolu... |

2 | Exploring the monodic fragment of first-order temporal logic using clausal temporal resolution
- Degtyarev, Fisher, et al.
- 2003
(Show Context)
Citation Context ...niversal part). Then the conclusion of the rule, :A , is added to U resulting in a new universal part U 0 . Suppose that the 2 Note that the non-ground eventuality :P(x) is not used. It was shown in [=-=4]-=- that if all step clauses are ground, for constant flooded problems we can neglect non-ground eventualities. step resolution rule is applied to another merged e-derived step clause, A 0 ) g B 0 . The ... |

1 | Monodic temporal resolution. Submitted. Available as Technical report ULCS-03-001 from http://www.csc.liv.ac.uk/research/techreports - Degtyarev, Fisher, et al. - 2003 |

1 | Temporalising tableaux. Accepted to Studia Logica - Kontchakov, Lutz, et al. |