Normal Forms and Proofs in Combined Modal and Temporal Logics

In an earlier paper by the author, Åqvist (1999), I presented an approach to the logic of historical necessity, or inevitability, in the sense of a “two‐dimensional” combination of tense and modal logic for worlds, or histories, with the same time order, known as T × W logic. Distinctive features of that approach were, apart from its two‐dimensionality, its being based on discrete and finite time, and its use of so‐called systematic frame constants in order to enable us to indicate longitudes (x‐values) and latitudes (y‐values) of any points in the co‐ordinate systems under consideration. This led us to study and axiomatize an infinite hierarchy HTWxy of two‐dimensional modal tense logics with the characteristic operators for historical necessity and possibility added to the original basic vocabulary. The main purpose of the present paper is then twofold: (A) to extend the logics HTWxy to the interesting branch of philosophical logic constituted by deontic logic as combined with tense (or temporal) logic; and (B) to deal with a curious puzzle known as the so‐called epistemic obligation paradox – a well known stumbling‐block in this area of research in philosophical logic. We argue for a solution to both these problems, which appeals to a new infinite hierarchy DHTWxym of extensions of the HTWxy in the sense of logics combining dyadic deontic modalities with temporal ones such as those for historical necessity and other two‐dimensional modalities.

The generation of too much information prohibits efficient resolution proof search in classical logics. Subsumption is used to discard redundant information and strategies have been developed to guide the proof search avoiding irrelevant information. The extension of the resolution method to temporal logics, further magnifies this problem. Here we develop an algorithm for the removal of irrelevant information during resolution proofs for a particular resolution proof system for propositional linear-time temporal logic. The resolution system operates on formulae in a normal form. Following a phase of classical style resolution, temporal resolution is carried out by detecting sets of formulae that together represent an □-formula to be resolved with a ◊-formula. It is before the application of the temporal resolution rule that the new algorithm is applied to efficiently remove irrelevant information. The completeness of the new algorithm is discussed and its complexity considered. Its practical efficiency is demonstrated by giving timings for a set of examples.

17 Automata-theoretic techniques for temporal reasoning

In this note, we consider the problem of introducing variables in temporal logic programs under the formalism of Temporal Equilibrium Logic, an extension of Answer Set Programming for dealing with linear-time modal operators. To this aim, we provide a definition of a first-order version of Temporal Equilibrium Logic that shares the syntax of first-order Linear-time Temporal Logic but has different semantics, selecting some Linear-time Temporal Logic models we call temporal stable models. Then, we consider a subclass of theories (called splittable temporal logic programs) that are close to usual logic programs but allowing a restricted use of temporal operators. In this setting, we provide a syntactic definition of safe variables that suffices to show the property of domain independence – that is, addition of arbitrary elements in the universe does not vary the set of temporal stable models. Finally, we present a method for computing the derivable facts by constructing a non-temporal logic program with variables that is fed to a standard Answer Set Programming grounder. The information provided by the grounder is then used to generate a subset of ground temporal rules which is equivalent to (and generally smaller than) the full program instantiation.

We will begin our study of modal logic with a basic system called K in honor of the famous logician Saul Kripke. K serves as the foundation for a whole family of systems. Each member of the family results from strengthening K in some way. Each of these logics uses its own symbols for the expressions it governs. For example, modal (or alethic) logics use □ for necessity, tense logics use H for what has always been, and deontic logics use O for obligation. The rules of K characterize each of these symbols and many more. Instead of rewriting K rules for each of the distinct symbols of modal logic, it is better to present K using a generic operator. Since modal logics are the oldest and best known of those in the modal family, we will adopt □ for this purpose. So □ need not mean necessarily in what follows. It stands proxy for many different operators, with different meanings. In case the reading does not matter, you may simply call □A ‘box A’.

There are two typical ways to formalize reasoning over time. The first is in a modal logic where time is characterized implicitly through tense operators. The second addresses time explicitly in a first-order theory, where the reasoning is itself essentially first order. In this paper we shall introduce an amalgamated temporal logic which combines the tense operators of the modal approach with the referential aspects of the first-order approach.The modal operators of the new logic, called IQ-C, include the familiar tense operators plus a few new ones; together they allow complex temporal propositions to be represented in a compact form. Unlike standard modal temporal logics, IQ-C contains terms which refer to times; this allows the logic to characterize certain temporal properties and operators which cannot be axiomatized or defined by conventional modal operators. IQ-C is expressively complete over any connected flow of time, and has several other interesting theoretical properties, some of which will be developed in this paper.Most interval logics take one of two approaches to temporal assertions; either they analyze assertions for intervals in terms of assertions at points, or they treat intervals as primitive, allowing assertions for intervals but not at points. IQ-C allows assertions both for intervals and at points, and treats them as independent; this yields a special degree of semantic discrimination and at the same time simplifies the representation of several temporal properties. In this paper we shall outline the IQ-C framework, specifying its special features and defining its semantics. We shall also extend a method of skolemization, which is designed to support a resolution based approach to the proof theory. This will be sketched in some detail, with a view to providing a preliminary account of the basic proof theory. A complete specification of the deductive theory, together with relevant proofs, will be developed elsewhere.KeywordsIntervalsPointsTemporal Representation and ExpressivenessHomogeneitySkolemizationand Modal Theorem Proving

Chapter 4 - Modal Logics and mu-Calculi: An Introduction

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

Logics and Multi-agents: towards a new symbolic model of cognition

This paper generalises and complements the work on combining temporal logics started by Finger and Gabbay [11, 10]. We present proofs of transference of soundness, completeness and decidability for the temporalisation of logics T(L) for any flow of time, eliminating the original restriction that required linear time for the transference of those properties through logic combination. We also generalise such results to the external application of a multi-modal system containing any number of connectives with arbitrary arity, that respect normality. This generalisation over generic flows of time propagates to other combinations of logics that can be interpreted in terms of temporalisations. In this way, the independent combination (also called fusion) of temporal logics is studied over generic flows of time. We show the transfer of soundness, completeness and decidability for independent combination of temporal logics. Finally, we also discuss the independent combination of any nite number of normal multi-modal logics.

In this paper a technique is presented which provides us with a means to develop resolution-based calculi for (first-order) modal and temporal logics. The approach is based on three parts: A special translation technique from modal and temporal logic formulae into classical predicate logic, a certain kind of saturation technique which is to be applied to given background theories, and an extraction of either suitable “simpler” background theories or logic-specific inference rules. The former is interesting in case existing classical logic theorem provers are to be utilized; the latter gains importance if one is prepared to extend theorem provers that are already at hand.

The present paper has grown out of a need for improved understanding of the technical aspects of the systems of tense logic that have been proposed and applied by Stuttgart people like Franz Giunthner, Christian Rohrer and I myself in connection with the Deutsche Forschungsgemeinschaft (DFG) research project "Die Beschreibung mit hilfe der Zeitlogik von Zeitformen und Verbalperiphrasen im Franzosischen, Portugiesischen und Spanischen". The systems in question were presented and put to work in the following series of contributions (see the Bibliography at the end of this paper): Aqvist (1976), Aqvist and Giinthner (1977), Aqvist, Giunthner and Rohrer (1977), Aqvist, Giinthner and Rohrer (1977a), Aqvist (1977) and Aqvist (1977a). These papers are all somehow based on the Appendix to my essay Aqvist (1973), where a version of the so-called multiple indexing technique in modal logic is presented. In Aqvist (1976) this technique was used so as to yield a reconstruction of the Reichenbachian doctrine of verb tenses (see Reichenbach, 1947, Section 51) in the framework of a two-dimensional tense logic; in the five remaining papers a more powerful logic was

Temporal Logic has the potential to improve current methods for specifying and reasoning about asynchronous systems. However, standard temporal logics have been difficult to use, because they are rather low level for the design of realistic systems and are based on modal logics that are not intuitive to users. To be more appropriate for describing complex systems, Interval Logic has been designed as a higher-level temporal logic with a graphical representation. A not quite trivial proof is presented to demonstrate the ability to represent proofs graphically.KeywordsModal LogicTemporal LogicService SpecificationAsynchronous SystemOutput QueueThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Time is the grand stage where human activities take place (rational or otherwise). And the view of a branching temporal universe, or tree of possible events, with our actual history linearly advancing through it, is a widely shared cultural idea, not confined to Academia (cf. Borges brilliant 1941 essayEl Jardin de senderos que se bifurcan). Even though logic is often considered a study of timeless propositions, temporal languages and logics over tree structures have a long tradition. Philosophers have studied temporal structure and temporal reasoning since the 1950s, from the ‘tense logic’ of Prior (1967) to the ‘STIT’ system of Belnap et al. (2001). Moreover, starting from 1970s, computer scientists have joined in, and developed many further flavours of temporal logic, with major strands such as Pnueli on program correctness (Manna and Pnueli 1991), Emerson and Clarke on process specification and verification (Emerson and Clarke 1980, 1982), Reiter on the situation calculus in AI (Reiter 2001), and Thomas on the automata-theoretic foundations of computing (Thomas 1990). In addition, the pure firstand second-order logic of tree-like structures, starting from Rabin’s classic decidability result (Rabin 1969), provides deeper background (cf. Gradel et al. (2002)). The chapter by Hodkinson and Reynolds on ‘Temporal Logic’ in the Handbook of Modal Logic (Hodkinson and Reynolds 2006) brings together many of these trends in one mathematical narrative. But the field of reasoning in, and about, time can be mapped out in many further ways: Van Benthems chapter ‘Temporal Logic’ in the

Many temporal and modal logic languages can be regarded as subsets of first order logic, i.e. the semantics of a temporal logic formula is given as a first order condition on points of the underlying models (Kripke structures). Often the set of possible models is restricted to models which are trees. A temporal logic language is (first order) expressively complete, if for every first order condition for a node of a tree there exists an equivalent temporal formula which expresses the same condition. In this paper expressive completeness of the temporal logic language with the set of operators U (until), S (since), and X k (k-next) is proved, and the result is extended to various other tree-like structures.