A Modal Logic for a Subclass of Event Structures

A logical characterization of well branching event structures

We formulate the notion of a regular event structures and conjecture that they correspond exactly to finite 1-safe Petri nets. We offer a partial result in support of the conjecture. This result is in terms of a natural subclass of regular event structures that admit a sensible labeling with Mazurkiewicz trace alphabets.

17 Automata-theoretic techniques for temporal reasoning

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

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.

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.

In the last few years, policy-based management has begun to emerge as the dominant paradigm for developing network and systems management functions, primarily, since it can reduce complexity in management applications. Although attempts are underway to standardize policy-based management, significant research challenges remain. Collaboration and state maintenance are two hot spots of the research. A modal logic- based policy definition language is proposed and its syntax, execution model is defined. Modal logic-based policy management (MLPM) supports collaborative management operations based on system's states. A large scale distributed management system is divided into many management domains to autonomous management. Global state view is acquired through resource pool and the corresponding triggering update mechanism. Administrator can use modal logic-based policy to describe system's goal and decision functions. Finally, modal logic-based policy network monitoring and management model is presented to show that MPLM can meet the requirements of distributed networked systems and applications.

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.

In this paper we present a framework for the combination of modal and temporal logic. This framework allows us to combine different normal forms, in particular, a separated normal form for temporal logic and a first-order clausal form for modal logics. The calculus of the framework consists of temporal resolution rules and standard first-order resolution rules.

Some Modal Calculi Based On IC

In a very short time Saul Kripke provided a suitable and rigorous semantics for different axiomatic modal systems and established a series of related results. Many key ideas were already in the air in the late Fifties, but it was Kripkean articles’ merit to system atically introduce comprehensive devices and solutions. Later on, the spreading of possible-worlds semantics massively changed the approach to modal logic, which enormously increased in popularity after that. Since Kripke’s work in modal logic is central to the development of the discipline, the aim of this essay is to present the fundamental results published between 1959 and 1965. Indeed, it was in such a brief and early phase of his career that Kripke was able to conceive the main novelties that would become central to the subsequent academic debates about modality. Here, their presentation will follow the original historical progressive introduction. Particular attention will be given to the interconnection between articles, their similarities in structure and the unified analysis produced by means of them. It actually appears quite impressive that, already in 1959, Kripke seemed to have planned all the developments he would present, one after the other, in the following years. First, an overview of the background where Kripke’s ideas start to rise is given. Then, each text’s results are individually briefly analysed.

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \(\exists x(x=\alpha )\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account.