The Notion of Temporal Logic and the Problem of Priority

A fuzzy real-time temporal logic

After the seminal work of Kraus, Lehmann and Magidor (formally known as the KLM approach) on conditionals and preferential models, many aspects of defeasibility in more complex formalisms have been studied in recent years. Examples of these aspects are the notion of typicality in description logic and defeasible necessity in modal logic. We discuss a new aspect of defeasibility that can be expressed in the case of temporal logic, which is the normality in an execution. In this contribution, we take Linear Temporal Logic () as case study for this defeasible aspect. has found extensive applications in Computer Science and Artificial Intelligence, notably as a formal framework for representing and verifying computer systems that vary over time. However, some systems may presents exceptions at some innocuous time points where they can be tolerated, or conversely, exceptions at other crucial time points where they need to be addressed. In order to ensure the reliability of such systems, we study a preferential extension of , called defeasible linear temporal logic (). In the first part of this paper, we show how semantics of KLM's preferential models can be integrated with . We also discuss the addition of non-monotonic temporal operators as a way to formalise defeasible properties of these systems. The second part of this paper is a study of the satisfiability problem of sentences. Based on Sistla and Clarke's work on the complexity of the classical language, we show the bounded-model property of two fragments of language. Moreover, we provide a procedure to check the satisfiability of sentences in both of these fragments.

In this paper, we introduce the notion of a temporal logic to characterize sets of organizing principles that perpetuate particular orientations to the lived experience of time. We identify a dominant temporal logic, circumscribed time, which has legitimated time as chunkable, single-purpose, linear, and ownable. We juxtapose this logic with the temporal experiences of participants in three ethnographic datasets to identify a set of alternative understandings of time -- that it is also spectral, mosaic, rhythmic, and obligated. We call this understanding porous time. We posit porous time as an expansion of circumscribed time in order to provoke reflection on how temporal logics underpin the ways that people orient to each other, research and design technologies, and normalize visions of success in contemporary life.

Alice ter Meulen integrates current research in natural language semantics, with detailed analyses of English discourse, and logical tools from a variety of sources into an information theory that provides the foundation for computational systems to reason about change and the flow of time. The topic of temporal meaning in texts has received considerable attention in recent years from scholars in linguistics, logical semantics, cognitive science, and artificial intelligence. Representing Time in Natural Language offers a systematic and detailed account of how we use temporal information contained in a text or in discourse to reason about the flow of time, inferring the order in which events happened when this is not explicitly stated. A new representational toolkit is designed to formalize an appropriately context-dependent notion of situated inference. Dynamic Aspect Trees representing temporal dependencies constitute a novel and important dynamic temporal logic that makes it easy to see what follows when from the information given in an ordinary English text. Ter Meulen makes use of some of the fundamental assumptions of Situation Semantics and incorporates the dynamic methodology embodied in Discourse Representation Theory and in other dynamic logics into her temporal logic. The result is a computational inference system that can be applied across the board to fragments of natural languages. Bradford Books imprint

Decidable fragments of first-order temporal logics

If Gurevich’s thesis is valid, ASMs can be used to model any sequential algorithm on any abstraction level. But algorithms are defined in an operational way, and they operate step by step. Often, a higher abstraction level than this is wanted for describing properties of systems. Temporal logics of different kinds are proposed for this in computer science (see [Pnu77] for the introduction of the concept into computer science, and [MP92, Lam94b] for several well-developed approaches).

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’.

THE TENSE REVOLUTION. According to historians like Thomas Kuhn, the story of a scientific revolution is a chronicle of disenchantment with a paradigm, often nurtured by nostalgia for bygone ideals, culminating in the production of a rival paradigm which ultima:tely wins the allegiance of the relevant scientific community away from the older paradigm through a process akin to American politics.1 What is thus true of revolutions in empirical science is no less true of revolutions in the formal sciences of logic and mathematics, and it so happens that a logical revolution is well underway today. Disenchanted with the usual handling (mishandling?) of time in modern logic, a dedicated band of revolutionaries under the aegis of Arthur Prior impugn the legitimacy of certain central notions of modern logic and have begun to develop and explore alternative logics, so-called tense logics, reminiscent of certain ancient and medieval logical developments.2 I propose in this paper to examine some of the principal charges directed against the standard treatment time receives in modern logic. We shall find that there are two breeds of tense logicians, identifiable by their stance vis4--vis what Prior has called the principle of comprehensive

An introduction to coalgebraic specification is presented via examples. A coalgebralc specification describes a collection of coalgebras satisfying certain assertions. It is thus an axiomatic description of a particular class of mathematical structures. Such specifications are especially suitable for state-based dynamical systems in general, and for classes in object-oriented programming languages in particular. This chapter will gradually introduce the notions of bisimilarity, invariance, component classes, temporal logic and refinement in a coalgebraic setting. Besides the running example of the coalgebraic specification of (possibly infinite) binary trees, a specification of Peterson's mutual exclusion algorithm is elaborated in detail.

We propose a notion of an abstract logic. Based on this notion, we define abstract logic programs to be sets of sentences of an abstract logic. When these abstract logics possess certain logical properties (some properties considered are compactness, finitariness, and monotone consequence relations) we show how to develop a fixed point, model-state-theoretic and proof theoretic semantics for such programs. The work of Fitting on developing a generalized semantics for multivalued logic programming is extended here to arbitrary abstract logics. We present examples to show how our semantics is robust enough to be applicable to various non-classical logics like temporal logic and multivalued logics, as well as to extensions of classical logic programming such as disjunctive logic programming. We also show how some aspects of the declarative semantics of distributed logic programming, particularly the work of Ramanujam, can be incorporated into our framework.

Although a significant body of research in the area of formal verification and model checking tools of software and hardware systems exists, the acceptance of these tools by industry and end-users is rather limited. Beside the technical problem of state space explosion, one of the main reasons for this limited acceptance is the unfamiliarity of users with the required specification notation. Requirements have to be typically expressed as temporal logic formalisms and notations. Property specification patterns were successfully introduced to bridge this gap between users and model checking tools. They also enable non-experts to write formal specifications that can be used for automatic model checking. In this paper, we propose an abstract high level pattern-based approach to the description of property specifications based on Use Case Maps (UCM). We present a set of commonly used properties with their specifications that are described in terms of occurrence, ordering and temporal scopes of actions. Furthermore, our approach also supports the description of properties with respect to their architectural scope. We provide a mapping of our UCM property specification patterns in terms of CTL, TCTL and Architectural TCTL (ArTCTL), an extension to TCTL, introduced in this research that provides temporal logics with architectural scopes. We illustrate the use of our pattern system for requirement specifications of an IP Header compression feature.

Necessity and sufficiency are well-established notions in logic and causality analysis, but have barely received attention in the formal methods community. In this paper, we present temporal logic characterizations of necessary and sufficient causes in terms of state sets in operational system models. We introduce degrees of necessity and sufficiency as quality measures for sufficient and necessary causes, respectively, along with a versatile weight-based approach to find “good causes”. The resulting optimization problems of finding optimal causes are shown to be solvable in polynomial time.

A rigorous formalization of desired system requirements is indispensable when performing any verification task. This often limits the application of verification techniques, as writing formal specifications is an error-prone and time-consuming manual task. To facilitate this, we present , a framework for applying Large Language Models (LLMs) to derive formal specifications (in temporal logics) from unstructured natural language. In particular, we introduce a new methodology to detect and resolve the inherent ambiguity of system requirements in natural language: we utilize LLMs to map subformulas of the formalization back to the corresponding natural language fragments of the input. Users iteratively add, delete, and edit these sub-translations to amend erroneous formalizations, which is easier than manually redrafting the entire formalization. The framework is agnostic to specific application domains and can be extended to similar specification languages and new neural models. We perform a user study to obtain a challenging dataset, which we use to run experiments on the quality of translations. We provide an open-source implementation, including a web-based frontend.

Automatic synthesis from temporal logic specifications is an attractive alternative to manual system design, due to its ability to generate correct-by-construction implementations from high-level specifications. Due to the high complexity of the synthesis problem, significant research efforts have been directed at developing practically efficient approaches for restricted specification language fragments. In this paper we focus on the fragment of Linear Temporal Logic (LTL) syntactically extended with bounded temporal operators. We propose a new synthesis approach with the primary motivation to solve efficiently the synthesis problem for specifications with bounded temporal operators, in particular those with large bounds. The experimental evaluation of our method shows that for this type of specifications it outperforms state-of-art synthesis tools, demonstrating that it is a promising approach to efficiently treating quantitative timing constraints in safety specifications.

The present paper explores applications of Display Logic as defined in [1] to modal logic. Acquaintance with that paper is presupposed, although we will give all necessary definitions. Display Logic is a rather elegant proof-theoretic system that was developed to explore in depth the possibility of total Gentzenization of various propositional logics. By Gentzenization I understand the strategy to replace connectives by structures. Gentzenization is something of an ingenious optical trick because it uses a single symbol to mean different things depending on the place it occupies in the sequent. In the original Gentzen system it was the comma that had to be interpreted as and when to the left of the turnstile and as or when to the right. The interpretation of the structures oscillates between two logical symbols depending on whether it is in the antecedent or in the consequent. This is why we call symbols like comma Gentzen toggles. These two symbols between which this toggle switches are the Gentzen duals of each other. So, and and or are Gentzen duals. The strength of Display logic lies in a rather general cut-elimination theorem. In [10] and [9], Heinrich Wansing has refined these methods for modal logics; he showed that contrary to Belnap’s own Gentzenization of modal operators as binary structure operators, a unary one is more appropriate (not only from an esthetical point of view) and makes perfect sense semantically as well. The Gentzen dual of the modal operator □ is actually not — as one might expect — the. possibility operator ◇, but the backward looking possibility operator, denoted here by ◇. (To be consistent with that we write □ instead of □ and ◇ instead of ◇.) The corresponding toggle is denoted by •. The reason why this is so natural lies in the fact that it is the exact Display or Gentzen dual, for we have that the sequent •B ⊢ A and the sequent B ⊢ • A are equivalent if • is read as if in the antecedent and if it is read as 0 if in the consequent. Wansing uses this fact to display various modal and tense logics à la Belnap by providing some formula introduction rules and basic structural rules for K and Kt and then Gentzenizing the additional axioms. The benefit lies not only in the homogeneity with which all these systems are now handled and the rather clear intuitive background. The benefit lies in the possibility to use the general cut-elimination theorem of [1].