17 Automata-theoretic techniques for temporal reasoning

: The question of whether branching-time temporal logic or linear-time temporal logic is best for reasoning about concurrent programs is one of the most controversial issues in logics of programs. Concurrent programs are usually modelled by labelled state-transition graphs in which some state is designated as the initial state. For historical reasons such graphs are called Kripke structures. In linear temporal logic operators are provided for describing events along a single time path (i.e., along a single path in a Kripke structure). In a branching-time logic the temporal operators quantify over the futures that are possible from a given state (i.e., over the possible paths that lead from a state). It is well known that the two types of temporal logic have different expressive powers. Linear temporal logic, for example, can express certain fairness properties that cannot be expressed in branching-time temporal logic. On the other hand, certain practical decision problems like model checking are easier for branching-time temporal logic than for linear temporal logic. This paper provides further insight on which type of logic is best. It is shown that if two finite Kripke structures can be distinguished by some formula that contains both branching-time and linear-time operators, then the structures can be distinguished by a formula that contains only branching time operators. Specifically, we show that if two finite Kripke structures can be distinguished by some formula of the logic CTL (i.e., if there is some CTL formula that is true in one but not in the other), then they can be distinguished by some formula of the logic CTL.

In temporal-logic model checking, we verify the correctness of a program with respect to a desired behaviour by checking whether a structure that models the program satisfies a temporal logic formula that specifies this s behaviour. One of the ways to overcome the expressiveness limitation of temporal logics is to augment them with quantification over atomic propositions. In this paper we consider the extension of branching temporal logics with existential quantification over atomic propositions. Once we add existential quantification to a branching temporal logic, it becomes sensitive to unwinding. That is, unwinding a structure into an infinite tree does not preserve the set of formulas it satisfies. Accordingly, we distinguish between two semantics, two practices as specification languages, and two versions of the model-checking problem for such a logic. One semantics refers to the structure that models the program, and the second semantics refers to the infinite computation tree that the program induces. We examine the complexity of the model-checking problem in the two semantics for the logics CTL and CTL* augmented with existential quantification over atomic propositions. Following the cheerless results that we get, we examine also the program complexity of model checking; i.e. the complexity of this problem in terms of the program, assuming the formula is fixed. We show that while fixing the formula dramatically reduces model-checking complexity in the tree semantics its influence on the structure semantics is poor.

In the last two decades, modal and description logics have provided a theoretical framework for important applications in many areas of computer science. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In this chapter we show how efficient Boolean reasoning techniques have been imported, used and integrated into reasoning tools for modal and description logics. To this extent, we focus on modal logics, and in particular mainly on K(m). In particular, we provide some background in modal logics; we describe a basic theoretical framework and we present and analyze the basic tableau-based and DPLL-based techniques; we describe optimizations and extensions of the DPLL-based procedures; we introduce the automata-theoretic/OBDD-based approach; finally, we present the eager approach.

1 Modal logic: a semantic perspective

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

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

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

Fragments of Monadic Second-Order Logics Over Word Structures

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.

Software-controlled bots, also called social bots, are computer programs that act like human users on social media platforms. Recent work on detection of social bots is dominated by machine learning approaches. In this paper we explore bot detection as a model checking problem. We introduce Temporal Network Logic (TNL) which we use to specify social networks where agents can post and follow each other. In this logic we formalize different types of social bot behavior. These are formulas that are satisfied in a model of a network with bots. We provide a simple algorithm to extract a logical model from a real-life social network. We show that we can reduce TNL to a fragment of linear temporal logic with past and use this to establish the computational efficiency of model checking for social bot detection.

Interval temporal logics are expressive formalisms for temporal representation and reasoning, which use time intervals as primitive temporal entities. They have been extensively studied for the past two decades and successfully applied in AI and computer science. Unfortunately, they lack the ability of expressing promptness conditions, as it happens with the commonly-used temporal logics, e.g., LTL: whenever we deal with a liveness request, such as “something good eventually happens”, there is no way to impose a bound on the delay with which it is fulfilled. In the last years, such an issue has been addressed in automata theory, game theory, and temporal logic. In this paper, we approach it in the interval temporal logic setting. First, we introduce PROMPT- PNL, a prompt extension of the well-studied interval temporal logic PNL, and we prove the undecidability of its satisfiability problem; then, we show how to recover decidability (NEXPTIME-completeness) by imposing a natural syntactic restriction on it.

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

Analytic proof calculi are introduced for box and diamond fragments of basic modal fuzzy logics that combine the Kripke semantics of modal logic K with the many-valued semantics of G\"odel logic. The calculi are used to establish completeness and complexity results for these fragments.