First-order logics of branching time

Studia Logica, 2010

In this paper we present BTC, which is a complete logic for branchingtime whose modal operator quantifies over histories and whose temporal operators involve a restricted quantification over histories in a given possible choice. This is a technical novelty, since the operators of the usual logics for branching-time such as CTL express an unrestricted quantification over histories and moments. The value of the apparatus we introduce is connected to those logics of agency that are interpreted on branching-time, as for instance Stit Logics.

2003

Suitable extensions of monadic second-order theories of k successors have been proposed in the literature to specify in a concise way reactive systems whose behaviour can be naturally modeled with respect to a (possibly infinite) set of differently-grained temporal domains. This is the case, for instance, of the wide-ranging class of real-time reactive systems whose components have dynamic behaviours regulated by very different time constants, e.g., days, hours, and seconds. In this paper, we focus on the theory of k-refinable downward unbounded layered structures MSO[< tot , (↓ i ) k−1 i=0 ], that is, the theory of infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, whose satisfiability problem is nonelementarily decidable. We define a propositional temporal logic counterpart of MSO[< tot , (↓ i ) k−1 i=0 ] with set quantification restricted to infinite paths, called CTSL * k , which features an original mix of linear and branching temporal operators. We prove the expressive completeness of CTSL * k with respect to such a path fragment of MSO[<tot, (↓i) k−1 i=0 ] and show that its satisfiability problem is 2EXPTIME-complete.

Lecture Notes in Computer Science, 1999

Lecture Notes in Computer Science, 2009

Temporal logics are a well investigated formalism for the specification and verification of reactive systems. Using formal verification techniques, we can ensure the correctness of a system with respect to its desired behavior (specification), by verifying whether a model of the system satisfies a temporal logic formula modeling the specification. From a practical point of view, a very challenging issue in using temporal logic in formal verification is to come out with techniques that automatically allow to select small critical parts of the system to be successively verified. Another challenging issue is to extend the expressiveness of classical temporal logics, in order to model more complex specifications. In this paper, we address both issues by extending the classical branching-time temporal logic CTL * with minimal model quantifiers (MCTL * ). These quantifiers allow to extract, from a model, minimal submodels on which we check the specification (also given by an MCTL * formula). We show that MCTL * is strictly more expressive than CTL * . Nevertheless, we prove that the model checking problem for MCTL * remains decidable and in particular in PSPACE. Moreover, differently from CTL * , we show that MCTL * does not have the tree model property, is not bisimulation-invariant and is sensible to unwinding. As far as the satisfiability concerns, we prove that MCTL * is highly undecidable. We further investigate the model checking and satisfiability problems for MCTL * sublogics, such as MPML, MCTL, and MCTL + , for which we obtain interesting results. Among the others, we show that MPML retains the finite model property and the decidability of the satisfiability problem. 6 6 J J J J J J i 6 6 J J J J J J

1999

For branching-time temporal logic based on an Ockhamist semantics, we explore a temporal language extended with two additional syntactic tools. For reference to the set of all possible futures at a moment of time we use syntactically designated "restricted variables" called fan-names. For reference to all possible futures alternative to the actual one we use a modification of a difference modality, localized to the set of all possible futures at the actual moment of time.

Theoretical Aspects of Computing – ICTAC 2018, 2018

We show that Branching-time temporal logics CTL and CTL * , as well as Alternating-time temporal logics ATL and ATL * , are as semantically expressive in the language with a single propositional variable as they are in the full language, i.e., with an unlimited supply of propositional variables. It follows that satisfiability for CTL, as well as for ATL, with a single variable is EXPTIME-complete, while satisfiability for CTL * , as well as for ATL * , with a single variable is 2EXPTIME-complete,-i.e., for these logics, the satisfiability for formulas with only one variable is as hard as satisfiability for arbitrary formulas.

Theoretical Computer Science, 1992

In this paper we solve some open problems raised in recent publications of the Computer Science Temporal Logic school represented by Manna-Pnueli [11], [12], Abadi-Manna [5], Abadi [1]-[4]. These problems concern the proof theoretic powers of the following inference systems: T 0 introduced in [11], [12] and reformulated in [1]-[4]; the resolution system R of [5]; and T 1 , T 2 of [1]-[4]. We use first-order temporal logic (FTL) with modalities , [F ], and U denoting "nexttime", "always-in-the-future", and "until" respectively. Given a first-order similarity type or language L, the usual predicate etc. symbols of L are considered to be rigid, i.e. their meanings do not change in time. Similarly, individual variables x i (i ∈ ω) are rigid. To this we add an infinity y i (i ∈ ω) of flexible constants. That is, the meaning of y i is allowed to change in time. Other authors, see e.g. Abadi [1]-[4], add flexible predicates too, but we will not need them here though we will mention them occasionally. Our theorems remain true even if we allow flexible predicateand function symbols, as it will be very easy to see. F m(F T L) denotes the set of all FTL-formulas (of some fixed similarity type L) defined above. For semantic purposes, we use classical two-sorted models M = < T, D, f 0 ,. .. , f i ,. .. > i∈ω where D is a classical first-order structure of similarity type L, T = < T, 0, suc, ≤, +, × > is a structure of the same similarity type as the standard model N = < ω, 0, suc, ≤, +, × > of arithmetic, and for i ∈ ω, f i , a function from T into D, serves to interpret the flexible constant y i. T is called the time-frame of M, and, except for its language, is arbitrary. M od denotes the class of all models M of the above kind. (The members of M od are basically the same as Kripke models known from the

Information Processing Letters, 1987

2002

Temporal logic can be used to describe processes: their behaviour is characterized by a set of temporal models axiomatized by a temporal theory. Two types of models are most often used for this purpose: linear and branching time models. In this paper a third approach, based on socalled joint closure models, is studied using models which incorporate all possible behaviour in one model. Relations between this approach and the other two are studied. In order to define constructions needed to relate branching time models, appropriate algebraic notions are defined (in a category theoretical manner) and exploited. In particular, the notion of joint closure is used to construct one model subsuming a set of models. Using this universal algebraic construction we show that a set of linear models can be merged to a unique branching time model. Logical properties of the described algebraic constructions are studied. The proposed approach has been successfully aplied to obtain an appropriate semantics for nonmonotonic reasoning processes based on default logic. References are discussed that show the details of these applications.

Annals of Pure and Applied Logic, 2000

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator (Since or Until) have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment