Branching-time Temporal Logic Research Articles

Sequent calculi for branching time temporal logics of knowledge and beliefwith awareness: Completeness and decidability

In this paper, we consider branching time temporal logic CT L with epistemic modalities for knowledge (belief) and with awareness operators. These logics involve the discrete-time linear temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. In addition, the temporal logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”). In a language of these logics, there are awareness operators. For these logics, we present sequent calculi with a restricted cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness for these calculi are proved.

Sequent Calculi for Branching Time Temporal Logics of Knowledge and Belief

In this paper we consider branching time temporal logics of knowledge and belief. These logics involve the discrete time linear temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. The latter operator is interpreted with respect to a version of the bundle semantics. In addition the temporal logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”) and it contains the modality of common knowledge (belief). For these logics we present sequent calculi with a restricted cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness for these calculi are proved.

Axiomatic characterization of the AGM theory of belief revision in a temporal logic

Since belief revision deals with the interaction of belief and information over time, branching-time temporal logic seems a natural setting for a theory of belief change. We propose two extensions of a modal logic that, besides the next-time temporal operator, contains a belief operator and an information operator. The first logic is shown to provide an axiomatic characterization of the first six postulates of the AGM theory of belief revision, while the second, stronger, logic provides an axiomatic characterization of the full set of AGM postulates.

Specification and verification of data-driven Web applications

We study data-driven Web applications provided by Web sites interacting with users or applications. The Web site can access an underlying database, as well as state information updated as the interaction progresses, and receives user input. The structure and contents of Web pages, as well as the actions to be taken, are determined dynamically by querying the underlying database as well as the state and inputs. The properties to be verified concern the sequences of events (inputs, states, and actions) resulting from the interaction, and are expressed in linear or branching-time temporal logics. The results establish under what conditions automatic verification of such properties is possible and provide the complexity of verification. This brings into play a mix of techniques from logic and model checking.

A Tableau for Bundled CTL

We present a sound, complete and relatively straightforward tableau method for deciding valid formulas in the propositional version of the bundled (or suffix and fusion closed) computation tree logic BCTL*. This proves that BCTL* is decidable. It is also moderately useful to have a tableau available for a reasonably expressive branching-time temporal logic. However, the main interest in this should be that it leads us closer to being able to devise a tableau-based technique for theorem-proving in the important full computational tree logic CTL*.

From Linear to Branching-Time Temporal Logics: Transfer of Semantics and Definability

This paper investigates logical aspects of combining linear orders as semantics for modal and temporal logics, with modalities for possible paths, resulting in a variety of branching time logics over classes of trees. Here we adopt a unified approach to the Priorean, Peircean and Ockhamist semantics for branching time logics, by considering them all as fragments of the latter, obtained as combinations, in various degrees, of languages and semantics for linear time with a modality for possible paths. We then consider a hierarchy of natural classes of trees and bundled trees arising from a given class of linear orders and show that in general they provide different semantics. We also discuss transfer of definability from linear orders to trees and introduce a uniform translation from Priorean to Peircean formulae which transfers definability of properties of linear orders to definability of properties of all paths in trees.

Verifying data refinements using a model checker

Abstract In this paper, we consider how refinements between state-based specifications (e.g., written in Z) can be checked by use of a model checker. Specifically, we are interested in the verification of downward and upward simulations which are the standard approach to verifying refinements in state-based notations. We show how downward and upward simulations can be checked using existing temporal logic model checkers. In particular, we show how the branching time temporal logic CTL can be used to encode the standard simulation conditions. We do this for both a blocking, or guarded, interpretation of operations (often used when specifying reactive systems) as well as the more common non-blocking interpretation of operations used in many state-based specification languages (for modelling sequential systems). The approach is general enough to use with any state-based specification language, and we illustrate how refinements between Z specifications can be checked using the SAL CTL model checker using a small example.

Action and Knowledge in Alternating-Time Temporal Logic

Alternating-time temporal logic (ATL) is a branching time temporal logic in which statements about what coalitions of agents can achieve by strategic cooperation can be expressed. Alternating-time temporal epistemic logic (ATEL) extends ATL by adding knowledge modalities, with the usual possible worlds interpretation. This paper investigates how properties of agents’ actions can be expressed in ATL in general, and how properties of the interaction between action and knowledge can be expressed in ATEL in particular. One commonly discussed property is that an agent should know about all available actions, i.e., that the same actions should be available in indiscernible states. Van der Hoek and Wooldridge suggest a syntactic expression of this semantic property. This paper shows that this correspondence in fact does not hold. Furthermore, it is shown that the semantic property is not expressible in ATEL at all. In order to be able to express common and interesting properties of action in general and of the interaction between action and knowledge in particular, a generalization of the coalition modalities of ATL is proposed. The resulting logics, ATL-A and ATEL-A, have increased expressiveness without loosing ATL’s and ATEL’s tractability of model checking.

A clausal resolution method for branching-time logic ECTL+

We expand the applicability of the clausal resolution technique to the branching-time temporal logic ECTL+. ECTL+ is strictly more expressive than the basic computation tree logic CTL and its extension, ECTL, as it allows Boolean combinations of fairness and single temporal operators. We show how any ECTL+ formula can be translated to a normal form the structure of which was initially defined for CTL and then used for ECTL. This enables us to apply to ECTL+ a resolution technique defined over the set of clauses. Both correctness of the method and complexity of the transformation procedure are given.

Formal Analysis of Multiparty Contract Signing

We analyze the multiparty contract-signing protocols of Garay and MacKenzie (GM) and of Baum and Waidner (BW). We use a finite-state tool, Mocha, which allows specification of protocol properties in a branching-time temporal logic with game semantics. While our analysis does not reveal any errors in the BW protocol, in the GM protocol we discover serious problems with fairness for four signers and an oversight regarding abuse-freeness for three signers. We propose a complete revision of the GM subprotocols in order to restore fairness.

A clausal resolution method for extended computation tree logic ECTL

A temporal clausal resolution method was originally developed for linear time temporal logic and further extended to the branching-time framework of Computation Tree Logic (CTL). In this paper, following our general idea to expand the applicability of this efficient method to more expressive formalisms useful in a variety of applications in computer science and AI requiring branching time logics, we define a clausal resolution technique for Extended Computation Tree Logic (ECTL). The branching-time temporal logic ECTL is strictly more expressive than CTL in allowing fairness operators. The key elements of the resolution method for ECTL, namely the clausal normal form, the concepts of step resolution and a temporal resolution, are introduced and justified with respect to this new framework. Although in developing these components we incorporate many of the techniques defined for CTL, we need novel mechanisms in order to capture fairness together with the limit closure property of the underlying tree models. We accompany our presentation of the relevant techniques by examples of the application of the temporal resolution method. Finally, we provide a correctness argument and consider future work discussing an extension of the method yet further, to the logic CTL ∗, the most powerful logic of this class.

Model Checking Downward Simulations

This paper shows how downward simulation can be checked using existing temporal logic model checkers. In particular, we show how the branching time temporal logic CTL can be used to encode the standard downward simulation conditions. We do this for both a blocking, or guarded, interpretation of operations (often used when specifying reactive systems) as well as the more common non-blocking interpretation of operations used in many state-based specification languages (for modelling sequential systems). The approach is general enough to use with any state-based specification language, and any CTL model checker in which the language can be encoded.

Model Checking Functional and Performability Properties of Stochastic Fluid Models

Model checking of stochastic processes has been introduced to verify functional as well as performa- bility properties of formally specified systems. In this paper, model checking for a wide class of stochastic fluid models (SFMs) is considered. We present a branching time temporal logic, which is similar to the continuous stochastic logic (CSL), for expressing real-time probabilistic properties of SFMs. The model checking problem for this logic can be tackled through repetitive application of transient and steady state analysis of a modified version of the SFM under study. The complexity of the analysis techniques developed for SFMs depends on the size of the state space. We present techniques that allow to reduce the state space in the solution of model checking problems. A case study illustrates the logic and the model checking procedure.

From linear time to branching time

Model checking is a method for the verification of systems with respect to their specifications. Symbolic model-checking, which enables the verification of large systems, proceeds by calculating fixed-point expressions over the system's set of states. The μ-calculus is a branching-time temporal logic with fixed-point operators. As such, it is a convenient logic for symbolic model-checking tools. In particular, the alternation-free fragment of μ-calculus has a restricted syntax, making the symbolic evaluation of its formulas computationally easy. Formally, it takes time that is linear in the size of the system. On the other hand, specifiers find the μ-calculus inconvenient. In addition, specifiers often prefer to use linear-time formalisms. Such formalisms, however, cannot in general be translated to the alternation-free μ-calculus, and their symbolic evaluation involves nesting of fixed-points, resulting in time complexity that is quadratic in the size of the system. In this article, we characterize linear-time properties that can be specified in the alternation-free μ-calculus. We show that a linear-time property can be specified in the alternation-free μ-calculus iff it can be recognized by a deterministic Büchi automaton. We study the problem of deciding whether a linear-time property, specified by either an automaton or an LTL formula, can be translated to an alternation-free μ-calculus formula, and describe the translation, when possible.

A LOGIC BASED APPROACH TO PLANNING FOR SPECIFYING A CLASS OF TEMPORALLY EXTENDED GOALS

Planning with temporally extended goals has recently been the focus of much attention to researchers in the planning community. We study a class of planning goals where in addition to a main goal there exist other goals, which we call auxiliary goals, that act as constraints to the main goal. Both these type of goals can, in general, be a temporally extended goal. Linear temporal logic (LTL) is inadequate for specification of the overall goals of this type, although, for some situations, it is capable of expressing them separately. A branching-time temporal logic, like CTL, on the other hand, can be used for specifying these goals. However, we are interested in situations where an auxiliary goal has to be satisfiable within a fixed bound. We show that CTL becomes inadequate for capturing these situations. We bring out an existing logic, called min-max CTL, and show how it can effectively be used for the planning purpose. We give a logical framework for expressing the overall planning goals. We propose a sound and complete planning procedure that incorporates a model checking technology. Doing so, we can answer such planning queries as plan existence at the onset besides producing an optimal plan (if any) in polynomial time.

The power of first-order quantification over states in branching and linear time temporal logics

In this paper, we investigate the power of extending first-order quantification over states to branching and linear time temporal logics. We show that an unrestricted extension significantly enriches the expressive power of μ-calculus, but also leads to a significant jump in model checking complexity. However, by restricting the scope of the extension, we are able to present a powerful extension of μ-calculus that is richer than μ-calculus, but is in the same complexity class as μ-calculus in terms of model checking complexity. In the case of linear time temporal logic, we find that first-order quantification over states is more computationally expensive. We show that even under the most restricted scope of quantification, the program complexity of model checking linear temporal logic (LTL) is NP-hard and coNP-hard. However, we also show that model checking LTL with this generic extension remains PSPACE-complete.

On Non-local Propositional and Weak Monodic Quantified CTL*

In this paper we prove decidability of two kinds of branching time temporal logics. First we show that the non-local version of propositional PCTL*, in which truth values of atoms may depend on the branch of evaluation, is decidable. Then we use this result to establish decidability of various fragments of quantified PCTL*, where the next-time operator can be applied only to formulas with at most one free variable, all other temporal operators and path quantifiers are applicable only to sentences, and the first-order constructs follow the pattern of any of several decidable fragments of first-order logic.

DP lower bounds for equivalence-checking and model-checking of one-counter automata

We present a general method for proving DP-hardness of problems related to formal verification of one-counter automata. For this we show a reduction of the Sat-Unsat problem to the truth problem for a fragment of (Presburger) arithmetic. The fragment contains only special formulas with one free variable, and is particularly apt for transforming to simulation-like equivalences on one-counter automata. In this way we show that the membership problem for any relation subsuming bisimilarity and subsumed by simulation preorder is DP-hard (even) for one-counter nets (where the counter cannot be tested for zero). We also show DP-hardness for deciding simulation between one-counter automata and finite-state systems (in both directions), and for the model-checking problem with one-counter nets and the branching-time temporal logic EF.

On the expressivity and complexity of quantitative branching-time temporal logics

We investigate extensions of CTL allowing to express quantitative requirements about an abstract notion of time in a simple discrete-time framework, and study the expressive power of several relevant logics. When only subscripted modalities are used, polynomial-time model checking is possible even for the largest logic we consider, while the introduction of freeze quantifiers leads to a complexity blow-up.

Cooperation, Knowledge, and Time: Alternating-Time Temporal Epistemic Logic and Its Applications

Branching-time temporal logics have proved to be an extraordinarily successful tool in the formal specification and verification of distributed systems. Much of their success stems from the tractability of the model checking problem for the branching time logic CTL, which has made it possible to implement tools that allow designers to automatically verify that systems satisfy requirements expressed in CTL. Recently, CTL was generalised by Alur, Henzinger, and Kupferman in a logic known as “Alternating-time Temporal Logic” (ATL). The key insight in ATL is that the path quantifiers of CTL could be replaced by “cooperation modalities”, of the form 《Γ》, where Γ is a set of agents. The intended interpretation of an ATL formula 《Γ》ϕ is that the agents Γ can cooperate to ensure that ϕ holds (equivalently, that Γ have a winning strategy for ϕ). In this paper, we extend ATL with knowledge modalities, of the kind made popular in the work of Fagin, Halpern, Moses, Vardi and colleagues. Combining these knowledge modalities with ATL, it becomes possible to express such properties as “group Γ can cooperate to bring about ϕ iff it is common knowledge in Γ that ψ”. The resulting logic — Alternating-time Temporal Epistemic Logic (ATEL) — shares the tractability of model checking with its ATL parent, and is a succinct and expressive language for reasoning about game-like multiagent systems.