Real-time and Probabilistic Temporal Logics: An Overview

Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed

Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+

Rich Counter-Examples for Temporal-Epistemic Logic Model Checking

Model checking verifies that a model of a system satisfies a given property, and otherwise produces a counter-example explaining the violation. The verified properties are formally expressed in temporal logics. Some temporal logics, such as CTL, are branching: they allow to express facts about the whole computation tree of the model, rather than on each single linear computation. This branching aspect is even more critical when dealing with multi-modal logics, i.e. logics expressing facts about systems with several transition relations. A prominent example is CTLK, a logic that reasons about temporal and epistemic properties of multi-agent systems. In general, model checkers produce linear counter-examples for failed properties, composed of a single computation path of the model. But some branching properties are only poorly and partially explained by a linear counter-example. This paper proposes richer counter-example structures called tree-like annotated counter-examples (TLACEs), for properties in Action-Restricted CTL (ARCTL), an extension of CTL quantifying paths restricted in terms of actions labeling transitions of the model. These counter-examples have a branching structure that supports more complete description of property violations. Elements of these counter-examples are annotated with parts of the property to give a better understanding of their structure. Visualization and browsing of these richer counter-examples become a critical issue, as the number of branches and states can grow exponentially for deeply-nested properties. This paper formally defines the structure of TLACEs, characterizes adequate counter-examples w.r.t. models and failed properties, and gives a generation algorithm for ARCTL properties. It also illustrates the approach with examples in CTLK, using a reduction of CTLK to ARCTL. The proposed approach has been implemented, first by extending the NuSMV model checker to generate and export branching counter-examples, secondly by providing an interactive graphical interface to visualize and browse them.Comment: In Proceedings IWIGP 2012, arXiv:1202.422

Linear Time Logics - A Coalgebraic Perspective

We describe a general approach to deriving linear time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching behaviour and linear behaviour. Concretely, we define logics whose syntax is determined by the choice of linear behaviour and whose domain of truth values is determined by the choice of branching, and we provide two equivalent semantics for them: a step-wise semantics amenable to automata-based verification, and a path-based semantics akin to those of standard linear time logics. We also provide a semantic characterisation of the associated notion of logical equivalence, and relate it to previously-defined maximal trace semantics for such systems. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear time property. We conclude with a generalisation of the logics, dual in spirit to logics with discounting, which increases their practical appeal in the context of resource-aware computation by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the results in the previous version, with new proofs; Section 6 contains new result

On the Complexity of ATL and ATL* Module Checking

Module checking has been introduced in late 1990s to verify open systems, i.e., systems whose behavior depends on the continuous interaction with the environment. Classically, module checking has been investigated with respect to specifications given as CTL and CTL* formulas. Recently, it has been shown that CTL (resp., CTL*) module checking offers a distinctly different perspective from the better-known problem of ATL (resp., ATL*) model checking. In particular, ATL (resp., ATL*) module checking strictly enhances the expressiveness of both CTL (resp., CTL*) module checking and ATL (resp. ATL*) model checking. In this paper, we provide asymptotically optimal bounds on the computational cost of module checking against ATL and ATL*, whose upper bounds are based on an automata-theoretic approach. We show that module-checking for ATL is EXPTIME-complete, which is the same complexity of module checking against CTL. On the other hand, ATL* module checking turns out to be 3EXPTIME-complete, hence exponentially harder than CTL* module checking.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

From Quantified CTL to QBF

QCTL extends the temporal logic CTL with quantifications over atomic propositions. This extension is known to be very expressive: QCTL allows us to express complex properties over Kripke structures (it is as expressive as MSO). Several semantics exist for the quantifications: here, we work with the structure semantics, where the extra propositions label the Kripke structure (and not its execution tree), and the model-checking problem is known to be PSPACE-complete in this framework. We propose a model-checking algorithm for QCTL based on a reduction to QBF. We consider several reduction strategies, and we compare them with a prototype (based on the SMT-solver Z3) on several examples

Hybrid Branching-Time Logics

Hybrid branching-time logics are introduced as extensions of CTL-like logics with state variables and the downarrow-binder. Following recent work in the linear framework, only logics with a single variable are considered. The expressive power and the complexity of satisfiability of the resulting logics is investigated. As main result, the satisfiability problem for the hybrid versions of several branching-time logics is proved to be 2EXPTIME-complete. These branching-time logics range from strict fragments of CTL to extensions of CTL that can talk about the past and express fairness-properties. The complexity gap relative to CTL is explained by a corresponding succinctness result. To prove the upper bound, the automata-theoretic approach to branching-time logics is extended to hybrid logics, showing that non-emptiness of alternating one-pebble Buchi tree automata is 2EXPTIME-complete.Comment: An extended abstract of this paper was presented at the International Workshop on Hybrid Logics (HyLo 2007