Specification and Verification using Temporal Logics

International audienceThis chapter illustrates two aspects of automata theory related to linear-time temporal logic LTL used for the verification of computer systems. First, we present a translation from LTL formulae to Büchi automata. The aim is to design an elementary translation which is reasonably efficient and produces small automata so that it can be easily taught and used by hand on real examples. Our translation is in the spirit of the classical tableau constructions but is optimized in several ways. Secondly, we recall how temporal operators can be defined from regular languages and we explain why adding even a single operator definable by a context-free language can lead to undecidability

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+

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

A Flexible and Efficient Temporal Logic Tool for Python: PyTeLo

Temporal logic is an important tool for specifying complex behaviors of systems. It can be used to define properties for verification and monitoring, as well as goals for synthesis tools, allowing users to specify rich missions and tasks. Some of the most popular temporal logics include Metric Temporal Logic (MTL), Signal Temporal Logic (STL), and weighted STL (wSTL), which also allow the definition of timing constraints. In this work, we introduce PyTeLo, a modular and versatile Python-based software that facilitates working with temporal logic languages, specifically MTL, STL, and wSTL. Applying PyTeLo requires only a string representation of the temporal logic specification and, optionally, the dynamics of the system of interest. Next, PyTeLo reads the specification using an ANTLR-generated parser and generates an Abstract Syntax Tree (AST) that captures the structure of the formula. For synthesis, the AST serves to recursively encode the specification into a Mixed Integer Linear Program (MILP) that is solved using a commercial solver such as Gurobi. We describe the architecture and capabilities of PyTeLo and provide example applications highlighting its adaptability and extensibility for various research problems

Efficient First-Order Temporal Logic for Infinite-State Systems

In this paper we consider the specification and verification of infinite-state systems using temporal logic. In particular, we describe parameterised systems using a new variety of first-order temporal logic that is both powerful enough for this form of specification and tractable enough for practical deductive verification. Importantly, the power of the temporal language allows us to describe (and verify) asynchronous systems, communication delays and more complex properties such as liveness and fairness properties. These aspects appear difficult for many other approaches to infinite-state verification.Comment: 16 pages, 2 figure

Complexity and Expressivity of Branching- and Alternating-Time Temporal Logics with Finitely Many Variables

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.Comment: Prefinal version of the published pape

Efficient Automata-based Planning and Control under Spatio-Temporal Logic Specifications

The use of spatio-temporal logics in control is motivated by the need to impose complex spatial and temporal behavior on dynamical systems, and to control these systems accordingly. Synthesizing correct-by-design control laws is a challenging task resulting in computationally demanding methods. We consider efficient automata-based planning for continuous-time systems under signal interval temporal logic specifications, an expressive fragment of signal temporal logic. The planning is based on recent results for automata-based verification of metric interval temporal logic. A timed signal transducer is obtained accepting all Boolean signals that satisfy a metric interval temporal logic specification, which is abstracted from the signal interval temporal logic specification at hand. This transducer is modified to account for the spatial properties of the signal interval temporal logic specification, characterizing all real-valued signals that satisfy this specification. Using logic-based feedback control laws, such as the ones we have presented in earlier works, we then provide an abstraction of the system that, in a suitable way, aligns with the modified timed signal transducer. This allows to avoid the state space explosion that is typically induced by forming a product automaton between an abstraction of the system and the specification.Comment: 8 pages - Accepted for Publication at ACC 202