Verification of Non-Regular Program Properties

Most temporal logics which have been introduced and studied in the past decades can be embedded into the modal mu-calculus. This is the case for e.g. PDL, CTL, CTL*, ECTL, LTL, etc. and entails that these logics cannot express non-regular program properties. In recent years, some novel approaches towards an increase in expressive power have been made: Fixpoint Logic with Chop enriches the mu-calculus with a sequential composition operator and thereby allows to characterise context-free processes. The Modal Iteration Calculus uses inflationary fixpoints to exceed the expressive power of the mu-calculus. Higher-Order Fixpoint Logic (HFL) incorporates a simply typed lambda-calculus into a setting with extremal fixpoint operators and even exceeds the expressive power of Fixpoint Logic with Chop. But also PDL has been equipped with context-free programs instead of regular ones. In terms of expressivity there is a natural demand for richer frameworks since program property specifications are simply not limited to the regular sphere. Expressivity however usually comes at the price of an increased computational complexity of logic-related decision problems. For instance are the satisfiability problems for the above mentioned logics undecidable. We investigate in this work the model checking problem of three different logics which are capable of expressing non-regular program properties and aim at identifying fragments with feasible model checking complexity. Firstly, we develop a generic method for determining the complexity of model checking PDL over arbitrary classes of programs and show that the border to undecidability runs between PDL over indexed languages and PDL over context-sensitive languages. It is however still in PTIME for PDL over linear indexed languages and in EXPTIME for PDL over indexed languages. We present concrete algorithms which allow implementations of model checkers for these two fragments. We then introduce an extension of CTL in which the UNTIL- and RELEASE- operators are adorned with formal languages. These are interpreted over labeled paths and restrict the moments on such a path at which the operators are satisfied. The UNTIL-operator is for instance satisfied if some path prefix forms a word in the language it is adorned with (besides the usual requirement that until that moment some property has to hold and at that very moment some other property must hold). Again, we determine the computational complexities of the model checking problems for varying classes of allowed languages in either operator. It turns out that either enabling context-sensitive languages in the UNTIL or context-free languages in the RELEASE- operator renders the model checking problem undecidable while it is EXPTIME-complete for indexed languages in the UNTIL and visibly pushdown languages in the RELEASE- operator. PTIME-completeness is a result of allowing linear indexed languages in the UNTIL and deterministic context-free languages in the RELEASE. We do also give concrete model checking algorithms for several interesting fragments of these logics. Finally, we turn our attention to the model checking problem of HFL which we have already studied in previous works. On finite state models it is k-EXPTIME-complete for HFL(k), the fragment of HFL obtained by restricting functions in the lambda-calculus to order k. Novel in this work is however the generalisation (from the first-order case to the case for functions of arbitrary order) of an idea to improve the best and average case behaviour of a model checking algorithm by using partial functions during the fixpoint iteration guided by the neededness of arguments. This is possible, because the semantics of a closed HFL formula is not a total function but the value of a function at some argument. Again, we give a concrete algorithm for such an improved model checker and argue that despite the very high model checking complexity this improvement is very useful in practice and gives feasible results for HFL with lower order fuctions, backed up by a statistical analysis of the number of needed arguments on a concrete example. Furthermore, we show how HFL can be used as a tool for the development of algorithms. Its high expressivity allows to encode a wide variety of problems as instances of model checking already in the first-order fragment. The rather unintuitive -- yet very succinct -- problem encoding together with an analysis of the behaviour of the above sketched optimisation may give deep insights into the problem. We demonstrate this on the example of the universality problem for nondeterministic finite automata, where a slight variation of the optimised model checking algorithm yields one of the best known methods so far which was only discovered recently. We do also investigate typical model-theoretic properties for each of these logics and compare them with respect to expressive power

Games for Modal and Temporal Logics

Every logic comes with several decision problems. One of them is the model checking problem: does a given structure satisfy a given formula? Another is the satisfiability problem: for a given formula, is there a structure fulfilling it? For modal and temporal logics; tableaux, automata and games are commonly accepted as helpful techniques that solve these problems. The fact that these logics possess the tree model property makes tableau structures suitable for these tasks. On the other hand, starting with Büchi's work, intimate connections between these logics and automata have been found. A formula can describe an automaton's behaviour, and automata are constructed to accept exactly the word or tree models of a formula. In recent years the use of games has become more popular. There, an existential and a universal player play on a formula (and a structure) to decide whether the formula is satisfiable, resp. satisfied. The logical problem at hand is then characterised by the question of whether or not the existential player has a winning strategy for the game. These three methodologies are closely related. For example the non-emptiness test for an alternating automaton is nothing more than a 2-player game, while winning strategies for games are very similar to tableaux. Game-theoretic characterisations of logical problems give rise to an interactive semantics for the underlying logics. This is particularly useful in the specification and verification of concurrent systems where games can be used to generate counterexamples to failing properties in a very natural way. We start by defining simple model checking games for Propositional Dynamic Logic, PDL, in Chapter 4. These allow model checking for PDL in linear running time. In fact, they can be obtained from existing model checking games for the alternating free µ-calculus. However, we include them here because of their usefulness in proving correctness of the satisfiability games for PDL later on. Their winning strategies are history-free. Chapter 5 contains model checking games for branching time logics. Beginning with the Full Branching Time Logic CTL* we introduce the notion of a focus game. Its key idea is to equip players with a tool that highlights a particular formula in a set of formulas. The winning conditions for these games consider the players' behaviours regarding the change of the focus. This proves to be useful in capturing the regeneration of least and greatest fixed point constructs in CTL*. Deciding the winner of these games can be done using space which is polynomial in the size of the input. Their winning strategies are history-free, too. We also show that model checking games for CTL+ arise from those for CTL* by disregarding the focus. This does not affect the polynomial space complexity. These can be further optimised to obtain model checking games for the Computation Tree Logic CTL which coincide with the model checking games for the alternating free µ-calculus applied to formulas translated from CTL into it. This optimisation improves the games' computational complexity, too. As in the PDL case, deciding the winner of such a game can be done in linear running time. The winning strategies remain history-free. Focus games are also used to give game-based accounts of the satisfiability problem for Linear Time Temporal Logic LTL, CTL and PDL in Chapter 6. They lead to a polynomial space decision procedure for LTL, and exponential time decision procedures for CTL and PDL. Here, winning strategies are only history-free for the existential player. The universal player s strategies depend on a finite part of the history of a play. In spite of the strong connections between tableaux, automata and games their differences are more than simply a matter of taste. Complete axiomatisations for LTL, CTL and PDL can be extracted from the satisfiability focus games in an elegant way. This is done in Chapter 7 by formulating the game rules, the winning conditions and the winning strategies in terms of an axiom system. Completeness of this system then follows from the fact that the existential player wins the game on a consistent formula, i.e. it is satisfiable. We also introduce satisfiability games for CTL* based on the focus approach. They lead to a double exponential time decision procedure. As in the LTL, CTL and PDL case, only the existential player has history-free winning strategies. Since these strategies witness satisfiability of a formula and stay in close relation to its syntactical structure, it might be possible to derive a complete axiomatisation for CTL* from these games as well. Finally, Chapter 9 deals with Fixed Point Logic with Chop, FLC. It extends modal µ-calculus with a sequential composition operator. Satisfiability for FLC is undecidable but its model checking problem remains decidable. In fact it is hard for polynomial space. We give two different game-based solutions to the model checking problem for FLC. Deciding the winner for both types of games meets this polynomial space lower bound for formulas with fixed alternation (and sequential) depth. In the general case the winner can be determined using exponential time, resp. exponential space. The former result holds for games that give rise to global model checking whereas the latter describes the complexity of local FLC model checking. FLC is interesting for verification purposes since it --- unlike all the other logics discussed here --– can describe properties which are non-regular. The thesis concludes with remarks and comments on further research in the area of games for modal and temporal logics

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

Reasoning About the Transfer of Control

We present DCL-PC: a logic for reasoning about how the abilities of agents and coalitions of agents are altered by transferring control from one agent to another. The logical foundation of DCL-PC is CL-PC, a logic for reasoning about cooperation in which the abilities of agents and coalitions of agents stem from a distribution of atomic Boolean variables to individual agents -- the choices available to a coalition correspond to assignments to the variables the coalition controls. The basic modal constructs of DCL-PC are of the form coalition C can cooperate to bring about phi. DCL-PC extends CL-PC with dynamic logic modalities in which atomic programs are of the form agent i gives control of variable p to agent j; as usual in dynamic logic, these atomic programs may be combined using sequence, iteration, choice, and test operators to form complex programs. By combining such dynamic transfer programs with cooperation modalities, it becomes possible to reason about how the power of agents and coalitions is affected by the transfer of control. We give two alternative semantics for the logic: a direct semantics, in which we capture the distributions of Boolean variables to agents; and a more conventional Kripke semantics. We prove that these semantics are equivalent, and then present an axiomatization for the logic. We investigate the computational complexity of model checking and satisfiability for DCL-PC, and show that both problems are PSPACE-complete (and hence no worse than the underlying logic CL-PC). Finally, we investigate the characterisation of control in DCL-PC. We distinguish between first-order control -- the ability of an agent or coalition to control some state of affairs through the assignment of values to the variables under the control of the agent or coalition -- and second-order control -- the ability of an agent to exert control over the control that other agents have by transferring variables to other agents. We give a logical characterisation of second-order control

Discourses on social software

Can computer scientists contribute to the solution of societal problems? Can logic help to model social interactions? Are there recipes for making groups with diverging preferences arrive at reasonable decisions? Why is common knowledge important for social interaction? Does the rational pursuit of individual interests put the public interest in danger, and if so, why? Discourses on Social Software sheds light on these and similar questions. This book offers the reader an ideal introduction to the exciting new field of social software. It shows in detail the many ways in which the seemingly abstract sciences of logic and computer science can be put to use to analyse and solve contemporary social problems. The unusual format of a series of discussions among a logician, a computer scientist, a philosopher and some researchers from other disciplines encourages the reader to develop his own point of view. The only requirements for reading this book are a nodding familiarity with logic, a curious mind, and a taste for spicy debate.Kunnen de computerwetenschappers bijdragen aan een oplossing van sociale problemen? Kan logica gebruikt worden om sociale interactie te modelleren? Zijn er regels op te stellen om groepen met afwijkende voorkeuren tot redelijke besluiten te laten komen? Discourses on Social Software biedt de lezer een ideale inleiding op (nog nieuwe) gebied van sociale software. Het toont in detail de vele manieren waarin de schijnbaar abstracte wetenschappen van logica en computerwetenschap aan het werk kunnen worden gezet om eigentijdse sociale problemen te analyseren en op te lossen. Door de ongebruikelijke aanpak in dit boek, namelijk door discussies tussen een logicus, een computerwetenschapper, een filosoof en onderzoekers uit andere disciplines, wordt de lezer aangemoedigd zijn eigen standpunt te ontwikkelen. De enige vereisten om dit boek te lezen zijn enige vertrouwdheid met de logica, een nieuwsgierige geest, en liefde voor een pittig debat