Quantales and Temporal Logics

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

We consider monadic second order logic (MSO) and the modal μ-calculus (Lμ) over transition systems (Kripke structures). It is well known that every class of transition systems which is definable by a sentence of Lμ is definable by a sentence of MSO as well. It will be shown that the converse is also true for an important fragment of MSO: every class of transition systems which is MSO-definable and which is closed under bisimulation - i.e., the sentence does not distinguish between bisimilar models - is also Lμ-definable. Hence we obtain the following expressive completeness result: the bisimulation invariant fragment of MSO and Lμ are equivalent. The result was proved by David Janin and Igor Walukiewicz. Our presentation is based on their article [91]. The main step is the development of automata-based characterizations of Lμ over arbitrary transition systems and of MSO over transition trees (see also Chapter 16). It turns out that there is a general notion of automaton subsuming both characterizations, so we obtain a common ground to compare these two logics. Moreover we need the notion of the ω-unravelling for a transition system, on the one hand to obtain a bisimilar transition tree and on the other hand to increase the possibilities of choosing successors.

Model-Checking of Infinite Kripke Structures Defined by Simple Graph Grammars

In this thesis, we introduce and examine four new temporal logic formalisms that can be used as specification languages for the automated verification of the reliability of hardware and software designs with respect to a desired behavior. The work is organized in two parts. In the first one, we reason about two logics for computations, GCTL* and MCTL*, which are useful to describe a correct execution of monolithic closed systems. In the second one, instead, we focus on two logics for strategies, SL and mATL*, which are useful to formalize several interesting properties about interactive plays in multi-entities systems modeled as multi-agent games. In the “Logics for Computations” part, we first study the immersion of the idea of graded quantifications into the temporal-logic framework. In first order logic, existential and universal quantifiers express the concepts of the existence of at least one individual object satisfying a formula, or that all individual objects satisfy a formula. In other logics, these quantifiers have been generalized to express that, for a given non-negative integer n, at least n or all but n individuals satisfy a particular formula. Here, we consider GCTL, a temporal logic with graded path quantifiers, which allows to describe properties like “there exist at least n different classes of computational fluxes in which a system reaches a predetermined state”, where the classes over paths are computed by means of a predetermined equivalence relation. More precisely, we uniformly extend the classic concept of graded quantifiers from states to paths, through the use of a concept of path equivalence with respect to a given path formula. About this logic, in particular, we study the expressiveness and succinctness relationships with respect to GµCalculus and the complexity of the satisfiability problem, which results to be ExpTime-Complete. This research is partially based on the works [BMM09] “Graded Computation Tree Logic” and [BMM10] “Graded Computation Tree Logic with Binary Coding” published, respectively, in the proceedings of the “IEEE Symposium on Logic in Computer Science, 2009” and “EACSL Annual Conference on Computer Science Logic, 2010”. Preliminary results can be also found in [Mog07]. Furthermore, we consider special quantifiers over substructures, which allow to select, using parametric criteria, small critical parts of a system to be successively verified. In literature, there are some attempts to define a logic that allows to modify the underlying structure under exam and then to verify on it some assigned property. However, as far as we know, none of them is able to select minimal submodels of a given property describing the criteria on which then execute the verification process. Here, we base our work on the search of a new operator that merges the concept of quantifiers on structures with that one derived by a generalization of the concept of pruning. The results of this work, is a class of three different extensions of CTL* with minimal model quantifiers, which we name MCTL*. Regarding these logics, we study several reductions among them, as well as the satisfiability problem that we prove to be highly undecidability, i.e., Σ_1^1-Hard, for two out of the three cases. This research is partially based on the work [MM09] “Branching-Time Temporal Logics with Minimal Model Quantifiers” published in the proceedings of the “International Conference on Developments in Language Theory, 2009”. In the “Logics for Strategies” part, we first study the problem of defining a new specification language through which it is possible to express several important properties of multi-entities systems that are neither expressible using classical monolithic temporal logics, such as CTL*, nor using two-agent-teams temporal logics, such as ATL*. In literature, we can found some proposal of logics that try to achieve this goal, but unfortunately, none of them succeeds completely on all the aspects. Among them, one of the most important attempts is CHP-SL, a logics in which one can use variables over strategies. However, this logic has a deep weakness, since it does not allow to describe games with more than two players and even two-players concurrent games. Here, we introduce SL, a logic with a syntax similar in some aspects to the first order logic, in which the strategies of the agent building the game are treated as first order objects on which we can quantify. This logic generalizes CHP-SL, by allowing the specification of the correct behavior of multi-agent concurrent games. In SL, for example, we are able to express very complex but useful Nash equilibria that are not expressible with CHP-SL. We enlighten that Nash equilibrium is one of the most important concepts in game theory. For the introduced logic, we solve two problems left open in the work on CHP-SL. Precisely, we show that the related model-checking problem is 2ExpTime-Complete, thus not harder of the same problem for several subsumed logics, while we prove that its satisfiability problem is highly undecidable, i.e., Σ_1^1-Hard. This research is partially based on the work [MMV10a] “Reasoning About Strategies” published in the proceedings of the “IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2010”. Finally, we consider the concept of relentful strategic reasoning, i.e., a formalism that expresses the ability of a strategy to be used not only to achieve a first given goal, but also to change its final goal in dependence of the history of the play. In the context of planning, memoryful quantification, i.e., quantification over computations that does not lose information about the past along the time, is one of the principal way to express the fact that a system is able to achieve a desired result, and shift to a different goal if some event happens. However, this kind of quantification was not considered before in the context of multi-agent planning. Here, we introduce mATL*, a fusion of the classic alternating temporal logic ATL* with memoryful quantification, with the aim of covering the previous idea. About this logics, we prove that, although it is equivalent to ATL*, it is exponentially more succinct. Nevertheless, we prove that both the model-checking and the satisfiability problems remain 2ExpTime-Complete, as for ATL*. This research is partially based on the work [MMV10b] “Relentful Strategic Reasoning in Alternating-Time Temporal Logic” published in the proceedings of the “International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2010”.

Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $\mathbb{M}^Y$, where $Y$ is a finite set and $\mathbb{M}$ an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, metric transition systems, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.

In this paper, the synthesis of bounded Petri nets from deterministic modal transition systems is shown to be undecidable. The proof is built from three components. First, it is shown that the problem of synthesising bounded Petri nets satisfying a given formula of the conjunctive nu-calculus (a suitable fragment of the mu-calculus) is undecidable. Then, an equivalence between deterministic modal transition systems and a language-based formalism called modal specifications is developed. Finally, the claim follows from a known equivalence between the conjunctive nu-calculus and modal specifications.

The paper explores properties of {\L}ukasiewicz mu-calculus, a version of the quantitative/probabilistic modal mu-calculus containing both weak and strong conjunctions and disjunctions from {\L}ukasiewicz (fuzzy) logic. We show that this logic encodes the well-known probabilistic temporal logic PCTL. And we give a model-checking algorithm for computing the rational denotational value of a formula at any state in a finite rational probabilistic nondeterministic transition system.

Bisimilar linear systems

This paper adresses the use of the model-checking technique to check the CARE properties of input multimodal interactions. Transition systems and temporal logic are used to specify a multimodal system and its properties. The properties of the multimodal interface are specified in the CTL temporal logic and checked by the SMV model checker.

We revisit two well-established verification techniques,k-inductionandbounded model checking(BMC), in the more general setting of fixed point theory over complete lattices. Our main theoretical contribution islatticed k-induction, which (i) generalizes classicalk-induction for verifying transition systems, (ii) generalizes Park induction for bounding fixed points of monotonic maps on complete lattices, and (iii) extends from naturalskto transfinite ordinals$$\kappa $$κ, thus yielding$$\kappa $$κ-induction.The lattice-theoretic understanding ofk-induction and BMC enables us to apply both techniques to thefully automatic verification of infinite-state probabilistic programs. Our prototypical implementation manages to automatically verify non-trivial specifications for probabilistic programs taken from the literature that—using existing techniques—cannot be verified without synthesizing a stronger inductive invariant first.

We present PCFTL (Probabilistic CounterFactual Temporal Logic), a new probabilistic temporal logic for the verification of Markov Decision Processes (MDP). PCFTL introduces operators for causal inference, allowing us to express interventional and counterfactual queries. Given a path formula φ, an interventional property is concerned with the satisfaction probability of φ if we apply a particular change I to the MDP (e.g., switching to a different policy); a counterfactual formula allows us to compute, given an observed MDP path τ, what the outcome of φ would have been had we applied I in the past and under the same random factors that led to observing τ. Our approach represents a departure from existing probabilistic temporal logics that do not support such counterfactual reasoning. From a syntactic viewpoint, we introduce a counterfactual operator that subsumes both interventional and counterfactual probabilities as well as the traditional probabilistic operator. This makes our logic strictly more expressive than PCTL⋆. The semantics of PCFTL rely on a structural causal model translation of the MDP, which provides a representation amenable to counterfactual inference. We evaluate PCFTL in the context of safe reinforcement learning using a benchmark of grid-world models.

In a mixed approach to system verification using theorem provers with an interface to specialized model-checking tools, it may be necessary to simplify models by considering abstract versions of them. We report on work in progress that aims to develop support within LAMBDA for a systematic approach to abstraction. We give a formalization in LAMBDA of a notion of abstraction for transition systems; the abstract systems have two sorts of transition, and are related to specifications in modal process logic. We prove that formulae in the modal mu-calculus are satisfied in an abstract version of a model only if they are satisfied in the model itself. We illustrate how the proof of an inductive step in the verification of a satisfaction relation for an infinite model can be reduced to the verification of a satisfaction relation for a very small finite model.

Chapter 4 - Modal Logics and mu-Calculi: An Introduction

This paper presents IM-SPDL, a stochastic extension of the modal logic PDL, which supports the specification of complex performance and dependability requirements. The logic is interpreted over extended stochastic labelled transition systems (ESLTS), i.e. transition systems containing both immediate and Markovian transitions. We define the syntax and semantics of the new logic and show that IM-SPDL provides powerful means to specify path-based properties with timing restrictions. In general, paths can be characterised by regular expressions, also called programs, where the executability of a program may depend on the validity of test formulae. For the model checking of IM-SPDL time-bounded path formulae, a deterministic program automaton is constructed from the requirement. Afterwards the product transition system between this automaton and the ESLTS is built and subsequently transformed into a continuous time Markov Chain (CTMC) on which numerical analysis is performed. Empirical results given in the paper show that model checking IM-SPDL can be realised efficiently in practice.

Durational transition systems are finite transition systems where every transition is additionally equipped with a duration. We consider the problem of interpreting /spl mu/-formulas over durational transition systems. In case the formula contains only operations minimum, maximum, addition, and sequencing, we show that the interpretation ist not only computable but (up to a linear factor) as efficiently computable as the interpretation of /spl mu/-formulas over ordinary finite transition systems.