On the Characterization of Until as a Fixed Point Under Clocked Semantics

Temporal Logic has the potential to improve current methods for specifying and reasoning about asynchronous systems. However, standard temporal logics have been difficult to use, because they are rather low level for the design of realistic systems and are based on modal logics that are not intuitive to users. To be more appropriate for describing complex systems, Interval Logic has been designed as a higher-level temporal logic with a graphical representation. A not quite trivial proof is presented to demonstrate the ability to represent proofs graphically.KeywordsModal LogicTemporal LogicService SpecificationAsynchronous SystemOutput QueueThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The paper describes an implementation in the HOL theorem prover of the quantitative Temporal Logic (qTL) of Morgan and McIver [18,4]. qTL is a branching-time temporal logic suitable for reasoning about probabilistic nondeterministic systems. The interaction between probabilities and nondeterminism, which is generally very difficult to reason about, is handled by interpreting the logic over real- rather than boolean-valued functions. In theqTL framework many laws of standard branching-time temporal logic generalize nicely giving access to a number of logical tools for reasoning about quantitative aspects of randomized algorithms.

There are two typical ways to formalize reasoning over time. The first is in a modal logic where time is characterized implicitly through tense operators. The second addresses time explicitly in a first-order theory, where the reasoning is itself essentially first order. In this paper we shall introduce an amalgamated temporal logic which combines the tense operators of the modal approach with the referential aspects of the first-order approach.The modal operators of the new logic, called IQ-C, include the familiar tense operators plus a few new ones; together they allow complex temporal propositions to be represented in a compact form. Unlike standard modal temporal logics, IQ-C contains terms which refer to times; this allows the logic to characterize certain temporal properties and operators which cannot be axiomatized or defined by conventional modal operators. IQ-C is expressively complete over any connected flow of time, and has several other interesting theoretical properties, some of which will be developed in this paper.Most interval logics take one of two approaches to temporal assertions; either they analyze assertions for intervals in terms of assertions at points, or they treat intervals as primitive, allowing assertions for intervals but not at points. IQ-C allows assertions both for intervals and at points, and treats them as independent; this yields a special degree of semantic discrimination and at the same time simplifies the representation of several temporal properties. In this paper we shall outline the IQ-C framework, specifying its special features and defining its semantics. We shall also extend a method of skolemization, which is designed to support a resolution based approach to the proof theory. This will be sketched in some detail, with a view to providing a preliminary account of the basic proof theory. A complete specification of the deductive theory, together with relevant proofs, will be developed elsewhere.KeywordsIntervalsPointsTemporal Representation and ExpressivenessHomogeneitySkolemizationand Modal Theorem Proving

Temporal logic model checking is a useful method for verifying properties of finite-state concurrent systems. However, due to the state explosion problem modular methods like compositional minimisation based on semantic congruences are essential in making the verification task manageable. In this paper we show that the so-called CFFD-equivalence defined by initial stability, infinite traces, divergence traces and stable failures is exactly the weakest compositional equivalence preserving nexttimeless linear temporal logic with an extra operator distinguishing deadlocks from divergences. Furthermore, a slight modification of CFFD, called the NDFD-equivalence, is exactly the weakest compositional equivalence preserving standard nexttimeless linear temporal logic.

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”.

A new logic, sequence-indexed linear-time temporal logic (SLTL), is obtained semantically from the standard linear-time temporal logic LTL by adding a sequence modal operator which represents a sequence of symbols. By the sequence modal operator of SLTL, we can appropriately express “sequential information” in temporal reasoning. A Gentzen-type sequent calculus for SLTL is introduced, and the completeness and cut-elimination theorems for this calculus are proved. SLTL is also shown to be PSPACE-complete and embeddable into LTL.KeywordsInduction HypothesisProof SystemSequence ModelLinear Temporal LogicPropositional VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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A locative inconsistency-tolerant hierarchical probabilistic computation tree logic (LIHpCTL) is introduced in this paper to establish the logical foundation of a new model checking paradigm. This logic is an extension of several previously proposed extensions of the standard temporal logic known as CTL, which is widely used for model checking. The extended model checking paradigm proposed is intended to appropriately verify locative (spatial), inconsistent, hierarchical, probabilistic (randomized), and time-dependent concurrent systems. Additionally, a survey of various studies on probabilistic, inconsistency-tolerant, and hierarchical temporal logics and their applications in model checking is conducted.

In order to handle the increasing complexity of hardware/software system designs, we propose an efficient model checker for the propositional temporal logic denoted by PTL. This logic is known to be well suited to verify electronic circuits and reactive systems. A typical verification problem consists of establishing formally a relationship between the specification of a software/hardware system and its implementation. We show also how a hardware designer should proceed to specify his design and prove its correctness using a PTL module under Maude a rewriting logic-based system. A series of experiments have been conducted successfully on a well-known benchmark to prove the effectiveness of mixing temporal logic and rewriting logic techniques.

Hyperproperties, which generalize trace properties by relating multiple traces, are widely studied in information-flow security. Recently, a number of logics for hyperproperties have been proposed, and there is a need to understand their decidability and relative expressiveness. The new logics have been obtained from standard logics with two principal extensions: temporal logics, like LTL and CTL$^*$, have been generalized to hyperproperties by adding variables for traces or paths. First-order and second-order logics, like monadic first-order logic of order and MSO, have been extended with the equal-level predicate. We study the impact of the two extensions across the spectrum of linear-time and branching-time logics, in particular for logics with quantification over propositions. The resulting hierarchy of hyperlogics differs significantly from the classical hierarchy, suggesting that the equal-level predicate adds more expressiveness than trace and path variables. Within the hierarchy of hyperlogics, we identify new boundaries on the decidability of the satisfiability problem. Specifically, we show that while HyperQPTL and HyperCTL$^*$ are both undecidable in general, formulas within their $\exists^*\forall^*$ fragments are decidable.

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \(\exists x(x=\alpha )\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account.

We introduce emph{Normative Temporal Logic} (acro{ntl}), a logic for reasoning about normative systems. acro{ntl} is a generalisation of the well-known branching-time temporal logic acro{ctl}, in which the path quantifiers $Apath$ (``on all pathsldots'') and $Epath$ (``on some pathldots'') are replaced by the indexed deontic operators $O{ s}$ and $P{ s}$, where for example $O{ s}phi$ means ``$phi$ is obligatory in the context of normative system $ s$''. After defining the logic, we give a sound and complete axiomatisation, and discuss the logic's relationship to standard deontic logics. We present a symbolic representation language for models and normative systems, and identify four different model checking problems, corresponding to whether or not a model is represented symbolically or explicitly, and whether or not we are given an interpretation for the normative systems named in formulae to be checked. We show that the complexity of model checking varies from acro{p}-complete up to acro{exptime}-hard for these variations.

Modern software and hardware designs are mostly hierarchical. Moreover, while the design specification is defined up-down, the design implementation and verification are done down-up. In such a case, as a rule, coverage properties for simulation-based verification are defined inconsistently for different stages of the design flow. The fact leads to the well known explosion of bug rate, when we pass from the lower design stage to the upper one. In this paper, we propose a new approach that allows propagation of quantitative properties from the upper stage of the design flow to the lower ones and their incremental computation on the components. The approach may be applied to any design, modeled as a Finite State Machine (FSM) or other formalisms, which eventually lead to weighted labeled trees. We use Weighted Monadic-Second Order Logic (WMSOL) to describe the desired families of quantitative properties and sum-like weighted labeled trees to describe the decomposition of the behaviour of the FSM. The last notion is based on a generalization of disjoint unions of structures with additional links between the components. Our approach shows how computation of a quantitative property, definable as a WMSOL formula on the upper stage of the design may be reduced for certain cost to incremental computations of effectively derivable WMSOL-definable properties on the components. We provide several examples of families of such properties and discuss different aspects, related to the applicability of our approach. The approach is new and provides a uniform theoretical basis for analyzing WMSOL-definable properties on hierarchical structures.

Contracts-refinement proof system for component-based embedded systems

In this article, we propose a language, Asynchronous Propositional Temporal Logic (APTL), for the specification and verification of distributed hard real-time sytems. APTL extends the logic TPTL by dealing explicitly with multiple local clocks. We propose a distributed-system model which permits definition of inequalities asserting the temporal precedence of local clock readings. We show the expressiveness of APTL through two nontrivial examples. Our logic can be used to specify and reason about such important properties as bounded clock rate drifting. We then give a 2 2 0(n) tableau-based decision procedure for determining APTL satisfiability, where n is the size (number of bits) of the input formula.