Extension Of Linear Temporal Logic Research Articles

From Non-punctuality to Non-adjacency: A Quest for Decidability of Timed Temporal Logics with Quantifiers

Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent real-time extensions of Linear Temporal Logic (LTL). In general, the satisfiability checking problem for these extensions is undecidable when both the future (Until, U) and the past (Since, S) modalities are used (denoted by MTL[U,S] and TPTL[U,S]). In a classical result, the satisfiability checking for Metric Interval Temporal Logic (MITL[U,S]), a non-punctual fragment of MTL[U,S], is shown to be decidable with EXPSPACE complete complexity. A straightforward adoption of non-punctuality does not recover decidability in the case of TPTL[U,S]. Hence, we propose a more refined notion called non-adjacency for TPTL[U,S] and focus on its 1-variable fragment, 1-TPTL[U,S]. We show that non-adjacent 1-TPTL[U,S] is strictly more expressive than MITL. As one of our main results, we show that the satisfiability checking problem for non-adjacent 1-TPTL[U,S] is decidable with EXPSPACE complete complexity. Our decidability proof relies on a novel technique of anchored interval word abstraction and its reduction to a non-adjacent version of the newly proposed logic called PnEMTL. We further propose an extension of MSO [<] (Monadic Second Order Logic of Orders) with Guarded Metric Quantifiers (GQMSO) and show that it characterizes the expressiveness of PnEMTL. That apart, we introduce the notion of non-adjacency in the context of GQMSO (NA-GQMSO), which is a syntactic generalization of logic Q2MLO due to Hirshfeld and Rabinovich and show the decidability of satisfiability checking for NA-GQMSO.

Bounded Model Checking for Metric Temporal Logic Properties of Timed Automata with Digital Clocks

Metric temporal logic (MTL) is a popular real-time extension of linear temporal logic (LTL). This paper presents a new simple SAT-based bounded model-checking (SAT-BMC) method for MTL interpreted over discrete infinite timed models generated by discrete timed automata with digital clocks. We show a new translation of the existential part of MTL to the existential part of linear temporal logic with a new set of atomic propositions and present the details of the new translation. We compare the new method's advantages to the old method based on a translation of the hard reset LTL (HLTL). Our method does not need new clocks or new transitions. It uses only one path and requires a smaller number of propositional variables and clauses than the HLTL-based method. We also implemented the new method, and as a case study, we applied the technique to analyze several systems. We support the theoretical description with the experimental results demonstrating the method's efficiency.

Satisfiability and Model Checking for One Parameterized Extension of Linear Temporal Logic

Satisfiability and Model Checking for One Parameterized Extension of Linear Temporal Logic

MTL and TPTL for One-Counter Machines

Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are quantitative extensions of Linear Temporal Logic (LTL) that are prominent and widely used in the verification of real-timed systems. We study MTL and TPTL as specification languages for one-counter machines. It is known that model checking one-counter machines against formulas of Freeze LTL (FLTL), a strict fragment of TPTL, is undecidable. We prove that in our setting, MTL is strictly less expressive than TPTL, and incomparable in expressiveness to FLTL, so undecidability for MTL is not implied by the result for FLTL. We show, however, that the model-checking problem for MTL is undecidable. We further prove that the satisfiability problem for the unary fragments of TPTL and MTL are undecidable; for TPTL, this even holds for the fragment in which only one register and the finally modality is used. This is opposed to a known decidability result for the satisfiability problem for the same fragment of FLTL.

On the Expressive Power of Some Extensions of Linear Temporal Logic

On the Expressive Power of Some Extensions of Linear Temporal Logic

On the Expressive Power of Some Extensions of Linear Temporal Logic

One of the most simple models of computation which is suitable for representation of reactive systems behaviour is a finite state transducer which operates over an input alphabet of control signals and an output alphabet of basic actions. The behaviour of such a reactive system displays itself in the correspondence between flows of control signals and compositions of basic actions performed by the system. We believe that the behaviour of this kind requires more suitable and expressive means for formal specifications than the conventionalLT L. In this paper, we define some new (as far as we know) extensionLP-LT Lof Linear Temporal Logic specifically intended for describing the properties of transducers computations. In this extension the temporal operators are parameterized by sets of words (languages) which represent distinguished flows of control signals that impact on a reactive system. Basic predicates in our variant of the temporal logic are also languages in the alphabet of basic actions of a transducer; they represent the expected response of the transducer to the specified environmental influences. In our earlier papers, we considered a model checking problem forLP-LT LandLP-CT Land showed that this problem has effective solutions. The aim of this paper is to estimate the expressive power ofLP-LT Lby comparing it with some well known logics widely used in the computer science for specification of reactive systems behaviour. We discovered that a restricted variant LP-1-LT Lof our logic is more expressive thanLTLand another restricted variantLP-n-LT Lhas the same expressive power as monadic second order logic S1S.

Parameterized linear temporal logics meet costs: still not costlier than LTL

We continue the investigation of parameterized extensions of linear temporal logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for parametric LDL (PLDL), which extends PLTL to be able to express all $$\omega $$ -regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.

Approximating Optimal Bounds in Prompt-LTL Realizability in Doubly-exponential Time

We consider the optimization variant of the realizability problem for Prompt Linear Temporal Logic, an extension of Linear Temporal Logic (LTL) by the prompt eventually operator whose scope is bounded by some parameter. In the realizability optimization problem, one is interested in computing the minimal such bound that allows to realize a given specification. It is known that this problem is solvable in triply-exponential time, but not whether it can be done in doubly-exponential time, i.e., whether it is just as hard as solving LTL realizability. We take a step towards resolving this problem by showing that the optimum can be approximated within a factor of two in doubly-exponential time. Also, we report on a proof-of-concept implementation of the algorithm based on bounded LTL synthesis, which computes the smallest implementation of a given specification. In our experiments, we observe a tradeoff between the size of the implementation and the bound it realizes. We investigate this tradeoff in the general case and prove upper bounds, which reduce the search space for the algorithm, and matching lower bounds.

Parametric Linear Dynamic Logic

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by adding temporal operators equipped with parameters that bound their scope. LDL is an extension of Linear Temporal Logic (LTL) to all ω-regular specifications, while maintaining a translation into exponentially-sized non-deterministic Büchi automata. Since LDL cannot express timing constraints, we add parameterized operators and subsume parameterized extensions of LTL like Parametric LTL and PROMPT-LTL. Our contribution is a translation of PLDL into exponentially-sized non-deterministic Büchi automata via alternating automata. This yields PSPACE algorithms for model checking and assume-guarantee model checking and a 2EXPTIME realizability algorithm. The problems are complete for their complexity classes. We give tight bounds on optimal parameter values for model checking and realizability and present a PSPACE procedure for model checking optimization and a 3EXPTIME algorithm for realizability optimization. Our results show that these PLDL problems are no harder than their (parametric) LTL counterparts.

Formally Reasoning About Quality

In recent years, there has been a growing need and interest in formally reasoning about the quality of software and hardware systems. As opposed to traditional verification, in which one considers the question of whether a system satisfies a given specification or not, reasoning about quality addresses the question of how well the system satisfies the specification. We distinguish between two approaches to specifying quality. The first, propositional quality , extends the specification formalism with propositional quality operators, which prioritize and weight different satisfaction possibilities. The second, temporal quality , refines the “eventually” operators of the specification formalism with discounting operators, whose semantics takes into an account the delay incurred in their satisfaction. In this article, we introduce two quantitative extensions of Linear Temporal Logic (LTL), one by propositional quality operators and one by discounting operators. In both logics, the satisfaction value of a specification is a number in [0, 1], which describes the quality of the satisfaction. We demonstrate the usefulness of both extensions and study the decidability and complexity of the decision and search problems for them as well as for extensions of LTL that combine both types of operators.

Parameterized Linear Temporal Logics Meet Costs: Still not Costlier than LTL

We continue the investigation of parameterized extensions of Linear Temporal Logic (LTL) that retain the attractive algorithmic properties of LTL: a polynomial space model checking algorithm and a doubly-exponential time algorithm for solving games. Alur et al. and Kupferman et al. showed that this is the case for Parametric LTL (PLTL) and PROMPT-LTL respectively, which have temporal operators equipped with variables that bound their scope in time. Later, this was also shown to be true for Parametric LDL (PLDL), which extends PLTL to be able to express all omega-regular properties. Here, we generalize PLTL to systems with costs, i.e., we do not bound the scope of operators in time, but bound the scope in terms of the cost accumulated during time. Again, we show that model checking and solving games for specifications in PLTL with costs is not harder than the corresponding problems for LTL. Finally, we discuss PLDL with costs and extensions to multiple cost functions.

Parametric Linear Dynamic Logic

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL was proposed as an extension of Linear Temporal Logic (LTL) that is able to express all $\omega$-regular specifications while still maintaining many of LTL's desirable properties like an intuitive syntax and a translation into non-deterministic B\"uchi automata of exponential size. But LDL lacks capabilities to express timing constraints. By adding parameterized operators to LDL, we obtain a logic that is able to express all $\omega$-regular properties and that subsumes parameterized extensions of LTL like Parametric LTL and PROMPT-LTL. Our main technical contribution is a translation of PLDL formulas into non-deterministic B\"uchi word automata of exponential size via alternating automata. This yields a PSPACE model checking algorithm and a realizability algorithm with doubly-exponential running time. Furthermore, we give tight upper and lower bounds on optimal parameter values for both problems. These results show that PLDL model checking and realizability are not harder than LTL model checking and realizability.

A decidable timeout-based extension of linear temporal logic

We develop a timeout extension of propositional linear temporal logic (which we call ToLTL) to specify timing properties of timeout-based models of real-time systems. A timeout is used to model the execution of an action marking the end of a delay. With a view to expressing such timeout constraints, ToLTL uses a dynamic variable to abstract the timeout behaviour in addition to a variable which captures the global clock and some static timing variables which record time instances when discrete events occur. We propose a corresponding timeout tableau for satisfiability checking of the ToLTL formulas. Further we prove that the validity checking for such an extended logic remains PSPACE-complete. Under discrete time semantics, with bounded timeout increments, the model-checking problem which verifies if a ToLTL-formula holds in a timeout Kripke structure is also shown to be PSPACE complete. We further prove that ToLTL, when interpreted over discrete time, can be embedded in the monadic second-order logic with time and timeout and show an expressive completeness result for ToLTL. However, when ToLTL is interpreted over dense time without non-zenoness, the resulting logic becomes -complete (i.e., highly undecidable).

On the Expressiveness of TPTL and MTL over ω-Data Words

Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent extensions of Linear Temporal Logic to specify properties about data languages. In this paper, we consider the class of data languages of non-monotonic data words over the natural numbers. We prove that, in this setting, TPTL is strictly more expressive than MTL. To this end, we introduce Ehrenfeucht-Fraisse (EF) games for MTL. Using EF games for MTL, we also prove that the MTL definability decision problem ("Given a TPTL-formula, is the language defined by this formula definable in MTL?") is undecidable. We also define EF games for TPTL, and we show the effect of various syntactic restrictions on the expressiveness of MTL and TPTL.

Linear Temporal Logic for Regular Cost Functions

Regular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities, which retains strong closure, equivalence, and decidability properties. The specificity of cost functions is that exact values are not considered, but only estimated. In this paper, we define an extension of Linear Temporal Logic (LTL) over finite words to describe cost functions. We give an explicit translation from this new logic to two dual form of cost automata, and we show that the natural decision problems for this logic are PSPACE-complete, as it is the case in the classical setting. We then algebraically characterize the expressive power of this logic, using a new syntactic congruence for cost functions introduced in this paper.

Quantitative Analysis of Lattice-valued Kripke Structures

To model and analyze systems with multi-valued information, in this paper, we present an extension of Kripke structures in the framework of complete residuted lattices, which we will refer to as lattice-valued Kripke structures (LKSs). We then show how the traditional trace containment and equivalence relations, can be lifted to the lattice-valued setting, and we introduce two families of lattice-valued versions of the relations. Further, we explore some interesting properties of these relations. Finally, we provide logical characterizations of our relations by a natural extension of linear temporal logic.

Optimal bounds in parametric LTL games

Parameterized linear temporal logics are extensions of Linear Temporal Logic (LTL) by temporal operators equipped with variables that bound their scope. In model-checking, such specifications were introduced as “PLTL” by Alur et al. and as “PROMPT-LTL” by Kupferman et al. We show how to determine in doubly-exponential time, whether a player wins a game with PLTL winning condition with respect to some, infinitely many, or all variable valuations. Hence, these problems are not harder than solving LTL games. Furthermore, we present an algorithm with triply-exponential running time to determine optimal variable valuations that allow a player to win a game. Finally, we give doubly-exponential upper and lower bounds on the values of such optimal variable valuations.

Runtime Enforcement of Web Service Message Contracts with Data

An increasing number of popular SOAP web services exhibit a stateful behavior, where a successful interaction is determined as much by the correct format of messages as by the sequence in which they are exchanged with a client. The set of such constraints forms a “message contract” that needs to be enforced on both sides of the transaction; it often includes constraints referring to actual data elements inside messages. We present an algorithm for the runtime monitoring of such message contracts with data parameterization. Their properties are expressed in LTL-FO+, an extension of Linear Temporal Logic that allows first-order quantification over the data inside a trace of XML messages. An implementation of this algorithm can transparently enforce an LTL-FO+ specification using a small and invisible Java applet. Violations of the specification are reported on-the-fly and prevent erroneous or out-of-sequence XML messages from being exchanged. Experiments on commercial web services from Amazon.com and Google indicate that LTL-FO+ is an appropriate language for expressing their message contracts, and that its processing overhead on sample traces is acceptable both for client-side and server-side enforcement architectures.

A dynamic assertion-based verification platform for validation of UML designs

Capacity limitations continue to impede widespread adoption of formal property verification in the design validation ow of software and hardware systems. The more popular choice (at least in the hardware domain) has been dynamic property verification (DPV), which is a semi-formal approach where the formal properties are checked over simulation runs. DPV is highly scalable and can support a rich specification language. The main contribution of this paper is to build an integrated DPV platform for validation of UML-based designs. Specifically, we present (a) a language, named Action-LTL (a simple extension of Linear Temporal Logic) for writing assertions over data attributes and events of UML models, and (b) an integrated dynamic assertion-verification platform for verification of UML designs. In view of the capacity limitations of existing formal property verification tools, we believe that the methods presented in this paper are of immediate practical value to the UML design community.

Method of Constructing Two-Way Alternating Automata for PSL and Translation to Nondeterministic Automata

PSL (property specification language) is a property specification language to describe parallel systems and can be divided into two parts,FL (foundation language) and OBE (optional branching extension). Since OBE is essentially the temporal logic CTL (computation tree logic),and PSL formulas with clock statements can be easily rewritten to unclocked formulas,this paper plays an emphasis on the unclocked FL logic. In order to be model-checked,each FL formula needs to be translated into a verifiable form,usually as an automaton (nondeterministic automaton). The translation into nondeterministic automata can be realized mainly by the construction of alternating automata. The translation rules for the two-way alternating automata from unclocked FL logic are explained in detail in this paper. The core logic of the construction rules is not only limited to an extension of LTL (linear temporal logic) with regular expressions,but considers overall FL operators adequately. A translation method from two-way alternating automata to nondeterministic automata is also provided. Finally,a translation tool from PSL formulas to the above two automata has been written. The complexity of the construction rules for the two-way alternating automata grows linearly with the length of the FL formulas,and at the same time,the correctness of the rules is verified. It is also proved that the two-way alternating automata and its corresponding nondeterministic automata accept the same language. The work above has important theoretical and application values for the modeling and model checking for the complex parallel systems.