Temporal Logic Programming Research Articles

Linear-Time Temporal Answer Set Programming

AbstractIn this survey, we present an overview on (Modal) Temporal Logic Programming in view of its application to Knowledge Representation and Declarative Problem Solving. The syntax of this extension of logic programs is the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). In the paper, we focus on the main recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL but involves a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). As a result, we obtain a proper extension of the stable models semantics for the general case of temporal formulas in the syntax of LTL. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between finite and infinite trace length. We also provide further useful results, such as the translation into other formalisms like Quantified Equilibrium Logic and Second-order LTL, and some techniques for computing temporal stable models based on automata constructions. In the remainder of the paper, we focus on practical aspects, defining a syntactic fragment called (modal) temporal logic programs closer to ASP, and explaining how this has been exploited in the construction of the solver telingo, a temporal extension of the well-known ASP solver clingo that uses its incremental solving capabilities.

Mining Road Traffic Rules with Signal Temporal Logic and Grammar-Based Genetic Programming

Traffic systems, where human and autonomous drivers interact, are a very relevant instance of complex systems and produce behaviors that can be regarded as trajectories over time. Their monitoring can be achieved by means of carefully stated properties describing the expected behavior. Such properties can be expressed using Signal Temporal Logic (STL), a specification language for expressing temporal properties in a formal and human-readable way. However, manually authoring these properties is a hard task, since it requires mastering the language and knowing the system to be monitored. Moreover, in practical cases, the expected behavior is not known, but it has instead to be inferred from a set of trajectories obtained by observing the system. Often, those trajectories come devoid of human-assigned labels that can be used as an indication of compliance with expected behavior. As an alternative to manual authoring, automatic mining of STL specifications from unlabeled trajectories would enable the monitoring of autonomous agents without sacrificing human-readability. In this work, we propose a grammar-based evolutionary computation approach for mining the structure and the parameters of an STL specification from a set of unlabeled trajectories. We experimentally assess our approach on a real-world road traffic dataset consisting of thousands of vehicle trajectories. We show that our approach is effective at mining STL specifications that model the system at hand and are interpretable for humans. To the best of our knowledge, this is the first such study on a set of unlabeled real-world road traffic data. Being able to mine interpretable specifications from this kind of data may improve traffic safety, because mined specifications may be helpful for monitoring traffic and planning safety promotion strategies.

Semantics of temporal constrained objects

This paper presents the declarative and operational semantics for the paradigm of temporal constrained objects (TCOB). This modeling paradigm is an extension of basic constrained objects where objects specify the structure of a system and constraints specify its behavior. Temporal constrained objects have been shown to provide a clear and high-level declarative specification of the time-dependent behavior of complex dynamic systems. Two key features of this paradigm are series variables and metric temporal operators. The main contribution of this paper lies in showing how we can define the semantics of temporal constrained objects in terms of a new paradigm called temporal constraint logic programs (TCLP), which extend temporal logic programs with constraint-solving. Compared with the several other logical languages incorporating constraints and temporal logic, the TCLP paradigm offers a clear and direct way of translating temporal constrained object programs using three simple temporal predicates: first p, next p, and prev p. The key approach is to map each class in a TCOB program to a predicate in the TCLP paradigm. Thus, the semantics of recursively defined classes is expressed in terms of the semantics of recursively defined predicates, a well-understood topic. Given the close connections between TCLP and CLP, we take advantage of the semantics of CLP for providing the declarative and operational semantics of TCLP and thereby also of temporal constrained objects.

A temporal logic programming approach to planning

This paper presents an approach to performing artificial intelligence planning through temporal logic programming with Search Control Knowledge (SCK). First, the planning problem described with Planning Domain Description Language is modeled as a program m in Modeling, Simulation and Verification Language (MSVL). Second, the SCK is also specified with an MSVL program $$m'$$ . Third, using the basic operation “and” in MSVL, a new MSVL program “ $$m~and~m'$$ ” is obtained. Forth, with the compiler MC of MSVL, an executable binary code of program “ $$m~and~m'$$ ” is obtained. Finally, planning result can be obtained via executing the executable code. Experimental results on selected benchmark planning domains from the International Planning Competition 2014 show that our approach is more effective in practice. Furthermore, the obtained plans are verified with the toolkit MSV so that a plan can be confirmed whether it is a reliable one.

Temporal Answer Set Programming on Finite Traces

AbstractIn this paper, we introduce an alternative approach to Temporal Answer Set Programming that relies on a variation of Temporal Equilibrium Logic (TEL) for finite traces. This approach allows us to even out the expressiveness of TEL over infinite traces with the computational capacity of (incremental) Answer Set Programming (ASP). Also, we argue that finite traces are more natural when reasoning about action and change. As a result, our approach is readily implementable via multi-shot ASP systems and benefits from an extension of ASP's full-fledged input language with temporal operators. This includes future as well as past operators whose combination offers a rich temporal modeling language. For computation, we identify the class of temporal logic programs and prove that it constitutes a normal form for our approach. Finally, we outline two implementations, a generic one and an extension of the ASP systemclingo.Under consideration for publication in Theory and Practice of Logic Programming (TPLP)

Evaluating Compliance: From LTL to Abductive Logic Programming

The compliance verification task amounts to establishing if the execution of a system, given in terms of observed events, does respect a given property. In the past both the frameworks of Temporal Logics and Logic Programming have been extensively exploited to assess compliance. In this work we review the LTL and the Abductive Logic Programming frameworks in the light of compliance evaluation, and formally investigate the relationship between the two approaches. We define a notion of compliance within each approach, and then we show that an arbitrary LTL formula can be expressed in SCIFF, by providing an automatic translation procedure from LTL to SCIFF which preserves compliance.

MSVL: a typed language for temporal logic programming

The development of types is an important but challenging issue in temporal logic programming. In this paper, we investigate how to formalize and implement types in the temporal logic programming language MSVL, which is an executable subset of projection temporal logic (PTL). Specifically, we extendMSVL with a few groups of types including basic data types, pointer types and struct types. On each type, we specify the domain of values and define some standard operations in terms of logic functions and predicates. Then, it is feasible to formalize statements of type declaration of program variables and statements of struct definitions as logic formulas. As the implementation of the theory, we extend theMSV toolkit with the support of modeling, simulation and verification of typedMSVL programs. Applications to the construction of AVL tree and ordered list show the practicality of the language.

Temporal logic programs with variables

AbstractIn this note, we consider the problem of introducing variables in temporal logic programs under the formalism of Temporal Equilibrium Logic, an extension of Answer Set Programming for dealing with linear-time modal operators. To this aim, we provide a definition of a first-order version of Temporal Equilibrium Logic that shares the syntax of first-order Linear-time Temporal Logic but has different semantics, selecting some Linear-time Temporal Logic models we call temporal stable models. Then, we consider a subclass of theories (called splittable temporal logic programs) that are close to usual logic programs but allowing a restricted use of temporal operators. In this setting, we provide a syntactic definition of safe variables that suffices to show the property of domain independence – that is, addition of arbitrary elements in the universe does not vary the set of temporal stable models. Finally, we present a method for computing the derivable facts by constructing a non-temporal logic program with variables that is fed to a standard Answer Set Programming grounder. The information provided by the grounder is then used to generate a subset of ground temporal rules which is equivalent to (and generally smaller than) the full program instantiation.

Game semantics for non-monotonic intensional logic programming

Intensional logic programming is an extension of logic programming based on intensional logic, which includes as special cases both temporal and modal logic programming. In [13], M. Orgun and W.W. Wadge provided a general framework for capturing the semantics of intensional logic programming languages. They demonstrated that if the intensional operators of a language obey some simple semantic properties, then the programs of the language are guaranteed to have a minimum model semantics. One key property involved in the construction of [13] is the monotonicity of intensional operators. In this paper we consider intensional logic programming from a game-theoretic perspective. In particular we define a two-person game and demonstrate that it can be used in order to define a model for any given intensional program of the class introduced in [13]. Moreover, this model is shown to be identical to the minimum model constructed in [13]. More importantly, we demonstrate that the game is even applicable to intensional languages with non-monotonic operators. In this way we provide the first (to our knowledge) general framework for capturing the semantics of non-monotonic intensional logic programming.

An infinitary encoding of temporal equilibrium logic

AbstractThis paper studies the relation between two recent extensions of propositional Equilibrium Logic, a well-known logical characterisation of Answer Set Programming. In particular, we show how Temporal Equilibrium Logic, which introduces modal operators as those typically handled in Linear-Time Temporal Logic (LTL), can be encoded into Infinitary Equilibrium Logic, a recent formalisation that allows the use of infinite conjunctions and disjunctions. We prove the correctness of this encoding and, as an application, we further use it to show that the semantics of the temporal logic programming formalism called TEMPLOG is subsumed by Temporal Equilibrium Logic.

Verifying safety critical task scheduling systems in PPTL axiom system

This paper presents a case study of formal verification of safety critical task scheduling systems. First, a scheduling algorithm described in a temporal logic programming language is presented; then a sufficient and necessary condition for the schedulability of task set is formalized. Further, the correctness of the condition is proved by means of theorem proving in the axiom system of Propositional Projection Temporal Logic.

Verification of distributed systems with the axiomatic system of MSVL

Abstract Since distributed systems are inherently concurrent and asynchronous, it is a challenge for us to verify distributed systems. MSVL is a useful temporal logic programming language and its axiomatic system has been established. However, the axiomatic system of MSVL lacks mechanisms to manage asynchronous communication, which makes it cannot deal with distributed systems. Thus, to verify distributed systems with MSVL in a deductive way, this paper is motivated to extend the axiomatic system of MSVL with new axioms for asynchronous communication. To this end, firstly we formalize state axioms regarding asynchronous communication commands and then prove the soundness and completeness. Further, to demonstrate how the extended axiomatic system of MSVL works for distributed systems, we apply it to the well-known Ricart–Agrawala (RA) algorithm, which is a distributed mutual exclusion algorithm and has an infinite state space. To do this, we model the RA algorithm with MSVL, specify the desired properties and then verify an instance of the RA algorithm with respect to the first-come-first-served property.

Super-Solutions

Annotated Probabilistic Temporal (APT) logic programs are a form of logic programs that allow users to state (or systems to automatically learn) rules of the form “formula G becomes true Δ t time units after formula F became true with ℓ to u % probability.” In this article, we deal with abductive reasoning in APT logic: given an APT logic program Π, a set of formulas H that can be “added” to Π, and a (temporal) goal g , is there a subset S of H such that Π ∪ S is consistent and entails the goal g ? In general, there are many different solutions to the problem and some of them can be highly repetitive, differing only in some unimportant temporal aspects. We propose a compact representation called super-solutions that succinctly represent sets of such solutions. Super-solutions are compact, but lossless representations of sets of such solutions. We study the complexity of existence of basic, super-, and maximal super-solutions as well as check if a set is a solution/super-solution/maximal super-solution. We then leverage a geometric characterization of the problem to suggest a set of pruning strategies and interesting properties that can be leveraged to make the search of basic and super-solutions more efficient. We propose correct sequential algorithms to find solutions and super-solutions. In addition, we develop parallel algorithms to find basic and super-solutions.

Logical foundations for more expressive declarative temporal logic programming languages

In this article, we present a declarative propositional temporal logic programming language called TeDiLog that is a combination of the temporal and disjunctive paradigms in logic programming. TeDiLog is, syntactically, a sublanguage of the well-known Propositional Linear-time Temporal Logic (PLTL). TeDiLog allows both eventualities and always-formulas to occur in clause heads and also in clause bodies. To the best of our knowledge, TeDiLog is the first declarative temporal logic programming language that achieves this high degree of expressiveness. We establish the logical foundations of our proposal by formally defining operational and logical semantics for TeDiLog and by proving their equivalence. The operational semantics of TeDiLog relies on a restriction of the invariant-free temporal resolution procedure for PLTL that was introduced by Gaintzarain et al. in [2013]. We define a fixpoint semantics that captures the reverse (bottom-up) operational mechanism and prove its equivalence with the logical semantics. We also provide illustrative examples and comparison with other proposals.

Editorial introduction to the special issue

The information available to intelligent agents is often uncertain, inconsistent and incomplete. It is then crucially important to develop logics for uncertainty to achieve different complex reasoning tasks. Uncertainty is present in many applications and may be represented using different frameworks: plausibility orderings, probability theory frameworks, argumentation systems, etc. This special issue gathers five contributions that cover different aspects of reasoning under uncertainty using non classical logics. Four of the five papers are fully revised and extended versions of contributions initially presented at the Fifth International Conference on Scalable Uncertainty Management (SUM’11) The two first papers concern extensions of logic-based programming languages to deal with uncertain information. The paper by Pere Pardo, Lluis Godo entitled “t-DeLP: an argumentation-based Temporal Defeasible Logic Programming framework” deals with handling temporal information using defeasible logic. The proposed framework, called t-DeLP, allows to extend argumentation-based defeasible logic in order to represent temporal processes. The idea is to associate temporal information to literals and to define temporal logic programs are sets of basic temporal facts and (strict or defeasible) durative rules. Concepts of argumentation systems, such as the one of undefeated arguments, are redefined in this t-DeLP framework. A procedure for determining a set of warranted literals is proposed. The paper also discusses rational properties satisfied by the t-DeLP framework. The paper by Georg Gottlob,Thomas Lukasiewicz, Maria Vanina Martinez and Gerardo I. Simari entitled “Query Answering under Probabilistic Uncertainty in Datalog+/ Ontologies” concerns an extension of Datalog+/− to deal with uncertainty. The authors use Markov Logic Networks to provide a probabilistic semantics for the probabilistic Datalog+/−. They provide scalable and sound algorithms for

Linear time-dependent constraints programming with MSVL

This paper investigates specifying and solving linear time-dependent constraints with an interval temporal logic programming language MSVL. To this end, linear constraint statements involving linear equality and non-strict inequality are first defined. Further, the time-dependent relations in the constraints are specified by temporal operators, such as [InlineEquation not available: see fulltext.]. Thus, linear time-dependent constraints can be smoothly incorporated into MSVL. Moreover, to solve the linear constraints within MSVL by means of reduction, the operational semantics for linear constraints is given. In particular, semantic equivalence rules and transition rules within a state are presented, which enable us to reduce linear equations, inequalities and optimization problems in a convenient way. Besides, the operational semantics is proved to be sound. Finally, a production scheduling application is provided to illustrate how our approach works in practice.

Towards a Common Categorical Semantics for Linear-Time Temporal Logic and Functional Reactive Programming

Linear-time temporal logic (LTL) and functional reactive programming (FRP) are related via a Curry–Howard correspondence. Based on this observation, we develop a common categorical semantics for a subset of LTL and its corresponding flavor of FRP. We devise a class of categorical models, called fan categories, that explicitly reflect the notion of time-dependent trueness of temporal propositions and a corresponding notion of time-dependent type inhabitance in FRP. Afterwards, we define the more abstract concept of temporal category by extending categorical models of intuitionistic S4. We show that fan categories are a special form of temporal categories.

Annotated Probabilistic Temporal Logic

Annotated Probabilistic Temporal (APT) logic programs support building applications where we wish to reason about statements of the form “Formula G becomes true with a probability in the range [ L , U ] within (or in exactly) Δt time units after formula F became true.” In this paper, we present a sound, but incomplete fixpoint operator that can be used to check consistency and entailment in APT logic programs. We present the first implementation of APT-logic programs and evaluate both its compute time and convergence on a suite of 23 ground APT-logic programs that were automatically learned from two real-world data sets. In both cases, the APT-logic programs contained up to 1,000 ground rules. In one data set, entailment problems were solved on average in under 0.1 seconds per ground rule, while in the other, it took up to 1.3 seconds per ground rule. Consistency was also checked in a reasonable amount of time. When discussing entailment of APT-logic formulas, convergence of the fixpoint operator refers to ( U − L ) being below a certain threshold. We show that on virtually all of the 23 automatically generated APT-logic programs, convergence was quick---often in just 2-3 iterations of the fixpoint operator. Thus, our implementation is a practical first step towards checking consistency and entailment in temporal probabilistic logics without independence or Markovian assumptions.

Reasoning about actions with Temporal Answer Sets

AbstractIn this paper, we combine Answer Set Programming (ASP) with Dynamic Linear Time Temporal Logic (DLTL) to define a temporal logic programming language for reasoning about complex actions and infinite computations. DLTL extends propositional temporal logic of linear time with regular programs of propositional dynamic logic, which are used for indexing temporal modalities. The action language allows general DLTL formulas to be included in domain descriptions to constrain the space of possible extensions. We introduce a notion of Temporal Answer Set for domain descriptions, based on the usual notion of Answer Set. Also, we provide a translation of domain descriptions into standard ASP and use Bounded Model Checking (BMC) techniques for the verification of DLTL constraints.

Annotated probabilistic temporal logic

The semantics of most logics of time and probability is given via a probability distribution over threads , where a thread is a structure specifying what will be true at different points in time (in the future). When assessing the probabilities of statements such as “Event a will occur within 5 units of time of event b ,” there are many different semantics possible, even when assessing the truth of this statement within a single thread. We introduce the syntax of annotated probabilistic temporal (APT) logic programs and axiomatically introduce the key notion of a frequency function (for the first time) to capture different types of intrathread reasoning, and then provide a semantics for intrathread and interthread reasoning in APT logic programs parameterized by such frequency functions. We develop a comprehensive set of complexity results for consistency checking and entailment in APT logic programs, together with sound and complete algorithms to check consistency and entailment. The basic algorithms use linear programming, but we then show how to substantially and correctly reduce the sizes of these linear programs to yield better computational properties. We describe a real world application we are developing using APT logic programs.