Linear Temporal Logic LTL Research Articles

LTL-specification for development and verification of control programs

This work continues the series of articles on development and verification of control programs based on the LTL-specification. The essence of the approach is to describe the behavior of programs using formulas of linear temporal logic LTL of a special form. The developed LTL-specification can be directly verified by using a model checking tool. Next, according to the LTL-specification, the program code in the imperative programming language is unambiguously built. The translation of the specification into the program is carried out using a template. The novelty of the work consists in the proposal of two LTL-specifications of a new form — declarative and imperative, as well as in a more strict formal justification for this approach to program development and verification. A transition has been made to a more modern verification tool for finite and infinite systems — nuXmv. It is proposed to describe the behavior of control programs in a declarative style. For this purpose, a declarative LTL-specification is intended, which defines a labelled transition system as a formal model of program behavior. This method of describing behavior is quite expressive — the theorem on the Turing completeness of the declarative LTL-specification is proved. Next, to construct program code in an imperative language, the declarative LTL-specification is converted into an equivalent imperative LTL-specification. An equivalence theorem is proved, which guarantees that both specifications specify the same behavior. The imperative LTL-specification is translated into imperative program code according to the presented template. The declarative LTL-specification, which is subject to verification, and the control program built on it are guaranteed to specify the same behavior in the form of a corresponding transition system. Thus, during verification, a model is used that is adequate to the real behavior of the control program.

Deciding FO-rewritability of Regular Languages and Ontology-Mediated Queries in Linear Temporal Logic

Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,≡)-rewritability using unary predicates x ≡ 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSᴘᴀᴄᴇ-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,≡)- and FO(<,MOD)-definability is also PSᴘᴀᴄᴇ-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,≡)- and FO(<,MOD)-rewritability of LTL OMQs is ExᴘSᴘᴀᴄᴇ-complete, and that these problems become PSᴘᴀᴄᴇ-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,≡)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSᴘᴀᴄᴇ-, Π2p- and coNP-complete.

First-Order Rewritability and Complexity of Two-Dimensional Temporal Ontology-Mediated Queries

Aiming at ontology-based data access to temporal data, we design two-dimensional temporal ontology and query languages by combining logics from the (extended) DL-Lite family with linear temporal logic LTL over discrete time (Z,<). Our main concern is first-order rewritability of ontology-mediated queries (OMQs) that consist of a 2D ontology and a positive temporal instance query. Our target languages for FO-rewritings are two-sorted FO(<) -- first-order logic with sorts for time instants ordered by the built-in precedence relation < and for the domain of individuals---its extension FO(<, ≡) with the standard congruence predicates t ≡ 0 (mod n), for any fixed n > 1, and FO(RPR) that admits relational primitive recursion. In terms of circuit complexity, FO(<, ≡)- and FO(RPR)-rewritability guarantee answering OMQs in uniform AC0 and NC1, respectively. We proceed in three steps. First, we define a hierarchy of 2D DL-Lite/LTL ontology languages and investigate the FO-rewritability of OMQs with atomic queries by constructing projections onto 1D LTL OMQs and employing recent results on the FO-rewritability of propositional LTL OMQs. As the projections involve deciding consistency of ontologies and data, we also consider the consistency problem for our languages. While the undecidability of consistency for 2D ontology languages with expressive Boolean role inclusions might be expected, we also show that, rather surprisingly, the restriction to Krom and Horn role inclusions leads to decidability (and ExpSpace-completeness), even if one admits full Booleans on concepts. As a final step, we lift some of the rewritability results for atomic OMQs to OMQs with expressive positive temporal instance queries. The lifting results are based on an in-depth study of the canonical models and only concern Horn ontologies.

LTL-Specification of Bounded Counter Machines

The article revises the results of the work devoted to the problem of representing the behaviour of a program system as a set of formulas of the linear temporal logic LTL, followed by the use of this representation to verify the satisfiability of the program system properties through the procedure of proving the validity of logical inferences, expressed in terms of the LTL logic. The LTL logic is applied to bounded Minsky counter machines that are considered as program systems of which we need to get the specification of its behaviour. Earlier, when working with the temporal logic LTL as a program logic, a special pseudo-operator was actually introduced to refer to the previous values of variables involved in elementary propositions. Despite the fact that this pseudo-operator is easily implemented in the Cadence SMV verifier when proving the validity of logical LTL-inferences, the classical definition of the LTL logic does not imply its presence. In this article, only binary variables will be used to build an LTL-specification for the behaviour of a bounded counter machine, and tracking of previous values of these variables will be carried out exclusively within the LTL logic itself through the appropriate formulas.

First-order rewritability of ontology-mediated queries in linear temporal logic

We investigate ontology-based data access to temporal data. We consider temporal ontologies given in linear temporal logic LTL interpreted over discrete time (Z,<). Queries are given in LTL or MFO(<), monadic first-order logic with a built-in linear order. Our concern is first-order rewritability of ontology-mediated queries (OMQs) consisting of a temporal ontology and a query. By taking account of the temporal operators used in the ontology and distinguishing between ontologies given in full LTL and its core, Krom and Horn fragments, we identify a hierarchy of OMQs with atomic queries by proving rewritability into either FO(<), first-order logic with the built-in linear order, or FO(<,≡), which extends FO(<) with the standard arithmetic predicates x≡0(modn), for any fixed n>1, or FO(RPR), which extends FO(<) with relational primitive recursion. In terms of circuit complexity, FO(<,≡)- and FO(RPR)-rewritability guarantee OMQ answering in uniform ▪ and, respectively, ▪.We obtain similar hierarchies for more expressive types of queries: positive LTL-formulas, monotone MFO(<)- and arbitrary MFO(<)-formulas. Our results are directly applicable if the temporal data to be accessed is one-dimensional; moreover, they lay foundations for investigating ontology-based access using combinations of temporal and description logics over two-dimensional temporal data.

LTL-Specification of Counter Machines

The article is written in support of the educational discipline “Non-classical logics”. Within the framework of this discipline, the objects of study are the basic principles and constructive elements, with the help of which the formal construction of various non-classical propositional logics takes place. Despite the abstractness of the theory of non-classical logics, in which the main attention is paid to the strict mathematical formalization of logical reasoning, there are real practical areas of application of theoretical results. In particular, languages of temporal modal logics are widely used for modeling, specification, and verification (correctness analysis) of logic control program systems. This article demonstrates, using the linear temporal logic LTL as an example, how abstract concepts of non-classical logics can be reƒected in practice in the field of information technology and programming. We show the possibility of representing the behavior of a software system in the form of a set of LTL-formulas and using this representation to verify the satisfiability of program system properties through the procedure of proving the validity of logical inferences, expressed in terms of the linear temporal logic LTL. As program systems, for the specification of the behavior of which the LTL logic will be applied, Minsky counter machines are considered. Minsky counter machines are one of the ways to formalize the intuitive concept of an algorithm. They have the same computing power as Turing machines. A counter machine has the form of a computer program written in a high-level language, since it contains variables called counters, and conditional and unconditional jump operators that allow to build loop constructions. It is known that any algorithm (hypothetically) can be implemented in the form of a Minsky three-counter machine.

On the expressiveness of the approach to constructing PLC-programs by LTL-specification

The article is devoted to an approach to constructing and verification of discrete PLC-programs by LTL-specification. This approach provides a possibility of analysing the correctness of PLCprograms by using the model checking method. The linear temporal logic LTL is used as a language of specification of the program behavior. The correctness analysis of LTL-specification is automatically performed by the symbolic model checking tool Cadence SMV. The article demonstrates the consistency of the approach to constructing and verification of PLC programs by LTL-specification from the point of view of Turing power. It is proved that in accordance with this approach for any Minsky counter machine an LTL-specification can be built, which is used for machine implementation in any PLC programming language of standard IEC 61131-3. Minsky machines are equipollent to Turing machines, and the considered approach also has the Turing power. The proof focuses on representation of a counter machine behavior in the form of a set of LTL-formulas and matching these formulas to constructions of ST and SFC languages. SFC is interesting as a specific graphical language. ST is considered as a basic language because the implementation of a counter machine on IL, FBD/CFC and LD languages is reduced to rewriting blocks of an ST-program. The idea of the proof is demonstrated by an example of a Minsky 3-counter machine, which implements a function of squaring.

On the Expressiveness of the Approach to Constructing PLC-programs by LTL-Specification

The article is devoted to the approach to constructing and verification of discrete PLC-programs by LTL-specification. This approach provides an ability of correctness analysis of PLC-programs by the model checking method. The linear temporal logic LTL is used as a language of specification of the program behavior. The correctness analysis of LTL-specification is automatically performed by the symbolic model checking tool Cadence SMV. The article demonstrates the consistency of the approach to constructing and verification of PLC programs by LTL-specification from the point of view of Turing power. It is proved, that in accordance with this approach for any Minsky counter machine can be built an LTL-specification, which is used for machine implementation in any PLC programming language of standard IEC 61131-3. Minsky machines equipollent Turing machines, and the considered approach also has Turing power. The proof focuses on representation of a counter machine behavior in the form of a set of LTL-formulas and matching these formulas to constructions of ST and SFC languages. SFC is interesting as a specific graphical language. ST is considered as a basic language because an implementation of a counter machine in IL, FBD/CFC and LD languages is reduced to rewriting blocks of ST-program. The idea of the proof is demonstrated by an example of a Minsky 3-counter machine, which implements a function of squaring.

Non-transitive linear temporal logic and logical knowledge operations

© 2015 The Author, 2015. Published by Oxford University Press. All rights reserved.We study a linear temporal logic LTLNT with non-transitive time (with NEXT and UNTIL) and possible interpretations for logical knowledge operations in this approach. We assume time to be non-transitive, linear and discrete, it is a major innovative part of our article. Motivation for our approach that time might be non-transitive and comments on possible interpretations of logical knowledge operations are given. The main result of Section 5 is a solution of the decidability problem for LTLNT, we find and describe in details the decision algorithm. In Section 6 we introduce non-transitive linear temporal logic LTLNT(m) with uniform bound (m) for non-transitivity. We compare it with standard linear temporal logic LTL and the logic LTLNT - where non-transitivity has no upper bound - and show that LTLNT may be approximated by logics LTLNT(m). Concluding part of the article contains a list of open interesting problems.

Linear temporal logic vehicle routing with applications to multi‐UAV mission planning

AbstractMissions with high combinatorial complexity involving several logical and temporal constraints often arise in cooperative control of multiple Uninhabited Aerial Vehicles. In this paper, we propose a new class of problems that generalizes the standard Vehicle Routing Problem (VRP) by addressing complex tasks and constraints on the mission, called the ‘mission specifications’, expressed in a high‐level specification language. In the generalized problem setup, these mission specifications are naturally specified using the Linear Temporal Logic language LTL−X. Using a novel systematic procedure, the LTL−Xspecification is converted to a set of constraints suitable to a Mixed‐Integer Linear Programming (MILP) formulation, which in turn can be incorporated into two widely‐used MILP formulations of the standard VRP. Solving the resulting MILP provides an optimal plan that satisfies the given mission specification. The paper also presents two mission planning applications. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Linear Temporal Logic LTL: Basis for Admissible Rules

The object of our study is the propositional linear temporal logic LTL with operations Until and Next.We deal with inference rules admissible in LTL. Earlier the decidability of LTLw.r.t. to admissibility was shown. That, in particular, allowed to suggest a recursive infinite basis for admissible rules. However, that result gives no information about the structure and properties of such basis, thereby not allowing to use it for inferences. The current article aims to cover this gap and to provide an explicit (infinite) basis for rules admissible in LTL.

Tableau Tool for Testing Satisfiability in LTL: Implementation and Experimental Analysis

We report on the implementation and experimental analysis of an incremental multi-pass tableau-based procedure à la Wolper for testing satisfiability in the linear time temporal logic LTL, based on a breadth-first search strategy. We describe the implementation and discuss the performance of the tool on several series of pattern formulae, as well as on some random test sets, and compare its performance with an implementation of Schwendimann's one-pass tableaux by Widmann and Goré on several representative series of pattern formulae, including eventualities and safety patterns. Our experiments have established that Schwendimann's algorithm consistently, and sometimes dramatically, outperforms the incremental tableaux, despite the fact that the theoretical worst-case upper-bound of Schwendimann's algorithm, 2EXPTIME, is worse than that of Wolper's algorithm, which is EXPTIME. This shows, once again, that theoretically established worst-case complexity results do not always reflect truly the practical efficiency, at least when comparing decision procedures.

A History of Until

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: AUB holds at the current time instant w iff either B holds at w or there exists a time instant w′ in the future at which B holds and such that A holds in all the time instants between the current one and w′. This “ambivalent” nature poses a significant challenge when attempting to give deduction rules for until. In this paper, in contrast, we make explicit this duality of until by introducing a new temporal operator ∇ that allows us to formalize the “history” of until, i.e., the “internal” universal quantification over the time instants between the current one and w′. This approach provides the basis for formalizing deduction systems for temporal logics endowed with the until operator. For concreteness, we give here a labeled natural deduction system for a linear-time logic endowed with the new history operator and show that, via a proper translation, such a system is also sound and complete with respect to the linear temporal logic LTL with until.

Back from the future

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: A∪B holds at the current time instant w iff either B holds at w or there exists a time instant w' in the future at which B holds and such that A holds in all the time instants between the current one and ẃ. This “ambivalent” nature poses a significant challenge when attempting to give deduction rules for until. In this paper, in contrast, we make explicit this duality of until by introducing a new temporal operator ∇ that allows us to formalize the “history” of until, i.e., the “internal” universal quantification over the time instants between the current one and ẃ. This approach provides the basis for formalizing deduction systems for temporal logics endowed with the until operator. For concreteness, we give here a labeled natural deduction system N(LTL∇) for a linear-time logic LTL∇ endowed with the new history operator. We show that LTL∇ is equivalent to the linear temporal logic LTL with until, which follows by formalizing back and forth translations between the two logics. We also define an indirect translation from LTL∇ into LTL via temporal logics with past operators; such a result provides an upper bound to the problem of satisfiability for LTL∇ formulas.

Algorithm for Decision Procedure in Temporal Logic Treating Uncertainty, Plausibility, Knowledge and Interacting Agents

Our article studies logic UIALTL, which is a combination of the linear temporal logic LTL, a multi-agent logic with operation for passing knowledge via agents’ interaction, and a suggested logic based on operation of logical uncertainty. The logical operations of UIALTL also include (together with operations from LTL) operations of strong and weak until, agents’ knowledge operations, operation of knowledge via interaction, operation of logical uncertainty, the operations for environmental and global knowledge. UIALTL is defined as a set of all formulas valid at all Kripke-Hintikka like models NC. Any frame NC represents possible unbounded (in time) computation with multi-processors (parallel computational units) and agents’ channels for connections between computational units. The main aim of our article is to determine possible ways for computation logical laws of UIALTL. Principal problems we are dealing with are decidability and the satisfiability problems for UIALTL. We find an algorithm which recognizes theorems of UIALTL (so we show that UIALTL is decidable) and solves satisfiability problem for UIALTL. As an instrument we use reduction of formulas to rules in the reduced normal form and a technique to contract models NC to special non-UIALTL-models, and, then, verification of validity these rules in models of bounded size. The article uses standard results from non-classical logics based on Kripke-Hintikka models.

非標準モデルに基づくハイブリッドシステムの記述と推論について

In this paper, we introduce a nonstandard analysis into a logical modeling of continuous dynamics and present a new framework called hyper-finite hybrid automaton (HHA). HHA is a nonstandard interpretation of hybrid automata in the domain of *R. We also enlarge the linear temporal logic LTL to *LTL to describe the system specification. In this framework, we examine the validation of the system consistency of the hybrid system based on model checking, especially the existence and reachability of Zeno point.