We introduce a temporal logic to reason on global applications in an asynchronous setting. First, we define the Distributed States Logic (DSL), a modal logic for localities that embeds the local theories of each component into a theory of the distributed states of the system. We provide the logic with a sound and complete axiomatization. The contribution is that it is possible to reason about properties that involve several components, even in the absence of a global clock. Then, we define the Distributed States Temporal Logic (DSTL) by introducing temporal operators a' la Unity. We support our proposal by working out a pair of examples: a simple secure communication system, and an algorithm for distributed leader election.
The motivation for this work is that the existing logics for distributed systems do not have the right expressive power to reason on the systems behaviour, when the communication is based on asynchronous message passing. On the other side, asynchronous communication is the most used abstraction when modelling global applications.
This article surveys some of the recent work in verification of temporal epistemic logic via symbolic model checking, focusing on OBDD-based and SAT-based approaches for epistemic logics built on discrete and real-time branching time temporal logics.
This article examines the interpretation of the LTL temporal operators over finite and infinite sequences. This is used as the basis for deriving a sound and complete axiomatization for Caret, a recent temporal logic for reasoning about programs with nested procedure calls and returns.
Decision trees are simple, yet powerful, classification models used to classify categorical and numerical data, and, despite their simplicity, they are commonly used in operations research and management, as well as in knowledge mining. From a logical point of view, a decision tree can be seen as a structured set of logical rules written in propositional logic. Since knowledge mining is rapidly evolving towards temporal knowledge mining, and since in many cases temporal information is best described by interval temporal logics, propositional logic decision trees may evolve towards interval tem…
We use the algebraic framework for languages of infinite trees introduced in [4] to derive effective characterisations of various temporal logics, in particular the logic EF (a fragment of CTL) and its counting variant cEF.
The language of linear temporal logic can be interpreted over the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${{\sf ITL}^{\sf c}}_{\Diamond,\forall}$, recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.
This paper presents a distributed agent-based automated theorem proving framework based on order-sorted first-order logic. Each agent in our framework has its own knowledge base, communicating to its neighboring agent(s) using message-passing algorithms. The communication language between agents is restricted in such a manner that each agent can only communicate to its neighboring agent(s) by means of their common language. In this paper we provide a refutation-complete report procedure for automated theorem proving in order-sorted first-order logic in a subclass of distributed agent-based net…
Temporal logics are an obvious high-level descriptive companion formalism to dynamical systems which model behavior as deterministic evolution of state over time. A wide variety of distinct temporal logics applicable to dynamical systems exists, and each candidate has its own pragmatic justification. Here, a systematic approach to the construction of temporal logics for dynamical systems is proposed: Firstly, it is noted that dynamical systems can be seen as coalgebras in various ways. Secondly, a straightforward standard construction of modal logics out of coalgebras, namely Moss's coalgebrai…
This paper surveys main and recent studies on temporal logics in a broad sense by presenting various logic systems, dealing with various time structures, and discussing important features, such as decidability (or undecidability) results, expressiveness and proof systems.
We consider a logic used to describe sets of configurations of distributed systems, whose network topologies can be changed at runtime, by reconfiguration programs. The logic uses inductive definitions to describe networks with an unbounded number of components and interactions, written using a multiplicative conjunction, reminiscent of Bunched Implications and Separation Logic. We study the complexity of the satisfiability and entailment problems for the configuration logic under consideration. Additionally, we consider robustness properties, such as tightness (are all interactions entirely c…
Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, mode…
We consider Gödel temporal logic ($\sf GTL$), a variant of linear temporal logic based on Gödel--Dummett propositional logic. In recent work, we have shown this logic to enjoy natural semantics both as a fuzzy logic and as a superintuitionistic logic. Using semantical methods, the logic was shown to be {\sc pspace}-complete. In this paper we provide a deductive calculus for $\sf GTL$, and show this calculus to be sound and complete for the above-mentioned semantics.
In this paper, we propose Probabilistic discrete-time Projection Temporal Logic (PrPTL), which extends Projection Temporal Logic (PTL) with probability. To this end, some useful formulas are derived and some logic laws are given. Further, we define Time Normal Form (TNF) for PrPTL as the standard form and prove that any PrPTL formulas can be rewritten to TNF. According to the TNF, we construct the time normal form graph which can be used for the probabilistic model checking on PrPTL.
Temporal logics over finite traces are not the same as temporal logics over potentially infinite traces. Roşu first proved completeness for linear temporal logic on finite traces (LTLf) with a novel coinductive axiom. We offer a different proof, with fewer, more conventional axioms. Our proof is a direct adaptation of Kröger and Merz's Henkin-Hasenjaeger-style proof. The essence of our adaption is that we "inject" finiteness: that is, we alter the proof structure to ensure that models are finite. We aim to present a thorough, accessible proof.
The paper is focused on temporal logics for the description of the behaviour of real-time pushdown reactive systems. The paper is motivated to bridge tractable logics specialized for expressing separately dense-time real-time properties and context-free properties by ensuring decidability and tractability in the combined setting. To this end we introduce two real-time linear temporal logics for specifying quantitative timing context-free requirements in a pointwise semantics setting: Event-Clock Nested Temporal Logic (ECNTL) and Nested Metric Temporal Logic (NMTL). The logic ECNTL is an extens…
We review the notion of perfect recall in the literature on interpreted systems, game theory, and epistemic logic. In the context of Epistemic Temporal Logic (ETL), we give a (to our knowledge) novel frame condition for perfect recall, which is local and can straightforwardly be translated to a defining formula in a language that only has next-step temporal operators. This frame condition also gives rise to a complete axiomatization for S5 ETL frames with perfect recall. We then consider how to extend and consolidate the notion of perfect recall in sub-S5 settings, where the various notions di…
We consider modeling the conception of knowledge in terms of temporal logic. The study of knowledge logical operations is originated around 1962 by representation of knowledge and belief using modalities. Nowadays, it is very good established area. However, we would like to look to it from a bit another point of view, our paper models knowledge in terms of linear temporal logic with {\em past}. We consider various versions of logical knowledge operations which may be defined in this framework. Technically, semantics, language and temporal knowledge logics based on our approach are constructed.…
Whereas standard treatments of temporal logic are adequate for closed systems, having no run-time interactions with their environment, they fall short for reactive systems, interacting with their environments through synchronisation of actions. This paper introduces reactive temporal logic, a form of temporal logic adapted for the study of reactive systems. I illustrate its use by applying it to formulate definitions of a fair scheduler, and of a correct mutual exclusion protocol. Previous definitions of these concepts were conceptually much more involved or less precise, leading to debates on…
Extending and generalizing the approach of 2-sequents (Masini, 1992), we present sequent calculi for the classical modal logics in the K, D, T, S4 spectrum. The systems are presented in a uniform way-different logics are obtained by tuning a single parameter, namely a constraint on the applicability of a rule. Cut-elimination is proved only once, since the proof goes through independently from the constraints giving rise to the different systems. A sequent calculus for the discrete linear temporal logic ltl is also given and proved complete. Leitmotiv of the paper is the formal analogy between…
The paper is focused on temporal logics for the description of the behaviour of real-time pushdown reactive systems. The paper is motivated to bridge tractable logics specialized for expressing separately dense-time real-time properties and context-free properties by ensuring decidability and tractability in the combined setting. To this end we introduce two real-time linear temporal logics for specifying quantitative timing context-free requirements in a pointwise semantics setting: Event-Clock Nested Temporal Logic (EC_NTL) and Nested Metric Temporal Logic (NMTL). The logic EC_NTL is an exte…
We investigate a non-classical version of linear temporal logic whose propositional fragment is Gödel--Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics, a real-valued semantics and a bi-relational semantics, and show that these indeed define one and the same logic. Although this Gödel temporal logic does not have any form of the finite model property for these two semantics, we show that every falsifiable formula is falsifiable on a finite quasimodel, which yields decidability of the logic. We then…
Justification logics are modal-like logics with the additional capability of recording the reason, or justification, for modalities in syntactic structures, called justification terms. Justification logics can be seen as explicit counterparts to modal logics. The behavior and interaction of agents in distributed system is often modeled using logics of knowledge and time. In this paper, we sketch some preliminary ideas on how the modal knowledge part of such logics of knowledge and time could be replaced with an appropriate justification logic.
In this paper we consider the specification and verification of infinite-state systems using temporal logic. In particular, we describe parameterised systems using a new variety of first-order temporal logic that is both powerful enough for this form of specification and tractable enough for practical deductive verification. Importantly, the power of the temporal language allows us to describe (and verify) asynchronous systems, communication delays and more complex properties such as liveness and fairness properties. These aspects appear difficult for many other approaches to infinite-state ve…
Our manuscript studies linear temporal (with UNTIL and NEXT) logic based at a conception of intransitive time. non-transitive time. In particular, we demonstrate how the notion of knowledge might be represented in such a framework (here we consider logical operation NN and the operation UNTIL (actually, the time overall) to be directed to past). The basic mathematical problems we study are the fundamental ones for any logical system - decidability and decidability w.r.t. admissible rules. First, we consider the logic with non-uniform non-transitivity, and describe how to solve the decidability…
There are numerous applications where we have to deal with temporal uncertainty associated with events. The Temporal Probabilistic (TP) Logic Programs should provide support for valid-time indeterminacy of events, by proposing the concept of an indeterminate instant, that is, an interval of time-points (event's time-window) with an associated, lower and upper, probability distribution. In particular, we propose the new semantics, for the TP Logic Programs of Dekhtyar and Subrahmanian. Our semantics, based on the possible world semantics is a generalization of the possible world semantics for (…
This paper describes information flow within logical environments. The theory of information flow, the logic of distributed systems, was first defined by Barwise and Seligman (Information Flow: The Logic of Distributed Systems. 1997). Logical environments are a semantic-oriented version of institutions. The theory of institutions, which was initiated by Goguen and Burstall (Institutions: Abstract Model Theory for Specification and Programming. 1992), is abstract model theory. Information flow is the flow of information in channels over distributed systems. The semantic integration of distribut…
Metric Temporal Logic (MTL) is a generalisation of Linear Temporal Logic in which the Until and Since modalities are annotated with intervals that express metric constraints. A seminal result of Hirshfeld and Rabinovich shows that over the reals, first-order logic with binary order relation < and unary function +1 is strictly more expressive than MTL with integer constants. Indeed they prove that no temporal logic whose modalities are definable by formulas of bounded quantifier depth can be expressively complete for FO(<,+1). In this paper we show the surprising result that if we allow unary f…
In this paper, we propose a new logic for expressing and reasoning about probabilistic hyperproperties. Hyperproperties characterize the relation between different independent executions of a system. Probabilistic hyperproperties express quantitative dependencies between such executions. The standard temporal logics for probabilistic systems, i.e., PCTL and PCTL* can refer only to a single path at a time and, hence, cannot express many probabilistic hyperproperties of interest. The logic proposed in this paper, \HyperPCTL, adds explicit and simultaneous quantification over multiple traces to P…
Suitable extensions of the monadic second-order theory of k successors have been proposed in the literature to capture the notion of time granularity. In this paper, we provide the monadic second-order theories of downward unbounded layered structures, which are infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, and of upward unbounded layered structures, which consist of a finest domain and an infinite number of coarser and coarser domains, with expressively complete and elementarily decidable temporal logic counterparts. We obta…
Two new logics for verification of hyperproperties are proposed. Hyperproperties characterize security policies, such as noninterference, as a property of sets of computation paths. Standard temporal logics such as LTL, CTL, and CTL* can refer only to a single path at a time, hence cannot express many hyperproperties of interest. The logics proposed here, HyperLTL and HyperCTL*, add explicit and simultaneous quantification over multiple paths to LTL and to CTL*. This kind of quantification enables expression of hyperproperties. A model checking algorithm for the proposed logics is given. For a…
In this paper, we focus our attention on the interval temporal logic of the Allen's relations "meets", "begins", and "begun by" (ABBar for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties,such as accomplishment conditions, to capture basic modalities of point-based temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for ABBar over natural numbers is decidable by providing a small model theorem based on an origi…
Temporal Equilibrium Logic (TEL) is a promising framework that extends the knowledge representation and reasoning capabilities of Answer Set Programming with temporal operators in the style of LTL. To our knowledge it is the first nonmonotonic logic that accommodates fully the syntax of a standard temporal logic (specifically LTL) without requiring further constructions. This paper provides a systematic complexity analysis for the (consistency) problem of checking the existence of a temporal equilibrium model of a TEL formula. It was previously shown that this problem in the general case lies …
We consider the temporal logic with since and until modalities. This temporal logic is expressively equivalent over the class of ordinals to first-order logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability problem over the class of ordinals. Among the consequences of our proof, we show that given the code of some countable ordinal alpha and a formula, we can decide in PSPACE whether the formula has a model over alpha. In order to show these results, we introduce a class of simple ordinal automata, as expressive as Büchi ordinal automata. The PSPACE upper bound for the …
We consider a temporal logic EF+F^-1 for unranked, unordered finite trees. The logic has two operators: EFϕ, which says "in some proper descendant ϕholds", and F^-1ϕ, which says "in some proper ancestor ϕholds". We present an algorithm for deciding if a regular language of unranked finite trees can be expressed in EF+F^-1. The algorithm uses a characterization expressed in terms of forest algebras.
The Halpern-Shoham logic is a modal logic of time intervals. Some effort has been put in last ten years to classify fragments of this beautiful logic with respect to decidability of its satisfiability problem. We contribute to this effort by showing - what we believe is quite an unexpected result - that the logic of subintervals, the fragment of the Halpern-Shoham where only the operator "during", or D, is allowed, is undecidable over discrete structures. This is surprising as this logic is decidable over dense orders and its reflexive variant is known to be decidable over discrete structures.
The paper proposes and studies temporal logics for attributed words, that is, data words with a (finite) set of (attribute,value)-pairs at each position. It considers a basic logic which is a semantical fragment of the logic $LTL^\downarrow_1$ of Demri and Lazic with operators for navigation into the future and the past. By reduction to the emptiness problem for data automata it is shown that this basic logic is decidable. Whereas the basic logic only allows navigation to positions where a fixed data value occurs, extensions are studied that also allow navigation to positions with different da…
A specification given as a formula in linear temporal logic (LTL) defines a system by its set of traces. However, certain features such as information flow security constraints are rather modeled as so-called hyperproperties, which are sets of sets of traces. One logical approach to this is team logic, which is a logical framework for the specification of dependence and independence of information. LTL with team semantics has recently been discovered as a logic for hyperproperties. We study the complexity theoretic aspects of LTL with so-called synchronous team semantics and Boolean negation, …
Formalisms based on temporal logics interpreted over finite strict linear orders, known in the literature as finite traces, have been used for temporal specification in automated planning, process modelling, (runtime) verification and synthesis of programs, as well as in knowledge representation and reasoning. In this paper, we focus on first-order temporal logic on finite traces. We first investigate preservation of equivalences and satisfiability of formulas between finite and infinite traces, by providing a set of semantic and syntactic conditions to guarantee when the distinction between r…
The molecular computing has been successfully employed to solve more and more complex computation problems. However, as an important complex problem, the model checking are still far from fully resolved under the circumstance of molecular computing, since it is still a lack of method. To address this issue, a model checking method is presented for checking the basic constructs in a given temporal logic using molecular computing. Through the design of the new encoding and calling this process, we get a molecule-based approach for checking all of the basic constructs of this logic.
The paper studies problems of satisfiability, decidability and admissibility of inference rules, conceptions of knowledge and agent's knowledge in non-transitive temporal linear logic LTL(Past,m). We find algorithms solving mentioned problems, justify our approach to consider linear non-transitive time with several examples. Main, most complicated, technical new result is decidability of LTL(Past,m) w.r.t. admissible rules. We discuss several ways to formalize conceptions of knowledge and agent's knowledge within given approach in non-transitive linear logic with models directed to past.
Coalition logic is one of the most popular logics for multi-agent systems. While epistemic extensions of coalition logic have received much attention, existence of their complete axiomatisations has so far been an open problem. In this paper we settle several of those problems. We prove completeness for epistemic coalition logic with common knowledge, with distributed knowledge, and with both common and distributed knowledge, respectively.
Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL) are often used to describe specifications of probabilistic properties for discrete time and continuous time, respectively. In PCTL and CSL, the possibility of executions satisfying some temporal properties can be quantitatively represented by the probabilistic extension of the path quantifiers in their basic Computation Tree Logic (CTL), however, path formulae of them are expressed via the same operators in CTL. For this reason, both of them cannot represent formulae with quantitative temporal properties, such as…
In the literature, two powerful temporal logic formalisms have been proposed for expressing information flow security requirements, that in general, go beyond regular properties. One is classic, based on the knowledge modalities of epistemic logic. The other one, the so called hyper logic, is more recent and subsumes many proposals from the literature; it is based on explicit and simultaneous quantification over multiple paths. In an attempt to better understand how these logics compare with each other, we consider the logic KCTL* (the extension of CTL* with knowledge modalities and synchronou…
First-order temporal logics are notorious for their bad computational behaviour. It is known that even the two-variable monadic fragment is highly undecidable over various linear timelines, and over branching time even one-variable fragments might be undecidable. However, there have been several attempts on finding well-behaved fragments of first-order temporal logics and related temporal description logics, mostly either by restricting the available quantifier patterns, or considering sub-Boolean languages. Here we analyse seemingly `mild' extensions of decidable one-variable fragments with c…
Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of…
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 prov…
Path checking, the special case of the model checking problem where the model under consideration is a single path, plays an important role in monitoring, testing, and verification. We prove that for linear-time temporal logic (LTL), path checking can be efficiently parallelized. In addition to the core logic, we consider the extensions of LTL with bounded-future (BLTL) and past-time (LTL+Past) operators. Even though both extensions improve the succinctness of the logic exponentially, path checking remains efficiently parallelizable: Our algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(lo…
The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal …
We show that Branching-time temporal logics CTL and CTL*, as well as Alternating-time temporal logics ATL and ATL*, are as semantically expressive in the language with a single propositional variable as they are in the full language, i.e., with an unlimited supply of propositional variables. It follows that satisfiability for CTL, as well as for ATL, with a single variable is EXPTIME-complete, while satisfiability for CTL*, as well as for ATL*, with a single variable is 2EXPTIME-complete,--i.e., for these logics, the satisfiability for formulas with only one variable is as hard as satisfiabili…
We define a fragment of metric first-order temporal logic formulas that guarantees the finiteness of their table representations. We extend our fragment's definition to cover the temporal dual operators trigger and release and show that our fragment is strictly larger than those previously used in the literature. We integrate these additions into an existing runtime verification tool and formally verify in Isabelle/HOL that the tool correctly outputs the table of constants that satisfy the monitored formula. Finally, we provide some example specifications that are now monitorable thanks to our…
