Improving HyLTL model checking of hybrid systems

The problem of model-checking hybrid systems is a long-time challenge in the scientific community. Most of the existing approaches and tools are either limited on the properties that they can verify, or restricted to simplified classes of systems. To overcome those limitations, a temporal logic called HyLTL has been recently proposed. The model checking problem for this logic has been solved by translating the formula into an equivalent hybrid automaton, that can be analized using existing tools. The original construction employs a declarative procedure that generates exponentially many states upfront, and can be very inefficient when complex formulas are involved. In this paper we solve a technical issue in the construction that was not considered in previous works, and propose a new algorithm to translate HyLTL into hybrid automata, that exploits optimized techniques coming from the discrete LTL community to build smaller automata.Comment: In Proceedings GandALF 2013, arXiv:1307.416

The Logic of Time: from Aristotle to Computer Science

Charla tipo conferencia-seminario dada para alumnos de un másterThis short course will explore that continuous thread which connects the discussion about time in philosophy with the modern use of temporal logic in computer science. It will go through the history of temporal logic to show how ideas developed by ancient and medieval philosophy have been rediscovered in modern times and applied to solve relevant problems in computer science. Part 1: An historical perspective on temporal logic • Synthesis: the nature of time is a central issue of classical and medieval phylosophy • Downfall: in the Renaissance the subject loses interest and is removed from the philo- sophical discussion • Rediscovery: in the 19th and 20th centory temporal logic become a central issue again Part 2: Time in Computer Science • Algorithms, states and computations • Imperative programs and Reactive programs • Temporal Logic for Computer Science: CTL and LTL • The satisfiability problem • The model checking problemUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

On Sub-Propositional Fragments of Modal Logic

In this paper, we consider the well-known modal logics $\mathbf{K}$, $\mathbf{T}$, $\mathbf{K4}$, and $\mathbf{S4}$, and we study some of their sub-propositional fragments, namely the classical Horn fragment, the Krom fragment, the so-called core fragment, defined as the intersection of the Horn and the Krom fragments, plus their sub-fragments obtained by limiting the use of boxes and diamonds in clauses. We focus, first, on the relative expressive power of such languages: we introduce a suitable measure of expressive power, and we obtain a complex hierarchy that encompasses all fragments of the considered logics. Then, after observing the low expressive power, in particular, of the Horn fragments without diamonds, we study the computational complexity of their satisfiability problem, proving that, in general, it becomes polynomial

Deterministic Timed Finite State Machines: Equivalence Checking and Expressive Power

There has been a growing interest in defining models of automata enriched with time. For instance, timed automata were introduced as automata extended with clocks. In this paper, we study models of timed finite state machines (TFSMs), i.e., FSMs enriched with time, which accept timed input words and generate timed output words. Here we discuss some models of TFSMs with a single clock: TFSMs with timed guards, TFSMs with timeouts, and TFSMs with both timed guards and timeouts. We solve the problem of equivalence checking for all three models, and we compare their expressive power, characterizing subclasses of TFSMs with timed guards and of TFSMs with timeouts that are equivalent to each other.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Fast(er) Reasoning in Interval Temporal Logic

Clausal forms of logics are of great relevance in Artificial Intelligence, because they couple a high expressivity with a low complexity of reasoning problems. They have been studied for a wide range of classical, modal and temporal logics to obtain tractable fragments of intractable formalisms. In this paper we show that such restrictions can be exploited to lower the complexity of interval temporal logics as well. In particular, we show that for the Horn fragment of the interval logic AA (that is, the logic with the modal operators for Allen’s relations meets and met by) without diamonds the complexity lowers from NExpTime-complete to P-complete. We prove also that the tractability of the Horn fragments of interval temporal logics is lost as soon as other interval temporal operators are added to AA, in most of the cases

An Optimal Decision Procedure for MPNL over the Integers

Interval temporal logics provide a natural framework for qualitative and quantitative temporal reason- ing over interval structures, where the truth of formulae is defined over intervals rather than points. In this paper, we study the complexity of the satisfiability problem for Metric Propositional Neigh- borhood Logic (MPNL). MPNL features two modalities to access intervals "to the left" and "to the right" of the current one, respectively, plus an infinite set of length constraints. MPNL, interpreted over the naturals, has been recently shown to be decidable by a doubly exponential procedure. We improve such a result by proving that MPNL is actually EXPSPACE-complete (even when length constraints are encoded in binary), when interpreted over finite structures, the naturals, and the in- tegers, by developing an EXPSPACE decision procedure for MPNL over the integers, which can be easily tailored to finite linear orders and the naturals (EXPSPACE-hardness was already known).Comment: In Proceedings GandALF 2011, arXiv:1106.081

Extracting Interval Temporal Logic Rules: A First Approach

Discovering association rules is a classical data mining task with a wide range of applications that include the medical, the financial, and the planning domains, among others. Modern rule extraction algorithms focus on static rules, typically expressed in the language of Horn propositional logic, as opposed to temporal ones, which have received less attention in the literature. Since in many application domains temporal information is stored in form of intervals, extracting interval-based temporal rules seems the natural choice. In this paper we extend the well-known algorithm APRIORI for rule extraction to discover interval temporal rules written in the Horn fragment of Halpern and Shoham\u27s interval temporal logic

Begin, After, and Later: a Maximal Decidable Interval Temporal Logic

Interval temporal logics (ITLs) are logics for reasoning about temporal statements expressed over intervals, i.e., periods of time. The most famous ITL studied so far is Halpern and Shoham's HS, which is the logic of the thirteen Allen's interval relations. Unfortunately, HS and most of its fragments have an undecidable satisfiability problem. This discouraged the research in this area until recently, when a number non-trivial decidable ITLs have been discovered. This paper is a contribution towards the complete classification of all different fragments of HS. We consider different combinations of the interval relations Begins, After, Later and their inverses Abar, Bbar, and Lbar. We know from previous works that the combination ABBbarAbar is decidable only when finite domains are considered (and undecidable elsewhere), and that ABBbar is decidable over the natural numbers. We extend these results by showing that decidability of ABBar can be further extended to capture the language ABBbarLbar, which lays in between ABBar and ABBbarAbar, and that turns out to be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite orders, the naturals, the integers). We also prove that the proposed decision procedure is optimal with respect to the complexity class

Semantics and computation of the evolution of hybrid systems with ariadne

In this talk we will present material on the semantics, computability, and algorithms for the evolution of hybrid dynamical systems, and an overview of the tool Ariadne for verification of hybrid systems [1]. Hybrid systems are characterised by undergoing continuous evolution interspersed by discrete jumps. They exhibit all the complexities of finite automata, nonlinear dynamic systems and differential equations, and are extremely difficult to analyze. We will consider hybrid systems in which the continuous dynamics is given by a differential equation x = f(x), with discrete jumps x' = ri(x) which occur as soon as a guard condition gi(x) = 0 is activated. It is clear that the evolution of a hybrid system undergoes discontinuities, but since only continuous functions are computable, it is not clear to what extent, if any, it is possible to perform a rigorous analysis of a hybrid system. We will first show that we can define lower and upper semantics of evolution under which it is possible to compute reachable sets, and that away from discontinuity points (such as grazing or corner collision points), these semantics agree [2]. In order to perform reachability analysis, it is necessary to define the evolution over bounded initial sets of states. We show that this can be done using the operations of range, compose, flow and solve operations on functions. We will see that constrained image sets of the form {f(x) | x ? D | g(x) ? C}, are sufficient to express the evolution exactly, except for the case of degenerate (non-transverse) cross- ings [3]. The flow operation is the most computationally demanding, and we will give some details of the implementation and efficiency considerations [4]. We will give examples of reachability analysis in Ariadne, including electrical power converters and heating systems. Finally, we will outline some areas of active research, including differential inclusions [5] and modular reasoning

Horn fragments of the Halpern-Shoham Interval Temporal Logic

We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics