Why Propositional Quantification Makes Modal and Temporal Logics on Trees Robustly Hard? (original) (raw)
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Why Does Propositional Quantification Make Modal and Temporal Logics on Trees Robustly Hard?
Cornell University - arXiv, 2021
Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (QCTL t) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of QCTL t as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that QCTL t restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When QCTL t restricted to EX is interpreted on N-bounded trees for some N ≥ 2, we prove that the satisfiability problem is AExppol-complete; AExppolhardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of QCTL t restricted to EF or to EXEF and of the well-known modal logics such as K, KD, GL, K4 and S4 with propositional quantification under a semantics based on classes of trees.
Why Does Propositional Quantification Make Logics on Trees Robustly Hard?
Logical Methods in Computer Science
Adding propositional quantification to the modal logics K, T or S4 is known to lead to undecidability but CTL with propositional quantification under the tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability problem. We investigate the complexity of strict fragments of tQCTL as well as of the modal logic K with propositional quantification under the tree semantics. More specifically, we show that tQCTL restricted to the temporal operator EX is already Tower-hard, which is unexpected as EX can only enforce local properties. When tQCTL restricted to EX is interpreted on N-bounded trees for some N >= 2, we prove that the satisfiability problem is AExpPol-complete; AExpPol-hardness is established by reduction from a recently introduced tiling problem, instrumental for studying the model-checking problem for interval temporal logics. As consequences of our proof method, we prove Tower-hardness of tQCTL restricted to EF or to EXEF and of the well-known modal logi...
On Decidability and Expressiveness of Propositional Interval Neighborhood Logics
Lecture Notes in Computer Science, 2007
Interval-based temporal logics are an important research area in computer science and artificial intelligence. In this paper we investigate decidability and expressiveness issues for Propositional Neighborhood Logics (PNLs). We begin by comparing the expressiveness of the different PNLs. Then, we focus on the most expressive one, namely, PNL π+ , and we show that it is decidable over various classes of linear orders by reducing its satisfiability problem to that of the two-variable fragment of first-order logic with binary relations over linearly ordered domains, due to Otto. Next, we prove that PNL π+ is expressively complete with respect to such a fragment. We conclude the paper by comparing PNL π+ expressiveness with that of other interval-based temporal logics. about these logics are the undecidability of HS and CDT over most of classes of frames, of PITL over dense and discrete frames, and of BE over dense frames. A comprehensive survey of the main developments, results, and open problems in the area of propositional interval temporal logics can be found in .
Branching-Time Temporal Logics with Minimal Model Quantifiers
Lecture Notes in Computer Science, 2009
Temporal logics are a well investigated formalism for the specification and verification of reactive systems. Using formal verification techniques, we can ensure the correctness of a system with respect to its desired behavior (specification), by verifying whether a model of the system satisfies a temporal logic formula modeling the specification. From a practical point of view, a very challenging issue in using temporal logic in formal verification is to come out with techniques that automatically allow to select small critical parts of the system to be successively verified. Another challenging issue is to extend the expressiveness of classical temporal logics, in order to model more complex specifications. In this paper, we address both issues by extending the classical branching-time temporal logic CTL * with minimal model quantifiers (MCTL * ). These quantifiers allow to extract, from a model, minimal submodels on which we check the specification (also given by an MCTL * formula). We show that MCTL * is strictly more expressive than CTL * . Nevertheless, we prove that the model checking problem for MCTL * remains decidable and in particular in PSPACE. Moreover, differently from CTL * , we show that MCTL * does not have the tree model property, is not bisimulation-invariant and is sensible to unwinding. As far as the satisfiability concerns, we prove that MCTL * is highly undecidable. We further investigate the model checking and satisfiability problems for MCTL * sublogics, such as MPML, MCTL, and MCTL + , for which we obtain interesting results. Among the others, we show that MPML retains the finite model property and the decidability of the satisfiability problem. 6 6 J J J J J J i 6 6 J J J J J J
A Model Checker for Interval Temporal Logic over Finite Structures
2017
Model checking is the process of establishing whether a certain formula is satisfied by a given structure, and it is usually associated with point-based temporal logics. Recently, the question of how to correctly define and study the model checking problem for interval-based temporal logics has been raised. In this paper, we focus on a very natural finite version of the model checking problem for Halpern and Shoham’s modal logic of time intervals, a.k.a. HS, for which an algorithm that behaves in a very efficient way (under certain conditions) can be designed. We present an implementation of such an algorithm and analyse its performance through a systematic series of tests.
Information and Computation, 2018
Some temporal properties of reactive systems, such as actions with duration, accomplishments, and temporal aggregations, which are inherently interval-based, can not be properly dealt with by the standard, point-based temporal logics LTL, CTL and CTL*, as they give a state-by-state account of system evolution. Conversely, interval temporal logics-which feature intervals, instead of points, as their primitive entities-are highly expressive formalisms for temporal representation and reasoning that naturally allow one to deal with them. In this paper, we study the model checking (MC) problem for Halpern and Shoham's modal logic of time intervals (HS), interpreted on Kripke structures, under the homogeneity assumption, according to which a proposition letter holds over a finite computation path (interval) if and only if it holds at all of its states. HS is the best known interval-based temporal logic, which has one modality for each of the 13 possible ordering relations between pairs of intervals (the so-called Allen's relations), apart from equality. We focus on the MC problem for some HS fragments featuring modalities for (a subset of) Allen's relations meet, met-by, started-by, and finished-by, showing that it is in P NP , a class to which other pointbased logics (e.g., CTL+ and FCTL) are known to belong. Additionally, we provide some complexity lower bounds to the problem. All the algorithms we propose can be efficiently implemented by means of a polynomial-time procedure which iteratively invokes a SATsolver, enabling us to directly exploit the great speed of SAT-solvers.
Propositional interval neighborhood temporal logics
2003
Logics for time intervals provide a natural framework for dealing with time in various areas of computer science and artificial intelligence, such as planning, natural language processing, temporal databases, and formal specification. In this paper we focus our attention on propositional interval temporal logics with temporal modalities for neighboring intervals over linear orders. We study the class of propositional neighborhood logics (PN L) over two natural semantics, respectively admitting and excluding point-intervals. First, we introduce interval neighborhood frames and we provide representation theorems for them; then, we develop complete axiomatic systems and semantic tableaux for logics in PN L. HS features four basic operators: B (begin) and E (end ), and their transposes B and E . Given a formula ϕ and an interval [d 0 , d 1 ], B ϕ holds at [d 0 , d 1 ] if ϕ holds at [d 0 , d 2 ], for some d 2 < d 1 , and E ϕ holds at [d 0 , d 1 ] if ϕ holds at [d 2 , d 1 ], for some d 2 > d 0 .
Propositional interval neighborhood logics: Expressiveness, decidability, and undecidable extensions
In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics (PNL), we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern-Shoham's interval logic HS.
The computational complexity of hybrid temporal logics
Logic Journal of IGPL, 2000
In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specific times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics.
Undecidability of Interval Temporal Logics with the Overlap Modality
2009 16th International Symposium on Temporal Representation and Reasoning, 2009
We investigate fragments of Halpern-Shoham's interval logic HS involving the modal operators for the relations of left or right overlap of intervals. We prove that most of these fragments are undecidable, by employing a non-trivial reduction from the octant tiling problem.