This paper shows that the satisfiability problem for probabilistic CTL (PCTL, for short) is undecidable. By a reduction from $1\frac{1}{2}$-player games with PCTL winning objectives, we establish that the PCTL satisfiability problem is $Σ_1^1$-hard. We present an exponential-time algorithm for the satisfiability of a bounded, negation-closed fragment of PCTL, and show that the satisfiability problem for this fragment is EXPTIME-hard.
We give a sufficient condition under which every finite-satisfiable formula of a given PCTL fragment has a model with at most doubly exponential number of states (consequently, the finite satisfiability problem for the fragment is in 2-EXPSPACE). The condition is semantic and it is based on enforcing a form of ``progress'' in non-bottom SCCs contributing to the satisfaction of a given PCTL formula. We show that the condition is satisfied by PCTL fragments beyond the reach of existing methods.
Probabilistic systems are an important theme in AI domain. As the specification language, the logic PCTL is now the default logic for reasoning about probabilistic properties. In this paper, we present a natural and succinct probabilistic extension of mu-calculus, another prominent logic in the concurrency theory. We study the relationship with PCTL. Interestingly, the expressiveness is highly orthogonal with PCTL. The proposed logic captures some useful properties which cannot be expressed in PCTL. We investigate the model checking and satisfiability problem, and show that the model checking …
While model checking PCTL for Markov chains is decidable in polynomial-time, the decidability of PCTL satisfiability, as well as its finite model property, are long standing open problems. While general satisfiability is an intriguing challenge from a purely theoretical point of view, we argue that general solutions would not be of interest to practitioners: such solutions could be too big to be implementable or even infinite. Inspired by bounded synthesis techniques, we turn to the more applied problem of seeking models of a bounded size: we restrict our search to implementable -- and therefo…
We investigate the satisfiability and finite satisfiability problem for probabilistic computation-tree logic (PCTL) where operators are not restricted by any step bounds. We establish decidability for several fragments containing quantitative operators and pinpoint the difficulties arising in more complex fragments where the decidability remains open.
We prove that the model checking and the satisfiability problem of both Dynamic Logic of Propositional Assignments DL-PA and Coalition Logic of Propositional Control and Delegation DCL-PC are in PSPACE. We explain why the proof of EXPTIME-hardness of the model checking problem of DL-PA presented in (Balbiani, Herzig, Troquard, 2013) is false. We also explain why the proof of membership in PSPACE of the model checking problem of DCL-PC given in (van der Hoek, Walther, Wooldridge, 2010) is wrong.
In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function) …
We provide a lower complexity bound for the satisfiability problem of a multi-agent justification logic, establishing that the general NEXP upper bound from our previous work is tight. We then use a simple modification of the corresponding reduction to prove that satisfiability for all multi-agent justification logics from there is hard for the Sigma 2 p class of the second level of the polynomial hierarchy - given certain reasonable conditions. Our methods improve on these required conditions for the same lower bound for the single-agent justification logics, proven by Buss and Kuznets in 200…
It is known that the satisfiability problems of the product logics K4xS5 and S4xS5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. We improve these lower bounds for the complexity of these problems by showing that all three problems are EXPSPACE-hard under logspace reduction. In another paper we show that these problems are in ESPACE. This shows that all three problems are EXPSPACE-complete.
Recently, the separated fragment (SF) has been introduced and proved to be decidable. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. The known upper bound on the time required to decide SF's satisfiability problem is formulated in terms of quantifier alternations: Given an SF sentence $\exists \vec{z} \forall \vec{x}_1 \exists \vec{y}_1 \ldots \forall \vec{x}_n \exists \vec{y}_n . ψ$ in which $ψ$ is quantifier free, satisfiability can be decided in nondeterministic $n$-fold exponential time. In the present paper, we conduct a …
In this paper, we present a Bayesian method for statistical model checking (SMC) of probabilistic hyperproperties specified in the logic HyperPCTL* on discrete-time Markov chains (DTMCs). While SMC of HyperPCTL* using sequential probability ratio test (SPRT) has been explored before, we develop an alternative SMC algorithm based on Bayesian hypothesis testing. In comparison to PCTL*, verifying HyperPCTL* formulae is complex owing to their simultaneous interpretation on multiple paths of the DTMC. In addition, extending the bottom-up model-checking algorithm of the non-probabilistic setting is …
The choice of the right trade-off between expressiveness and complexity is the main issue in interval temporal logic. In their seminal paper, Halpern and Shoham showed that the satisfiability problem for HS (the temporal logic of Allen's relations) is highly undecidable over any reasonable class of linear orders. In order to recover decidability, one can restrict the set of temporal modalities and/or the class of models. In the following, we focus on the satisfiability problem for HS fragments under the homogeneity assumption, according to which any proposition letter holds over an interval if…
United Technologies Research Center（联合技术研究中心）；SRI International（SRI国际）
Given a Boolean formula $ϕ(x)$ in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly $e$ clauses, for all values of $e$. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the \emph{hardness} of the problem instance. Consequently, in real-world scenarios, this problem is typically infeas…
Answer Set Programming (ASP) is a problem modeling and solving framework for several problems in KR with growing industrial applications. Also for studies of computational complexity and deeper insights into the hardness and its sources, ASP has been attracting researchers for many years. These studies resulted in fruitful characterizations in terms of complexity classes, fine-grained insights in form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Recently, this lead to a novel result establishing that for the measure treewidth, which captures structural d…
We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of Short-PA sentences with $m+2$ alternating quantifiers is $Σ_{P}^m$-complete or $Π_{P}^m$-complete, when the first quantifier is $\exists$ or $\forall$, respectively. Counting versions and restricted systems are also analyzed. Further application are given to hardne…
