Parametric LTL on Markov Chains

This paper is concerned with the verification of finite Markov chains against parametrized LTL (pLTL) formulas. In pLTL, the until-modality is equipped with a bound that contains variables; e.g., $\Diamond_{\le x}\ \varphi$ asserts that $\varphi$ holds within $x$ time steps, where $x$ is a variable on natural numbers. The central problem studied in this paper is to determine the set of parameter valuations $V_{\prec p} (\varphi)$ for which the probability to satisfy pLTL-formula $\varphi$ in a Markov chain meets a given threshold $\prec p$, where $\prec$ is a comparison on reals and $p$ a probability. As for pLTL determining the emptiness of $V_{> 0}(\varphi)$ is undecidable, we consider several logic fragments. We consider parametric reachability properties, a sub-logic of pLTL restricted to next and $\Diamond_{\le x}$, parametric B\"uchi properties and finally, a maximal subclass of pLTL for which emptiness of $V_{> 0}(\varphi)$ is decidable.Comment: TCS Track B 201

On the Expressive Power of Datalog: Tools and a Case Study

AbstractWe study here the language Datalog(≠), which is the query language obtained from Datalog by allowing equalities and inequalities in the bodies of the rules. We view Datalog(≠) as a fragment of an infinitary logic Lω and show that Lω can be characterized in terms of certain two-person pebble games. This characterization provides us with tools for investigating the expressive power of Datalog(≠). As a case study, we classify the expressibility of fixed subgraph homeomorphism queries on directed graphs. S. Fortune, J. Hopcroft, and J. Wyllie (Theoret. Comput. Sci.10 (1980), 111-121) classified the computational complexity of these queries by establishing two dichotomies, which are proper only if P ≠ NP. Without using any complexity-theoretic assumptions, we show here that the two dichotomies are indeed proper in terms of expressibility in Datalog(≠)

Verifying Temporal Heap Properties Specified via Evolution Logic

This paper addresses the problem of establishing temporal properties of programs written in languages, such as Java, that make extensive use of the heap to allocate--- and deallocate---new objects and threads. Establishing liveness properties is a particularly hard challenge. One of the crucial obstacles is that heap locations have no static names and the number of heap locations is unbounded. The paper presents a framework for the verification of Java-like programs. Unlike classical model checking, which uses propositional temporal logic, we use first-order temporal logic to specify temporal properties of heap evolutions; this logic allows domain changes to be expressed, which permits allocation and deallocation to be modelled naturally. The paper also presents an abstract-interpretation algorithm that automatically verifies temporal properties expressed using the logic

B\"uchi Complementation and Size-Change Termination

We compare tools for complementing nondeterministic B\"uchi automata with a recent termination-analysis algorithm. Complementation of B\"uchi automata is a key step in program verification. Early constructions using a Ramsey-based argument have been supplanted by rank-based constructions with exponentially better bounds. In 2001 Lee et al. presented the size-change termination (SCT) problem, along with both a reduction to B\"uchi automata and a Ramsey-based algorithm. The Ramsey-based algorithm was presented as a more practical alternative to the automata-theoretic approach, but strongly resembles the initial complementation constructions for B\"uchi automata. We prove that the SCT algorithm is a specialized realization of the Ramsey-based complementation construction. To do so, we extend the Ramsey-based complementation construction to provide a containment-testing algorithm. Surprisingly, empirical analysis suggests that despite the massive gap in worst-case complexity, Ramsey-based approaches are superior over the domain of SCT problems. Upon further analysis we discover an interesting property of the problem space that both explains this result and provides a chance to improve rank-based tools. With these improvements, we show that theoretical gains in efficiency of the rank-based approach are mirrored in empirical performance

Game Refinement Relations and Metrics

We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal, for example, reach a target state, the question of winning is thus a probabilistic one: what is the maximal probability of winning from a given state? On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state similarity. We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative mu-calculus. The quantitative mu-calculus can express a large set of goals, including reachability, safety, and omega-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation

Weak MSO+U with Path Quantifiers over Infinite Trees

This paper shows that over infinite trees, satisfiability is decidable for weak monadic second-order logic extended by the unbounding quantifier U and quantification over infinite paths. The proof is by reduction to emptiness for a certain automaton model, while emptiness for the automaton model is decided using profinite trees.Comment: version of an ICALP 2014 paper with appendice

Propositional Dynamic Logic with Converse and Repeat for Message-Passing Systems

The model checking problem for propositional dynamic logic (PDL) over message sequence charts (MSCs) and communicating finite state machines (CFMs) asks, given a channel bound $B$, a PDL formula $\varphi$ and a CFM $\mathcal{C}$, whether every existentially $B$-bounded MSC $M$ accepted by $\mathcal{C}$ satisfies $\varphi$. Recently, it was shown that this problem is PSPACE-complete. In the present work, we consider CRPDL over MSCs which is PDL equipped with the operators converse and repeat. The former enables one to walk back and forth within an MSC using a single path expression whereas the latter allows to express that a path expression can be repeated infinitely often. To solve the model checking problem for this logic, we define message sequence chart automata (MSCAs) which are multi-way alternating parity automata walking on MSCs. By exploiting a new concept called concatenation states, we are able to inductively construct, for every CRPDL formula $\varphi$, an MSCA precisely accepting the set of models of $\varphi$. As a result, we obtain that the model checking problem for CRPDL and CFMs is still in PSPACE

Qualitative Analysis of Partially-observable Markov Decision Processes

We study observation-based strategies for partially-observable Markov decision processes (POMDPs) with omega-regular objectives. An observation-based strategy relies on partial information about the history of a play, namely, on the past sequence of observations. We consider the qualitative analysis problem: given a POMDP with an omega-regular objective, whether there is an observation-based strategy to achieve the objective with probability~1 (almost-sure winning), or with positive probability (positive winning). Our main results are twofold. First, we present a complete picture of the computational complexity of the qualitative analysis of POMDP s with parity objectives (a canonical form to express omega-regular objectives) and its subclasses. Our contribution consists in establishing several upper and lower bounds that were not known in literature. Second, we present optimal bounds (matching upper and lower bounds) on the memory required by pure and randomized observation-based strategies for the qualitative analysis of POMDP s with parity objectives and its subclasses

Synthesis from Recursive-Components Libraries

Synthesis is the automatic construction of a system from its specification. In classical synthesis algorithms it is always assumed that the system is "constructed from scratch" rather than composed from reusable components. This, of course, rarely happens in real life. In real life, almost every non-trivial commercial software system relies heavily on using libraries of reusable components. Furthermore, other contexts, such as web-service orchestration, can be modeled as synthesis of a system from a library of components. In 2009 we introduced LTL synthesis from libraries of reusable components. Here, we extend the work and study synthesis from component libraries with "call and return"' control flow structure. Such control-flow structure is very common in software systems. We define the problem of Nested-Words Temporal Logic (NWTL) synthesis from recursive component libraries, where NWTL is a specification formalism, richer than LTL, that is suitable for "call and return" computations. We solve the problem, providing a synthesis algorithm, and show the problem is 2EXPTIME-complete, as standard synthesis.Comment: In Proceedings GandALF 2011, arXiv:1106.081