Jan Otop

From model checking to model measuring

We define the model-measuring problem: given a model M and specification ϕ, what is the maximal distance ρ such that all models M′ within distance ρ from M satisfy (or violate) ϕ. The model measuring problem presupposes a distance function on models. We concentrate on automatic distance functions, which are defined by weighted automata. The model-measuring problem subsumes several generalizations of the classical model-checking problem, in particular, quantitative model-checking problems that measure the degree of satisfaction of a specification, and robustness problems that measure how much a model can be perturbed without violating the specification. We show that for automatic distance functions, and ω-regular linear-time and branching-time specifications, the model-measuring problem can be solved. We use automata-theoretic model-checking methods for model measuring, replacing the emptiness question for standard word and tree automata by the optimal-weight question for the weighted versions of these automata. We consider weighted automata that accumulate weights by maximizing, summing, discounting, and limit averaging. We give several examples of using the model-measuring problem to compute various notions of robustness and quantitative satisfaction for temporal specifications.

Distributed synthesis for LTL fragments

We consider the distributed synthesis problem for temporal logic specifications. Traditionally, the problem has been studied for LTL, and the previous results show that the problem is decidable iff there is no information fork in the architecture. We consider the problem for fragments of LTL and our main results are as follows: (1) We show that the problem is undecidable for architectures with information forks even for the fragment of LTL with temporal operators restricted to next and eventually. (2) For specifications restricted to globally along with non-nested next operators, we establish decidability (in EXPSPACE) for star architectures where the processes receive disjoint inputs, whereas we establish undecidability for architectures containing an information fork-meet structure. (3) Finally, we consider LTL without the next operator, and establish decidability (NEXPTIME-complete) for all architectures for a fragment that consists of a set of safety assumptions, and a set of guarantees where each guarantee is a safety, reachability, or liveness condition.

The Target Discounted-Sum Problem

The target discounted-sum problem is the following: Given a rational discount factor 0 <; λ <; 1 and three rational values a, b, and t, does there exist a finite or an infinite sequence w ∈ {a, b}* or w ∈ {a, b}ω, such that Σi=0|w| w(i)λi equals t? The problem turns out to relate to many fields of mathematics and computer science, and its decidability question is surprisingly hard to solve. We solve the finite version of the problem, and show the hardness of the infinite version, linking it to various areas and open problems in mathematics and computer science: β-expansions, discounted-sum automata, piecewise affine maps, and generalizations of the Cantor set. We provide some partial results to the infinite version, among which are solutions to its restriction to eventually-periodic sequences and to the cases that λ ≥ 1/2 or λ =1/n, for every n ∈ N. We use our results for solving some open problems on discounted-sum automata, among which are the exact-value problem for nondeterministic automata over finite words and the universality and inclusion problems for functional automata.

Decidable Elementary Modal Logics

In this paper, the modal logic over classes of structures definable by universal first-order Horn formulas is studied. We show that the satisfiability problems for that logics are decidable, confirming the conjecture from [E. Hemaspaandra and H. Schnoor, On the Complexity of Elementary Modal Logics, STACS 08]. We provide a full classification of logics defined by universal first-order Horn formulas, with respect to the complexity of satisfiability of modal logic over the classes of frames they define. It appears, that except for the trivial case of inconsistent formulas for which the problem is in P, local satisfiability is either NP-complete or PSPACE-complete, and global satisfiability is NP-complete, PSPACE-complete, or EXPTIME-complete. While our results holds even if we allow to use equality, we show that inequality leads to undecidability.

Edit Distance for Pushdown Automata

The edit distance between two words

Modal Logics Definable by Universal Three-Variable Formulas

w_1, w_2

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

is the minimal number of word operations letter insertions, deletions, and substitutions necessary to transform