Udi Boker

What's decidable about weighted automata?

Weighted automata map input words to numerical values. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. In the 90s, Krob studied the decidability of problems on rational series, which strongly relate to weighted automata. In particular, it follows from Krobs results that the universality problem (that is, deciding whether the values of all words are below some threshold) is decidable for weighted automata with weights in N ∪ {∞}, and that the equality problem is undecidable when the weights are in N ∪ {∞}. In this paper we continue the study of the borders of decidability in weighted automata, describe alternative and direct proofs of the above results, and tighten them further. Unlike the proofs of Krob, which are algebraic in their nature, our proofs stay in the terrain of state machines, and the reduction is from the halting problem of a two-counter machine. This enables us to significantly simplify Krobs reasoning and strengthen the results to apply already to a very simple class of automata: all the states are accepting, there are no initial nor final weights, and all the weights are from the set {-1, 0, 1}. The fact we work directly with automata enables us to tighten also the decidability results and to show that the universality problem for weighted automata with weights in N ∪ {∞}, and in fact even with weights in Q≥0 ∪ {∞}, is PSPACE-complete. Our results thus draw a sharper picture about the decidability of decision problems for weighted automata, in both the front of equality vs. universality and the front of the N ∪ {∞} vs. the Z ∪ {∞} domains.

Formalizing and reasoning about quality

Traditional formal methods are based on a Boolean satisfaction notion: a reactive system satisfies, or not, a given specification. We generalize formal methods to also address the quality of systems. As an adequate specification formalism we introduce the linear temporal logic LTL[

Discounting in LTL

{\cal F}

Comparing Computational Power

]. The satisfaction value of an LTL[

Determinizing Discounted-Sum Automata

{\cal F}

Formally Reasoning About Quality

] formula is a number between 0 and 1, describing the quality of the satisfaction. The logic generalizes traditional LTL by augmenting it with a (parameterized) set

The Target Discounted-Sum Problem

{\cal F}

Battery transition systems

of arbitrary functions over the interval [0,1]. For example,

Exact and Approximate Determinization of Discounted-Sum Automata

{\cal F}

Three paths to effectiveness

may contain the maximum or minimum between the satisfaction values of subformulas, their product, and their average. The classical decision problems in formal methods, such as satisfiability, model checking, and synthesis, are generalized to search and optimization problems in the quantitative setting. For example, model checking asks for the quality in which a specification is satisfied, and synthesis returns a system satisfying the specification with the highest quality. Reasoning about quality gives rise to other natural questions, like the distance between specifications. We formalize these basic questions and study them for LTL[