42 research outputs found
Lazy Probabilistic Model Checking without Determinisation
The bottleneck in the quantitative analysis of Markov chains and Markov
decision processes against specifications given in LTL or as some form of
nondeterministic B\"uchi automata is the inclusion of a determinisation step of
the automaton under consideration. In this paper, we show that full
determinisation can be avoided: subset and breakpoint constructions suffice. We
have implemented our approach---both explicit and symbolic versions---in a
prototype tool. Our experiments show that our prototype can compete with mature
tools like PRISM.Comment: 38 pages. Updated version for introducing the following changes: -
general improvement on paper presentation; - extension of the approach to
avoid full determinisation; - added proofs for such an extension; - added
case studies; - updated old case studies to reflect the added extensio
Linear Time Logic Control of Discrete-Time Linear Systems
The control of complex systems poses new challenges that fall beyond the traditional methods of control theory. One of these challenges is given by the need to control, coordinate and synchronize the operation of several interacting submodules within a system. The desired objectives are no longer captured by usual control specifications such as stabilization or output regulation. Instead, we consider specifications given by linear temporal logic (LTL) formulas. We show that existence of controllers for discrete-time controllable linear systems and LTL specifications can be decided and that such controllers can be effectively computed. The closed-loop system is of hybrid nature, combining the original continuous dynamics with the automatically synthesized switching logic required to enforce the specification
Simulation-based simplification of omega-automata
We analyze simulation relations as heuristics for the simplification of omega-automata, i.e., of finite automata working on infinite strings. Our focus is on alternating omega-automata. We introduce direct, delayed, and fair simulation for alternating Büchi automata (ABA) and show that these relations are preorders, they imply language containment, and they can be computed in polynomial time. We introduce quotient constructions for ABA suited for direct and delayed simulation, and we show that our notions of simulation are compatible with the standard translation of ABA to non-alternating automata. We apply these results to the problem of translating formulas of propositional linear-time temporal logic (LTL) to nondeterministic automata. We develop a translation algorithm from LTL to equivalent nondeterministic automata with an on-the-fly use of simulation relations for simplification, and we compare our approach to tableau-based translation algorithms. We extend our notion of delayed simulation to alternating parity automata (APA), introduce variants of this relation suited for quotienting of APA, and develop a simulation-based simplification algorithm for APA. We give a sketch of how to apply these results to a fragment of the modal mu-calculus
The Covering Problem
An important endeavor in computer science is to understand the expressive
power of logical formalisms over discrete structures, such as words. Naturally,
"understanding" is not a mathematical notion. This investigation requires
therefore a concrete objective to capture this understanding. In the
literature, the standard choice for this objective is the membership problem,
whose aim is to find a procedure deciding whether an input regular language can
be defined in the logic under investigation. This approach was cemented as the
right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on
first-order logic and has been in use since then. However, membership questions
are hard: for several important fragments, researchers have failed in this
endeavor despite decades of investigation. In view of recent results on one of
the most famous open questions, namely the quantifier alternation hierarchy of
first-order logic, an explanation may be that membership is too restrictive as
a setting. These new results were indeed obtained by considering more general
problems than membership, taking advantage of the increased flexibility of the
enriched mathematical setting. This opens a promising research avenue and
efforts have been devoted at identifying and solving such problems for natural
fragments. Until now however, these problems have been ad hoc, most fragments
relying on a specific one. A unique new problem replacing membership as the
right one is still missing. The main contribution of this paper is a suitable
candidate to play this role: the Covering Problem. We motivate this problem
with 3 arguments. First, it admits an elementary set theoretic formulation,
similar to membership. Second, we are able to reexplain or generalize all known
results with this problem. Third, we develop a mathematical framework and a
methodology tailored to the investigation of this problem
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
Separating Regular Languages with First-Order Logic
Given two languages, a separator is a third language that contains the first
one and is disjoint from the second one. We investigate the following decision
problem: given two regular input languages of finite words, decide whether
there exists a first-order definable separator. We prove that in order to
answer this question, sufficient information can be extracted from semigroups
recognizing the input languages, using a fixpoint computation. This yields an
EXPTIME algorithm for checking first-order separability. Moreover, the
correctness proof of this algorithm yields a stronger result, namely a
description of a possible separator. Finally, we generalize this technique to
answer the same question for regular languages of infinite words
A Verified Compositional Algorithm for AI Planning
We report on our HOL4 verification of an AI planning algorithm. The algorithm is compositional in the following sense: a planning problem is divided into multiple smaller abstractions, then each of the abstractions is solved, and finally the abstractions\u27 solutions are composed into a solution for the given problem. Formalising the algorithm, which was already quite well understood, revealed nuances in its operation which could lead to computing buggy plans. The formalisation also revealed that the algorithm can be presented more generally, and can be applied to systems with infinite states and actions, instead of only finite ones.
Our formalisation extends an earlier model for slightly simpler transition systems, and demonstrates another step towards formal treatments of more and more of the algorithms and reasoning used in AI planning, as well as model checking