37,728 research outputs found
Disjunctive form and the modal alternation hierarchy
This paper studies the relationship between disjunctive form, a syntactic
normal form for the modal mu calculus, and the alternation hierarchy. First it
shows that all disjunctive formulas which have equivalent tableau have the same
syntactic alternation depth. However, tableau equivalence only preserves
alternation depth for the disjunctive fragment: there are disjunctive formulas
with arbitrarily high alternation depth that are tableau equivalent to
alternation-free non-disjunctive formulas. Conversely, there are
non-disjunctive formulas of arbitrarily high alternation depth that are tableau
equivalent to disjunctive formulas without alternations. This answers
negatively the so far open question of whether disjunctive form preserves
alternation depth. The classes of formulas studied here illustrate a previously
undocumented type of avoidable syntactic complexity which may contribute to our
understanding of why deciding the alternation hierarchy is still an open
problem.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Satisfiability Games for Branching-Time Logics
The satisfiability problem for branching-time temporal logics like CTL*, CTL
and CTL+ has important applications in program specification and verification.
Their computational complexities are known: CTL* and CTL+ are complete for
doubly exponential time, CTL is complete for single exponential time. Some
decision procedures for these logics are known; they use tree automata,
tableaux or axiom systems. In this paper we present a uniform game-theoretic
framework for the satisfiability problem of these branching-time temporal
logics. We define satisfiability games for the full branching-time temporal
logic CTL* using a high-level definition of winning condition that captures the
essence of well-foundedness of least fixpoint unfoldings. These winning
conditions form formal languages of \omega-words. We analyse which kinds of
deterministic {\omega}-automata are needed in which case in order to recognise
these languages. We then obtain a reduction to the problem of solving parity or
B\"uchi games. The worst-case complexity of the obtained algorithms matches the
known lower bounds for these logics. This approach provides a uniform, yet
complexity-theoretically optimal treatment of satisfiability for branching-time
temporal logics. It separates the use of temporal logic machinery from the use
of automata thus preserving a syntactical relationship between the input
formula and the object that represents satisfiability, i.e. a winning strategy
in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner
closure of the input formula only. Last but not least, the games presented here
come with an attempt at providing tool support for the satisfiability problem
of complex branching-time logics like CTL* and CTL+
MDP Optimal Control under Temporal Logic Constraints
In this paper, we develop a method to automatically generate a control policy
for a dynamical system modeled as a Markov Decision Process (MDP). The control
specification is given as a Linear Temporal Logic (LTL) formula over a set of
propositions defined on the states of the MDP. We synthesize a control policy
such that the MDP satisfies the given specification almost surely, if such a
policy exists. In addition, we designate an "optimizing proposition" to be
repeatedly satisfied, and we formulate a novel optimization criterion in terms
of minimizing the expected cost in between satisfactions of this proposition.
We propose a sufficient condition for a policy to be optimal, and develop a
dynamic programming algorithm that synthesizes a policy that is optimal under
some conditions, and sub-optimal otherwise. This problem is motivated by
robotic applications requiring persistent tasks, such as environmental
monitoring or data gathering, to be performed.Comment: Technical report accompanying the CDC2011 submissio
Adding modular predicates to first-order fragments
We investigate the decidability of the definability problem for fragments of
first order logic over finite words enriched with modular predicates. Our
approach aims toward the most generic statements that we could achieve, which
successfully covers the quantifier alternation hierarchy of first order logic
and some of its fragments. We obtain that deciding this problem for each level
of the alternation hierarchy of both first order logic and its two-variable
fragment when equipped with all regular numerical predicates is not harder than
deciding it for the corresponding level equipped with only the linear order and
the successor. For two-variable fragments we also treat the case of the
signature containing only the order and modular predicates.Relying on some
recent results, this proves the decidability for each level of the alternation
hierarchy of the two-variable first order fragmentwhile in the case of the
first order logic the question remains open for levels greater than two.The
main ingredients of the proofs are syntactic transformations of first order
formulas as well as the algebraic framework of finite categories
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