393 research outputs found
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+
Hybrid Branching-Time Logics
Hybrid branching-time logics are introduced as extensions of CTL-like logics
with state variables and the downarrow-binder. Following recent work in the
linear framework, only logics with a single variable are considered. The
expressive power and the complexity of satisfiability of the resulting logics
is investigated.
As main result, the satisfiability problem for the hybrid versions of several
branching-time logics is proved to be 2EXPTIME-complete. These branching-time
logics range from strict fragments of CTL to extensions of CTL that can talk
about the past and express fairness-properties. The complexity gap relative to
CTL is explained by a corresponding succinctness result.
To prove the upper bound, the automata-theoretic approach to branching-time
logics is extended to hybrid logics, showing that non-emptiness of alternating
one-pebble Buchi tree automata is 2EXPTIME-complete.Comment: An extended abstract of this paper was presented at the International
Workshop on Hybrid Logics (HyLo 2007
The Complexity of Enriched Mu-Calculi
The fully enriched μ-calculus is the extension of the propositional
μ-calculus with inverse programs, graded modalities, and nominals. While
satisfiability in several expressive fragments of the fully enriched
μ-calculus is known to be decidable and ExpTime-complete, it has recently
been proved that the full calculus is undecidable. In this paper, we study the
fragments of the fully enriched μ-calculus that are obtained by dropping at
least one of the additional constructs. We show that, in all fragments obtained
in this way, satisfiability is decidable and ExpTime-complete. Thus, we
identify a family of decidable logics that are maximal (and incomparable) in
expressive power. Our results are obtained by introducing two new automata
models, showing that their emptiness problems are ExpTime-complete, and then
reducing satisfiability in the relevant logics to these problems. The automata
models we introduce are two-way graded alternating parity automata over
infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite
forests. The former are a common generalization of two incomparable automata
models from the literature. The latter extend alternating automata in a similar
way as the fully enriched μ-calculus extends the standard μ-calculus.Comment: A preliminary version of this paper appears in the Proceedings of the
33rd International Colloquium on Automata, Languages and Programming (ICALP),
2006. This paper has been selected for a special issue in LMC
Enriched MU-Calculi Module Checking
The model checking problem for open systems has been intensively studied in
the literature, for both finite-state (module checking) and infinite-state
(pushdown module checking) systems, with respect to Ctl and Ctl*. In this
paper, we further investigate this problem with respect to the \mu-calculus
enriched with nominals and graded modalities (hybrid graded Mu-calculus), in
both the finite-state and infinite-state settings. Using an automata-theoretic
approach, we show that hybrid graded \mu-calculus module checking is solvable
in exponential time, while hybrid graded \mu-calculus pushdown module checking
is solvable in double-exponential time. These results are also tight since they
match the known lower bounds for Ctl. We also investigate the module checking
problem with respect to the hybrid graded \mu-calculus enriched with inverse
programs (Fully enriched \mu-calculus): by showing a reduction from the domino
problem, we show its undecidability. We conclude with a short overview of the
model checking problem for the Fully enriched Mu-calculus and the fragments
obtained by dropping at least one of the additional constructs
Finite-State Abstractions for Probabilistic Computation Tree Logic
Probabilistic Computation Tree Logic (PCTL) is the established temporal
logic for probabilistic verification of discrete-time Markov chains. Probabilistic
model checking is a technique that verifies or refutes whether a property
specified in this logic holds in a Markov chain. But Markov chains are often
infinite or too large for this technique to apply. A standard solution to
this problem is to convert the Markov chain to an abstract model and to
model check that abstract model. The problem this thesis therefore studies
is whether or when such finite abstractions of Markov chains for model
checking PCTL exist.
This thesis makes the following contributions. We identify a sizeable fragment
of PCTL for which 3-valued Markov chains can serve as finite abstractions;
this fragment is maximal for those abstractions and subsumes many
practically relevant specifications including, e.g., reachability. We also develop
game-theoretic foundations for the semantics of PCTL over Markov
chains by capturing the standard PCTL semantics via a two-player games.
These games, finally, inspire a notion of p-automata, which accept entire
Markov chains. We show that p-automata subsume PCTL and Markov
chains; that their languages of Markov chains have pleasant closure properties;
and that the complexity of deciding acceptance matches that of probabilistic
model checking for p-automata representing PCTL formulae. In addition,
we offer a simulation between p-automata that under-approximates
language containment. These results then allow us to show that p-automata
comprise a solution to the problem studied in this thesis
Complexity of branching temporal description logics
We study branching-time temporal description logics (TDLs) based on the DLs ALC and EL and the temporal logics CTL and CTL*. The main contributions are algorithms for satisfiability that are more direct than existing approaches, and (mostly) tight elementary complexity bounds that range from PTIME to 2EXPTIME and 3EXPTIME. A careful use of tree automata techniques allows us to obtain transparent and uniform algorithms, avoiding to deal directly with the intricacies of CTL*
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