6,580 research outputs found
On the decidability and complexity of Metric Temporal Logic over finite words
Metric Temporal Logic (MTL) is a prominent specification formalism for
real-time systems. In this paper, we show that the satisfiability problem for
MTL over finite timed words is decidable, with non-primitive recursive
complexity. We also consider the model-checking problem for MTL: whether all
words accepted by a given Alur-Dill timed automaton satisfy a given MTL
formula. We show that this problem is decidable over finite words. Over
infinite words, we show that model checking the safety fragment of MTL--which
includes invariance and time-bounded response properties--is also decidable.
These results are quite surprising in that they contradict various claims to
the contrary that have appeared in the literature
Visibly Linear Dynamic Logic
We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear
Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown
languages over finite words. In VLDL one can, e.g., express that a function
resets a variable to its original value after its execution, even in the
presence of an unbounded number of intermediate recursive calls. We prove that
VLDL describes exactly the -visibly pushdown languages. Thus it is
strictly more expressive than LTL and able to express recursive properties of
programs with unbounded call stacks.
The main technical contribution of this work is a translation of VLDL into
-visibly pushdown automata of exponential size via one-way alternating
jumping automata. This translation yields exponential-time algorithms for
satisfiability, validity, and model checking. We also show that visibly
pushdown games with VLDL winning conditions are solvable in triply-exponential
time. We prove all these problems to be complete for their respective
complexity classes.Comment: 25 Page
An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata
In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem
stating that every formula of Past LTL (the extension of LTL with past
operators) is equivalent to a formula of the form , where
and contain only past operators. Some years later, Chang,
Manna, and Pnueli built on this result to derive a similar normal form for LTL.
Both normalisation procedures have a non-elementary worst-case blow-up, and
follow an involved path from formulas to counter-free automata to star-free
regular expressions and back to formulas. We improve on both points. We present
a direct and purely syntactic normalisation procedure for LTL yielding a normal
form, comparable to the one by Chang, Manna, and Pnueli, that has only a single
exponential blow-up. As an application, we derive a simple algorithm to
translate LTL into deterministic Rabin automata. The algorithm normalises the
formula, translates it into a special very weak alternating automaton, and
applies a simple determinisation procedure, valid only for these special
automata.Comment: This is the extended version of the referenced conference paper and
contains an appendix with additional materia
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+
Truly On-The-Fly LTL Model Checking
We propose a novel algorithm for automata-based LTL model checking that
interleaves the construction of the generalized B\"{u}chi automaton for the
negation of the formula and the emptiness check. Our algorithm first converts
the LTL formula into a linear weak alternating automaton; configurations of the
alternating automaton correspond to the locations of a generalized B\"{u}chi
automaton, and a variant of Tarjan's algorithm is used to decide the existence
of an accepting run of the product of the transition system and the automaton.
Because we avoid an explicit construction of the B\"{u}chi automaton, our
approach can yield significant improvements in runtime and memory, for large
LTL formulas. The algorithm has been implemented within the SPIN model checker,
and we present experimental results for some benchmark examples
One Theorem to Rule Them All: A Unified Translation of LTL into {\omega}-Automata
We present a unified translation of LTL formulas into deterministic Rabin
automata, limit-deterministic B\"uchi automata, and nondeterministic B\"uchi
automata. The translations yield automata of asymptotically optimal size
(double or single exponential, respectively). All three translations are
derived from one single Master Theorem of purely logical nature. The Master
Theorem decomposes the language of a formula into a positive boolean
combination of languages that can be translated into {\omega}-automata by
elementary means. In particular, Safra's, ranking, and breakpoint constructions
used in other translations are not needed
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