16 research outputs found
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
A Simple Rewrite System for the Normalization of Linear Temporal Logic
In the mid 80s, Lichtenstein, Pnueli, and Zuck showed that every formula of
Past LTL (the extension of Linear Temporal Logic with past operators) is
equivalent to a conjunction of formulas of the form , where and contain
only past operators. Some years later, Chang, Manna, and Pnueli derived a
similar normal form for LTL. Both normalization 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. In
2020, Sickert and Esparza presented a direct and purely syntactic normalization
procedure for LTL yielding a normal form similar to the one by Chang, Manna,
and Pnueli, with a single exponential blow-up, and applied it to the problem of
constructing a succinct deterministic -automaton for a given formula.
However, their procedure had exponential time complexity in the best case. In
particular, it does not perform better for formulas that are almost in normal
form. In this paper we present an alternative normalization procedure based on
a simple set of rewrite rules
A Verified and Compositional Translation of LTL to Deterministic Rabin Automata
We present a formalisation of the unified translation approach from linear temporal logic (LTL) to omega-automata from [Javier Esparza et al., 2018]. This approach decomposes LTL formulas into "simple" languages and allows a clear separation of concerns: first, we formalise the purely logical result yielding this decomposition; second, we develop a generic, executable, and expressive automata library providing necessary operations on automata to re-combine the "simple" languages; third, we instantiate this generic theory to obtain a construction for deterministic Rabin automata (DRA). We extract from this particular instantiation an executable tool translating LTL to DRAs. To the best of our knowledge this is the first verified translation of LTL to DRAs that is proven to be double-exponential in the worst case which asymptotically matches the known lower bound
Efficient Normalization of Linear Temporal Logic
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 normalization 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
direct and purely syntactic normalization procedures for LTL, yielding a normal
form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a
single exponential blow-up. As an application, we derive a simple algorithm to
translate LTL into deterministic Rabin automata. The algorithm normalizes the
formula, translates it into a special very weak alternating automaton, and
applies a simple determinization procedure, valid only for these special
automata.Comment: Submitted to J. ACM. arXiv admin note: text overlap with
arXiv:2304.08872, arXiv:2005.0047
Refinement checking on parametric modal transition systems
Modal transition systems (MTS) is a well-studied specification formalism of reactive systems supporting a step-wise refinement methodology. Despite its many advantages, the formalism as well as its currently known extensions are incapable of expressing some practically needed aspects in the refinement process like exclusive, conditional and persistent choices. We introduce a new model called parametric modal transition systems (PMTS) together with a general modal refinement notion that overcomes many of the limitations. We investigate the computational complexity of modal and thorough refinement checking on PMTS and its subclasses and provide a direct encoding of the modal refinement problem into quantified Boolean formulae, allowing us to employ state-of-the-art QBF solvers for modal refinement checking. The experiments we report on show that the feasibility of refinement checking is more influenced by the degree of nondeterminism rather than by the syntactic restrictions on the types of formulae allowed in the description of the PMTS
On the Translation of Automata to Linear Temporal Logic
While the complexity of translating future linear temporal logic (LTL) into
automata on infinite words is well-understood, the size increase involved in
turning automata back to LTL is not. In particular, there is no known
elementary bound on the complexity of translating deterministic
-regular automata to LTL. Our first contribution consists of tight
bounds for LTL over a unary alphabet: alternating, nondeterministic and
deterministic automata can be exactly exponentially, quadratically and linearly
more succinct, respectively, than any equivalent LTL formula. Our main
contribution consists of a translation of general counter-free deterministic
-regular automata into LTL formulas of double exponential
temporal-nesting depth and triple exponential length, using an intermediate
Krohn-Rhodes cascade decomposition of the automaton. To our knowledge, this is
the first elementary bound on this translation. Furthermore, our translation
preserves the acceptance condition of the automaton in the sense that it turns
a looping, weak, B\"uchi, coB\"uchi or Muller automaton into a formula that
belongs to the matching class of the syntactic future hierarchy. In particular,
it can be used to translate an LTL formula recognising a safety language to a
formula belonging to the safety fragment of LTL (over both finite and infinite
words).Comment: Full version with appendix of a chapter with the same title that
appears in the FoSSaCS 2022 conference proceeding