CTL is the dominant temporal specification language in practice mainly due to the fact that it admits model checking in linear time. Logic programming and the database query language Datalog are often used as an implementation platform for logic languages. In this paper we present the exact relation between CTL and Datalog and moreover we build on this relation and known efficient algorithms for CTL to obtain efficient algorithms for fragments of stratified Datalog. The contributions of this paper are: a) We embed CTL into STD which is a proper fragment of stratified Datalog. Moreover we show that STD expresses exactly CTL -- we prove that by embedding STD into CTL. Both embeddings are linear. b) CTL can also be embedded to fragments of Datalog without negation. We define a fragment of Datalog with the successor build-in predicate that we call TDS and we embed CTL into TDS in linear time. We build on the above relations to answer open problems of stratified Datalog. We prove that query evaluation is linear and that containment and satisfiability problems are both decidable. The results presented in this paper are the first for fragments of stratified Datalog that are more general than those containing only unary EDBs.
We know that CTL can be embedded into Transitive Closure logic @cite_20 and into alternation-free @math --calculus @cite_47 . In @cite_17 the authors observe that CTL can be embedded into stratified Datalog. In this paper it is the first time that the exact fragment of stratified Datalog with the same expressive power with CTL has been identified.
@cite_47 @cite_20 @cite_17
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We provide a direct and modular translation from the temporal logics CTL, ETL, FCTL (CTL extended with the ability to express fairness) and the Modal µ-calculus to Monadic inf-Datalog with built-in predicates. We call it inf-Datalog because the semantics we provide is a little different from the conventional Datalog least fixed point semantics, in that some recursive rules (corresponding to least fixed points) are allowed to unfold only finitely many times, whereas others (corresponding to greatest fixed points) are allowed to unfold infinitely many times.We characterize the fragments of Monadic inf-Datalog that have the same expressive power as Modal Logic (resp. CTL, alternation-free Modal µ-calculus and Modal µ-calculus). Our translation is interesting because it is direct and succinct. Moreover the fragments of Monadic inf-Datalog that we have exhibited have very simple syntactic characterizations as subsets of what we call Modal inf-Datalog programs. Three temporal logics are introduced which induce on labeled transition systems the same identifications as branching bisimulation. The first is an extension of Hennessy-Milner logic with a kind of unit operator. The second is another extension of Hennessy-Milner logic which exploits the power of backward modalities. The third is CTL* with the next-time operator interpreted over all paths, not just over maximal ones. A relevant side effect of the last characterization is that it sets a bridge between the state- and event-based approaches to the semantics of concurrent systems. > We present Datalog LITE, a new deductive query language with a linear-time model-checking algorithm, that is, linear time data complexity and program complexity. Datalog LITE is a variant of Datalog that uses stratified negation, restricted variable occurrences and a limited form of universal quantification in rule bodies.Despite linear-time evaluation, Datalog LITE is highly expressive: It encompasses popular modal and temporal logics such as CTL or the alternation-free μ-calculus. In fact, these formalisms have natural presentations as fragments of Datalog LITE. Further, Datalog LITE is equivalent to the alternation-free portion of guarded fixed-point logic. Consequently, linear-time model checking algorithms for all mentioned logics are obtained in a unified way.The results are complemented by inexpressibility proofs to the effect that linear-time fragments of stratified Datalog have too limited expressive power. CTL synthesis [8] is a long-standing problem with applications to synthesising synchronization protocols and concurrent programs. We show how to formulate CTL model checking in terms of “monotonic theories”, enabling us to use the SAT Modulo Monotonic Theories (SMMT) [5] framework to build an efficient SAT-modulo-CTL solver. This yields a powerful procedure for CTL synthesis, which is not only faster than previous techniques from the literature, but also scales to larger and more difficult formulas. Additionally, because it is a constraint-based approach, it can be easily extended with further constraints to guide the synthesis. Moreover, our approach is efficient at producing minimal Kripke structures on common CTL synthesis benchmarks.