First-order fragments with successor over infinite words

We consider fragments of first-order logic and as models we allow finite andinfinite words simultaneously. The only binary relations apart from equalityare order comparison < and the successor predicate +1. We givecharacterizations of the fragments Sigma2 = Sigma2[<,+1] and FO2 = FO2[<,+1] interms of algebraic and topological properties. To this end we introduce thefactor topology over infinite words. It turns out that a language L is in theintersection of FO2 and Sigma2 if and only if L is the interior of an FO2language. Symmetrically, a language is in the intersection of FO2 and Pi2 ifand only if it is the topological closure of an FO2 language. The fragmentDelta2, which by definition is the intersection Sigma2 and Pi2 contains exactlythe clopen languages in FO2. In particular, over infinite words Delta2 is astrict subclass of FO2. Our characterizations yield decidability of themembership problem for all these fragments over finite and infinite words; andas a corollary we also obtain decidability for infinite words. Moreover, wegive a new decidable algebraic characterization of dot-depth 3/2 over finitewords. Decidability of dot-depth 3/2 over finite words was first shown by Glaßer andSchmitz in STACS 2000, and decidability of the membership problem for FO2 overinfinite words was shown 1998 by Wilke in his habilitation thesis whereasdecidability of Sigma2 over infinite words was not known before

Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

Quantified CTL: Expressiveness and Complexity

While it was defined long ago, the extension of CTL with quantification over atomic propositions has never been studied extensively. Considering two different semantics (depending whether propositional quantification refers to the Kripke structure or to its unwinding tree), we study its expressiveness (showing in particular that QCTL coincides with Monadic Second-Order Logic for both semantics) and characterise the complexity of its model-checking and satisfiability problems, depending on the number of nested propositional quantifiers (showing that the structure semantics populates the polynomial hierarchy while the tree semantics populates the exponential hierarchy)

A first-order logic characterization of safety and co-safety languages

Linear Temporal Logic (LTL) is one of the most popular temporal logics, that comes into play in a variety of branches of computer science. Among the various reasons of its widespread use there are its strong foundational properties: LTL is equivalent to counter-free omega-automata, to star-free omega-regular expressions, and (by Kamp's theorem) to the first-order theory of one successor (S1S[FO]). Safety and co-safety languages, where a finite prefix suffices to establish whether a word does not belong or belongs to the language, respectively, play a crucial role in lowering the complexity of problems like model checking and reactive synthesis for LTL. SafetyLTL (resp., coSafetyLTL) is a fragment of LTL where only universal (resp., existential) temporal modalities are allowed, that recognises safety (resp., co-safety) languages only. The main contribution of this paper is the introduction of a fragment of S1S[FO], called SafetyFO, and of its dual coSafetyFO, which are expressively complete with respect to the LTL-definable safety and co-safety languages. We prove that they exactly characterize SafetyLTL and coSafetyLTL, respectively, a result that joins Kamp's theorem, and provides a clearer view of the characterization of (fragments of) LTL in terms of first-order languages. In addition, it gives a direct, compact, and self-contained proof that any safety language definable in LTL is definable in SafetyLTL as well. As a by-product, we obtain some interesting results on the expressive power of the weak tomorrow operator of SafetyLTL, interpreted over finite and infinite words. Moreover, we prove that, when interpreted over finite words, SafetyLTL (resp. coSafetyLTL) devoid of the tomorrow (resp., weak tomorrow) operator captures the safety (resp., co-safety) fragment of LTL over finite words

Some Turing-Complete Extensions of First-Order Logic

We introduce a natural Turing-complete extension of first-order logic FO. The extension adds two novel features to FO. The first one of these is the capacity to add new points to models and new tuples to relations. The second one is the possibility of recursive looping when a formula is evaluated using a semantic game. We first define a game-theoretic semantics for the logic and then prove that the expressive power of the logic corresponds in a canonical way to the recognition capacity of Turing machines. Finally, we show how to incorporate generalized quantifiers into the logic and argue for a highly natural connection between oracles and generalized quantifiers.Comment: In Proceedings GandALF 2014, arXiv:1408.556

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+

On relating CTL to Datalog

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.Comment: 34 pages, 1 figure (file .eps

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.Comment: 37 page