Michael Vanden Boom

Weak cost monadic logic over infinite trees

Cost monadic logic has been introduced recently as a quantitative extension to monadic second-order logic. A sentence in the logic defines a function from a set of structures to N∪{∞}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. The rich theory associated with these functions has already been studied over finite words and trees. We extend the theory to infinite trees for the weak form of the logic (where second-order quantification is interpreted over finite sets). In particular, we show weak cost monadic logic is equivalent to weak cost automata, and finite-memory strategies suffice in the infinite two-player games derived from such automata. We use these results to provide a decision procedure for the logic and to show there is a function definable in cost monadic logic which is not definable in weak cost monadic logic.

The Complexity of Boundedness for Guarded Logics

Given a formula phi(x, X) positive in X, the bounded ness problem asks whether the fix point induced by phi is reached within some uniform bound independent of the structure (i.e. Whether the fix point is spurious, and can in fact be captured by a finite unfolding of the formula). In this paper, we study the bounded ness problem when phi is in the guarded fragment or guarded negation fragment of first-order logic, or the fix point extensions of these logics. It is known that guarded logics have many desirable computational and model theoretic properties, including in some cases decidable bounded ness. We prove that bounded ness for the guarded negation fragment is decidable in elementary time, and, making use of an unpublished result of Colcombet, even 2EXPTIME-complete. Our proof extends the connection between guarded logics and automata, reducing bounded ness for guarded logics to a question about cost automata on trees, a type of automaton with counters that assigns a natural number to each input rather than just a boolean.

Quasi-Weak Cost Automata: A New Variant of Weakness

Cost automata have a finite set of counters which can be manipulated on each transition but do not aect control flow. Based on the evolution of the counter values, these automata define functions from a domain like words or trees to N[{1}, modulo an equivalence relation which ignores exact values but preserves boundedness properties. These automata have been studied by Colcombet et al. as part of a “theory of regular cost functions”, an extension of the theory of regular languages which retains robust equivalences, closure properties, and decidability like the classical theory. We extend this theory by introducing quasi-weak cost automata. Unlike traditional weak automata which have a hard-coded bound on the number of alternations between accepting and rejecting states, quasi-weak automata bound the alternations using the counter values (which can vary across runs). We show that these automata are strictly more expressive than weak cost automata over infinite trees. The main result is a Rabin-style characterization theorem: a function is quasi-weak definable if and only if it is definable using two dual forms of nondeterministic Buchi cost automata. This yields a new decidability result for cost functions over infinite trees. 1998 ACM Subject Classification F.1.1 Models of Computation

Two-way cost automata and cost logics over infinite trees

Regular cost functions provide a quantitative extension of regular languages that retains most of their important properties, such as expressive power and decidability, at least over finite and infinite words and over finite trees. Much less is known over infinite trees. We consider cost functions over infinite trees defined by an extension of weak monadic second-order logic with a new fixed-point-like operator. We show this logic to be decidable, improving previously known decidability results for cost logics over infinite trees. The proof relies on an equivalence with a form of automata with counters called quasi-weak cost automata, as well as results about converting two-way alternating cost automata to one-way alternating cost automata.

A Step Up in Expressiveness of Decidable Fixpoint Logics

Guardedness restrictions are one of the principal means to obtain decidable logics — operators such as negation are restricted so that the free variables are contained in an atom. While guardedness has been applied fruitfully in the setting of first-order logic, the ability to add fixpoints while retaining decidability has been very limited. Here we show that one of the main restrictions imposed in the past can be lifted, getting a richer decidable logic by allowing fixpoints in which the parameters of the fixpoint can be unguarded. Using automata, we show that the resulting logics have a decidable satisfiability problem, and provide a fine study of the complexity of satisfiability. We show that similar methods apply to decide questions concerning the elimination of fixpoints within formulas of the logic.

Characterizing Definability in Decidable Fixpoint Logics

We look at characterizing which formulas are expressible in rich decidable logics such as guarded fixpoint logic, unary negation fixpoint logic, and guarded negation fixpoint logic. We consider semantic characterizations of definability, as well as effective characterizations. Our algorithms revolve around a finer analysis of the tree-model property and a refinement of the method of moving back-and-forth between relational logics and logics over trees.

Effective Interpolation and Preservation in Guarded Logics

Desirable properties of a logic include decidability, and a model theory that inherits properties of first-order logic, such as interpolation and preservation theorems. It is known that the Guarded Fragment (GF) of first-order logic is decidable and satisfies some preservation properties from first-order model theory; however, it fails to have Craig interpolation. The Guarded Negation Fragment (GNF), a recently defined extension, is known to be decidable and to have Craig interpolation. Here we give the first results on effective interpolation for extensions of GF. We provide an interpolation procedure for GNF whose complexity matches the doubly exponential upper bound for satisfiability of GNF. We show that the same construction gives not only Craig interpolation, but Lyndon interpolation and relativized interpolation, which can be used to provide effective proofs of some preservation theorems. We provide upper bounds on the size of GNF interpolants for both GNF and GF input, and complement this with matching lower bounds.

Interpolation with Decidable Fixpoint Logics

A logic satisfies Craig interpolation if whenever one formula φ1 in the logic entails another formula φ2 in the logic, there is an intermediate formula - one entailed by φ1 and entailing φ2 - using only relations in the common signature of φ1 and φ2. Uniform interpolation strengthens this by requiring the interpolant to depend only on φ1 and the common signature. A uniform interpolant can thus be thought of as a minimal upper approximation of a formula within a subsignature. For first-order logic, interpolation holds but uniform interpolation fails. Uniform interpolation is known to hold for several modal and description logics, but little is known about uniform interpolation for fragments of predicate logic over relations with arbitrary arity. Further, little is known about ordinary Craig interpolation for logics over relations of arbitrary arity that have a recursion mechanism, such as fixpoint logics. In this work we take a step towards filling these gaps, proving interpolation for a decidable fragment of least fixpoint logic called unary negation fixpoint logic. We prove this by showing that for any fixed k, uniform interpolation holds for the k-variable fragment of the logic. In order to show this we develop the technique of reducing questions about logics with tree-like models to questions about modal logics, following an approach by Gradel, Hirsch, and Otto. While this technique has been applied to expressivity and satisfiability questions before, we show how to extend it to reduce interpolation questions about such logics to interpolation for the μ-calculus.

Query Answering with Transitive and Linear-Ordered Data

We consider entailment problems involving powerful constraint languages such as guarded existential rules, in which additional semantic restrictions are put on a set of distinguished relations. We consider restricting a relation to be transitive, restricting a relation to be the transitive closure of another relation, and restricting a relation to be a linear order. We give some natural generalizations of guardedness that allow inference to be decidable in each case, and isolate the complexity of the corresponding decision problems. Finally we show that slight changes in our conditions lead to undecidability.