Francicleber Martins Ferreira

On Minimal Models

We investigate some logics which use the concept of minimal models in their definition. Minimal objects are widely used in Logic and Computer Science. They are applied in the context of Inductive Definitions, Logic Programming and Artificial Intelligence. An example of logic which uses this concept is the MIN(FO) logic due to van Benthem [20]. He shows that MIN(FO) is equivalent to the Least Fixed Point logic (LFP) in expressive power. In [6], we extended MIN(FO) to the MIN Logic and proved it is equivalent to second-order logic in expressive power. Here, we exhibit a fragment of MIN, the MIN Δ logic, which is more expressive than LFP, less expressive than MIN and closed under boolean connectives and first-order quantification. In order to do this, in the Section 2, we prove that the Downward Lowenheim-Skolem Theorem holds for arbitrary countable sets of LFP-formulas by showing that every infinite structure has a countable LFP-substructure. The method may be used to generalize this theorem to any set of LFP-formulas. We also analyse the expressive power of the Nested Abnormality Theories (NATs) of Lifschitz, another formalism based on minimal models used in Artificial Intelligence, and we demonstrate that for each second-order theory Γ there is a NAT which is a conservative extension of Γ. We give a translation from second-order sentences into such NATs which is linear in the size of the sentence in prenex normal form. Finally, we establish a hierarchy of expressiveness of these logics that deal with the concept of minimal models.

The Descriptive Complexity of Decision Problems through Logics with Relational Fixed-Point and Capturing Results

Abstract In this work, we generalize the classical fixed-point logics using relations instead of operators in order to capture the notion of nondeterminism. The basic idea is that we use loops in a relation instead of fixed-points of a function, that is, X is a fixed-point of the relation R in case the pair (X,X) belongs to R . We introduce the notion of initial fixed-point of an inflationary relation R and the associated operator rifp. We denote by RIFP the first-order logic with the inflationary relational fixed-point operator rifp and show that it captures the polynomial hierarchy using a translation to second-order logic. We also consider the fragment RIFP1 with the restriction that the rifp operator can be applied at most once. We show that RIFP1 captures the class NP and compare our logic with the nondeterministic fixed-point logic proposed by Abiteboul, Vianu and Vardi in [S. Abiteboul, M. Vardi, and V. Vianu. Fixpoint logics, relational machines, and computational complexity. Journal of the ACM, 44-1:30–56, 1997].

Polynomial Hierarchy Graph Properties in Hybrid Logic

Abstract In this article, we show that for each property of graphs G in the polynomial hierarchy (PH) there is a sequence ϕ 1 , ϕ 2 , … of formulas of the full hybrid logic which are satisfied exactly by the frames in G . Moreover, the size of ϕ n is bounded by a polynomial in n. These results lead to the definition of syntactically defined fragments of hybrid logic whose model checking problem is complete for each degree in the polynomial hierarchy.

Expressible preferential logics

We introduce expressible preferential logics, whose preference relations can be defined by some abstract logic. Abstract logics not only allow one to describe preference relations, but also classes of abstract preferential logics and give general proofs for properties common to all logics in these classes. Our approach follows that of Abstract Model Theory. We show that some well-known non-monotonic logics are preferential. We prove that they are elementary, which means that their preference relation can be defined in first-order logic. We study expressiveness and definability results for wide classes of abstract preferential logics in the spirit of Universal Logic. We present a collapse result for expressible preferential logics. We prove that, for a class of expressible preferential logics, if the class of minimal models of a finite set of sentences is Δ-L-expressible, then it is L-expressible, i.e. such class of models can be finitely axiomatized in L. Using this result, we show that under certain conditions one can axiomatize the class of minimal models of a finite set of sentences where some symbol P is defined using that set and an explicit definition for this symbol.

Hybrid logics and NP graph properties

We show that for each property of graphs G in NP there is a sequence O1, O2, . . . of formulas of the full hybrid logic which are satisfied exactly by the frames in G. Moreover, the size of On is bounded by a polynomial. We also show that the same holds for each graph property in the polynomial hierarchy.

Expressiveness and definability in circumscription

We investigate expressiveness and definability issues with respect to minimal models, particularly in the scope of Circumscription. First, we give a proof of the failure of the Lowenheim-Skolem Theorem for Circumscription. Then we show that, if the class of P; Z-minimal models of a first-order sentence is Δ-elementary, then it is elementary. That is, whenever the circumscription of a first-order sentence is equivalent to a first-order theory, then it is equivalent to a finitely axiomatizable one. This means that classes of models of circumscribed theories are either elementary or not Δ-elementary. Finally, using the previous result, we prove that, whenever a relation Pi is defined in the class of P; Z-minimal models of a first-order sentence Φ and whenever such class of P; Z-minimal models is Δ-elementary, then there is an explicit definition ψ for Pi such that the class of P; Z-minimal models of Φ is the class of models of Φ ∧ ψ. In order words, the circumscription of P in Φ with Z varied can be replaced by Φ plus this explicit definition ψ for Pi.

The predicate-minimizing logic MIN

The concept of minimization is widely used in several areas of Computer Science. Although this notion is not properly formalized in first-order logic, it is so with the logic MIN(FO) [13] where a minimal predicate P is defined as satisfying a given first-order description φ(P). We propose the MIN logic as a generalization of MIN(FO) since the extent of a minimal predicate P is not necessarily unique in MIN as it is in MIN(FO). We will explore two different possibilities of extending MIN(FO) by creating a new predicate defined as the union, the U-MIN logic, or intersection, the I-MIN logic, of the extent of all minimal P that satisfies φ(P). We will show that U-MIN and I-MIN are interdefinable. Thereafter, U-MIN will be just MIN. Finally, we will prove that simultaneous minimizations does not increase the expressiveness of MIN, and that MIN and second-order logic are equivalent in expressive power.