Unifying Decidable Entailments in Separation Logic with Inductive Definitions

The entailment problem $$\upvarphi \models \uppsi $$ φ ⊧ ψ in Separation Logic [12, 15], between separated conjunctions of equational ($$x \approx y$$ x ≈ y and $$x \not \approx y$$ x ≉ y ), spatial ($$x \mapsto (y_1,\ldots ,y_\upkappa )$$ x ↦ ( y 1 , … , y κ ) ) and predicate ($$p(x_1,\ldots ,x_n)$$ p ( x 1 , … , x n ) ) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called establishment [10, 13, 14] and restrictedness [8], respectively. Both classes are shown to be in $$\mathsf {2\text {EXPTIME}}$$ 2 EXPTIME by the independent proofs from [14] and [8], respectively, and a many-one reduction of established to restricted entailment problems has been given [8]. In this paper, we strictly generalize the restricted class, by distinguishing the conditions that apply only to the left- ($$\upvarphi $$ φ ) and the right- ($$\uppsi $$ ψ ) hand side of entailments, respectively. We provide a many-one reduction of this generalized class, called safe, to the established class. Together with the reduction of established to restricted entailment problems, this new reduction closes the loop and shows that the three classes of entailment problems (respectively established, restricted and safe) form a single, unified, $$\mathsf {2\text {EXPTIME}}$$ 2 EXPTIME -complete class.

Polynomial and Exponential Bounded Logic Programs with Function Symbols: Some New Decidable Classes

A logic program with function symbols is called finitely ground if there is a finite propositional logic program whose stable models are exactly the same as the stable models of this program. Finite groundability is an important property for logic programs with function symbols because it makes feasible to compute such programs' stable models using traditional ASP solvers. In this paper, we introduce new decidable classes of finitely ground programs called poly-bounded and k-EXP-bounded programs, which, to the best of our knowledge, strictly contain all other decidable classes of finitely ground programs discovered so far in the literature. We also study the relevant complexity properties for these classes of programs. We prove that the membership complexities for poly-bounded and k-EXP-bounded programs are EXPTIME-complete and (k+1)-EXPTIME-complete, respectively.

Finite Controllability of Conjunctive Query Answering with Existential Rules: Two Steps Forward

Reasoning with existential rules typically consists of checking whether a Boolean conjunctive query is satisfied by all models of a first-order sentence having the form of a conjunction of Datalog rules extended with existential quantifiers in rule-heads. To guarantee decidability, five basic decidable classes - linear, weakly-acyclic, guarded, sticky, and shy - have been singled out, together with several generalizations and combinations. For all basic classes, except shy, the important property of finite controllability has been proved, ensuring that a query is satisfied by all models of the sentence if, and only if, it is satisfied by all of its finite models. This paper takes two steps forward: (i) devise a general technique to facilitate the process of (dis)proving finite controllability of an arbitrary class of existential rules; and (ii) specialize the technique to complete the picture for the five mentioned classes, by showing that also shy is finitely controllable.

EXPLICIT MATHEMATICS AND OPERATIONAL SET THEORY: SOME ONTOLOGICAL COMPARISONS

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

On the complexity of stream equality

We study the complexity of deciding the equality of streams specified by systems of equations. There are several notions of stream models in the literature, each generating a different semantics of stream equality. We pinpoint the complexity of each of these notions in the arithmetical or analytical hierarchy. Their complexity ranges from low levels of the arithmetical hierarchy such as Π02 for the most relaxed stream models, to levels of the analytical hierarchy such as Π11 and up to subsuming the entire analytical hierarchy for more restrictive but natural stream models. Since all these classes properly include both the semi-decidable and co-semi-decidable classes, it follows that regardless of the stream semantics employed, there is no complete proof system or algorithm for determining equality or inequality of streams. We also discuss several related problems, such as the existence and uniqueness of stream solutions for systems of equations, as well as the equality of such solutions.