Babis Kostopoulos

Angelos Charalambidis, Babis Kostopoulos, Panos Rondogiannis

We consider a fragment of Higher-Order Datalog with negation and argue that it generalizes the familiar and important fragment of Linear Datalog. We investigate the expressive power of this fragment, establishing a tight connection with the hierarchy of space complexity classes. In particular, we demonstrate that for all $k \ge 1$, the $(k+1)$-order fragment of Stratified Linear Higher-Order Datalog$^\neg$ captures $(k-1)$-EXPSPACE. This result suggests that restricting programs to linear recursion shifts the expressive power of the corresponding fragments from time to space, generalizing the classical result that (Stratified) Linear Datalog captures NL. Unlike the first-order setting where an ordering assumption is required to capture NL, our results hold without any such assumption on the input database. The proof relies on simulating space-bounded Turing machines using Stratified Linear Higher-Order Datalog$^\neg$ programs and providing a space-efficient evaluation of the query program. We argue that identifying such computationally well-behaved fragments is a crucial step towards paving the way for practical implementations of Higher-Order Datalog.

Angelos Charalambidis, Giannos Chatziagapis, Babis Kostopoulos et al.

One of the most significant achievements of equilibrium logic was the characterization of strong equivalence, a property crucial for program transformation and optimization in Answer Set Programming (ASP). While ASP has recently been extended to a higher-order setting to enhance its expressive power, the lack of a comparable purely logical foundation has made verifying strong equivalence for higher-order programs or even proving the correctness of simple program transformations, a difficult challenge. This paper addresses this gap by developing a logical semantics for higher-order ASP by extending the equilibrium logic framework. Within this extended framework we demonstrate that every stratified higher-order logic program possesses a unique equilibrium model. Moreover, we establish definability results demonstrating that the syntax of our higher-order language is sufficiently expressive to capture its semantic domains. Finally, and most importantly, we generalize the classical theorem of strong equivalence to the higher-order setting: we prove that two programs are strongly equivalent if and only if they share the same higher-order models.

Bart Bogaerts, Angelos Charalambidis, Giannos Chatziagapis et al.

We propose a stable model semantics for higher-order logic programs. Our semantics is developed using Approximation Fixpoint Theory (AFT), a powerful formalism that has successfully been used to give meaning to diverse non-monotonic formalisms. The proposed semantics generalizes the classical two-valued stable model semantics of (Gelfond and Lifschitz 1988) as-well-as the three-valued one of (Przymusinski 1990), retaining their desirable properties. Due to the use of AFT, we also get for free alternative semantics for higher-order logic programs, namely supported model, Kripke-Kleene, and well-founded. Additionally, we define a broad class of stratified higher-order logic programs and demonstrate that they have a unique two-valued higher-order stable model which coincides with the well-founded semantics of such programs. We provide a number of examples in different application domains, which demonstrate that higher-order logic programming under the stable model semantics is a powerful and versatile formalism, which can potentially form the basis of novel ASP systems.