Utilizing Treewidth for Quantitative Reasoning on Epistemic Logic Programs

Abstract

Extending the popular answer set programming paradigm by introspective reasoning capacities has received increasing interest within the last years. Particular attention is given to the formalism of epistemic logic programs (ELPs) where standard rules are equipped with modal operators which allow to express conditions on literals for being known or possible, that is, contained in all or some answer sets, respectively. ELPs thus deliver multiple collections of answer sets, known as world views. Employing ELPs for reasoning problems so far has mainly been restricted to standard decision problems (complexity analysis) and enumeration (development of systems) of world views. In this paper, we take a next step and contribute to epistemic logic programming in two ways: First, we establish quantitative reasoning for ELPs, where the acceptance of a certain set of literals depends on the number (proportion) of world views that are compatible with the set. Second, we present a novel system that is capable of efficiently solving the underlying counting problems required to answer such quantitative reasoning problems. Our system exploits the graph-based measure treewidth and works by iteratively finding and refining (graph) abstractions of an ELP program. On top of these abstractions, we apply dynamic programming that is combined with utilizing existing search-based solvers like (e)clingo for hard combinatorial subproblems that appear during solving. It turns out that our approach is competitive with existing systems that were introduced recently.

IASCAR: Incremental Answer Set Counting by Anytime Refinement

“…Set (S3) includes a very small set of instances of combinatorial problems. The instances in sets (S1) and (S2) have been used in previous works on ASP and counting (Eiter et al 2021;Besin et al 2021;Hecher 2022). Set (S1) encodes finding extensions of an argumentation framework (Fichte et al 2022b;Dvořák et al 2020;Gaggl et al 2020).…”

Treewidth-aware reductions of normal ASP to SAT – Is normal ASP harder than SAT after all?

“…Analogously, we require Formulas (13), (14), and (15) for guiding provability of an auxiliary variable (x ≺ y) along the TD. and (e ≺ b) is not possible anyway due to Formulas (10).…”

“…This can be witnessed by a couple of solvers that adhere to parameterized complexity and in particular to treewidth. Some techniques for solvers on Boolean formulas at least seem to be particular wellsuited for counting problems, e.g., based on dynamic programming [49,48], hybrid solving [62,10], which also includes the winner [69,70,42] of two tracks of the most recent model counting competition [43], as well as compact knowledge compilation [32,33]. However, treewidth also allows to improve solving hard decision problems with the help of knowledge compilation [25].…”

“…, (p o ≺ x) ∈ M , one can reverse any (y ≺ y ) ∈ M . More concretely, for any (y ≺ y ) ∈ M with y, y ∈ χ(t), by Formulas (9), one can derive (y ≺ y) ∈ M , which is prohibited by Formulas (10). Consequently, cyclic, transitive consequences over variables in χ ≤t cannot occur in any model of F ≤t .…”

Exploiting Database Management Systems and Treewidth for Counting

“…Set (S3) includes a very small set of instances of combinatorial problems. The instances in sets (S1) and (S2) have been used in previous works on ASP and counting (Eiter et al, 2021;Besin et al, 2021;Hecher, 2022). Set (S1) encodes finding extensions of an argumentation framework (Fichte et al, 2022b;Dvořák et al, 2020;Gaggl et al, 2020).…”