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Probabilistic reasoning, parameter learning, and most probable explanation inference for answer set programming have recently received growing attention. They are only some of the problems that can be formulated as Algebraic Answer Set Counting (AASC) problems. The latter are however hard to solve, and efficient evaluation techniques are needed. Inspired by Vlasser et al.'s Tp-compilation (JAR, 2016), we introduce Tp-unfolding, which employs forward reasoning to break the cycles in the positive dependency graph of a program by unfolding them. Tp-unfolding is defined for any normal answer set program and unfolds programs with respect to unfolding sequences, which are akin to elimination orders in SAT-solving. Using "good" unfolding sequences, we can ensure that the increase of the treewidth of the unfolded program is small. Treewidth is a measure adhering to a program's tree-likeness, which gives performance guarantees for AASC. We give sufficient conditions for the existence of good unfolding sequences based on the novel notion of component-boosted backdoor size, which measures the cyclicity of the positive dependencies in a program. The experimental evaluation of a prototype implementation, the AASC solver aspmc, shows promising results.
Weighted Logic is a powerful tool for the specification of calculations over semirings that depend on qualitative information. Using a novel combination of Weighted Logic and Here-and-There (HT) Logic, in which this dependence is based on intuitionistic grounds, we introduce Answer Set Programming with Algebraic Constraints (ASP($\mathcal A \mathcal C$)), where rules may contain constraints that compare semiring values to weighted formula evaluations. Such constraints provide streamlined access to a manifold of constructs available in ASP, like aggregates, choice constraints, and arithmetic operators. They extend some of them and provide a generic framework for defining programs with algebraic computation, which can be fruitfully used e.g. for provenance semantics of datalog programs. While undecidable in general, expressive fragments of ASP($\mathcal A \mathcal C$) can be exploited for effective problem solving in a rich framework.
In this work, we introduce AXolotl, a self-study aid designed to guide students through the basics of formal reasoning and term manipulation. Unlike most of the existing study aids for formal reasoning, AXolotl is an Android-based application with a simple touch-based interface. Part of the design goal was to minimize the possibility of user errors which distract from the learning process. Such as typos or inconsistent application of the provided rules. The system includes a zoomable proof viewer which displays the progress made so far and allows for storage of the completed proofs as a JPEG or L A T E X file. The software is available on the google play store and comes with a small library of problems. Additional problems may be opened in AXolotl using a simple input language. Currently, AXolotl supports problems that can be solved using rules which transform a single expression into a set of expressions. This covers educational scenarios found in our first-semester introduction to logic course and helps bridge the gap between propositional and first-order reasoning. Future developments will include rewrite rules which take a set of expressions and return a set of expressions, as well as a quantified first-order extension.
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