Step-wise explanations can explain logic puzzles and other satisfaction problems by showing how to derive decisions step by step. Each step consists of a set of constraints that derive an assignment to one or more decision variables. However, many candidate explanation steps exist, with different sets of constraints and different decisions they derive. To identify the most comprehensible one, a user-defined objective function is required to quantify the quality of each step. However, defining a good objective function is challenging. Here, interactive preference elicitation methods from the wider machine learning community can offer a way to learn user preferences from pairwise comparisons. We investigate the feasibility of this approach for step-wise explanations and address several limitations that distinguish it from elicitation for standard combinatorial problems. First, because the explanation quality is measured using multiple sub-objectives that can vary a lot in scale, we propose two dynamic normalization techniques to rescale these features and stabilize the learning process. We also observed that many generated comparisons involve similar explanations. For this reason, we introduce MACHOP (Multi-Armed CHOice Perceptron), a novel query generation strategy that integrates non-domination constraints with upper confidence bound-based diversification. We evaluate the elicitation techniques on Sudokus and Logic-Grid puzzles using artificial users, and validate them with a real-user evaluation. In both settings, MACHOP consistently produces higher-quality explanations than the standard approach.

In the field of Explainable Constraint Solving, it is common to explain to a user why a problem is unsatisfiable. A recently proposed method for this is to compute a sequence of explanation steps. Such a step-wise explanation shows individual reasoning steps involving constraints from the original specification, that in the end explain a conflict. However, computing a step-wise explanation is computationally expensive, limiting the scope of problems for which it can be used. We investigate how we can use proofs generated by a constraint solver as a starting point for computing step-wise explanations, instead of computing them step-by-step. More specifically, we define a framework of abstract proofs, in which both proofs and step-wise explanations can be represented. We then propose several methods for converting a proof to a step-wise explanation sequence, with special attention to trimming and simplification techniques to keep the sequence and its individual steps small. Our results show our method significantly speeds up the generation of step-wise explanation sequences, while the resulting step-wise explanation has a quality similar to the current state-of-the-art.

We build on a recently proposed method for stepwise explaining the solutions to Constraint Satisfaction Problems (CSPs) in a human understandable way. An explanation here is a sequence of simple inference steps where simplicity is quantified by a cost function. Explanation generation algorithms rely on extracting Minimal Unsatisfiable Subsets (MUSs) of a derived unsatisfiable formula, exploiting a one-to-one correspondence between so-called non-redundant explanations and MUSs. However, MUS extraction algorithms do not guarantee subset minimality or optimality with respect to a given cost function. Therefore, we build on these formal foundations and address the main points of improvement, namely how to generate explanations efficiently that are provably optimal (with respect to the given cost metric). To this end, we developed (1) a hitting set-based algorithm for finding the optimal constrained unsatisfiable subsets; (2) a method for reusing relevant information across multiple algorithm calls; and (3) methods for exploiting domain-specific information to speed up the generation of explanation sequences. We have experimentally validated our algorithms on a large number of CSP problems. We found that our algorithms outperform the MUS approach in terms of explanation quality and computational time (on average up to 56 % faster than a standard MUS approach).

We build on a recently proposed method for stepwise explaining the solutions to Constraint Satisfaction Problems (CSPs) in a human understandable way. An explanation here is a sequence of simple inference steps where simplicity is quantified by a cost function. Explanation generation algorithms rely on extracting Minimal Unsatisfiable Subsets (MUSs) of a derived unsatisfiable formula, exploiting a one-to-one correspondence between so-called non-redundant explanations and MUSs. However, MUS extraction algorithms do not guarantee subset minimality or optimality with respect to a given cost function. Therefore, we build on these formal foundations and address the main points of improvement, namely how to generate explanations efficiently that are provably optimal (with respect to the given cost metric). To this end, we developed (1) a hitting set-based algorithm for finding the optimal constrained unsatisfiable subsets; (2) a method for reusing relevant information across multiple algorithm calls; and (3) methods for exploiting domain-specific information to speed up the generation of explanation sequences. We have experimentally validated our algorithms on a large number of CSP problems. We found that our algorithms outperform the MUS approach in terms of explanation quality and computational time (on average up to 56 % faster than a standard MUS approach).

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

The formalisms for knowledge representation and reasoning(KR&R) typically have a variety of semantics, each one having its particular application scenarios.

The study of groundedness was limited to exact lattice points; in this paper, we extend it to the bilattice: for an approximator A of O, we define A-groundedness. We show that all partial A-stable fixpoints are A-grounded and that the A-well-founded fixpoint is uniquely characterised as the least precise A-grounded fixpoint. We apply our theory to logic programming and study complexity.

The formalisms for knowledge representation and reasoning(KR&R) typically have a variety of semantics, each one having its particular application scenarios. However, the KR&Rcommunity cannot readily benefit from such a variety due toa lack of efficient solver technology. This is partly caused bythe fact that solver development is laborious and even accomplishing a working prototype can form a major effort.In this paper, we introduce a new framework that enables us todeclaratively specify a given semantics in second-order logicand to automatically generate a solver from that specification. Hence, KR&R researchers can rapidly develop a solverprototype for their new/existing semantics with a minimal effort. Technically, our framework builds on a recent approachfor nesting SAT solvers based on lazy clause generation.We evaluate our framework in the context of Dung’s argumentation frameworks, logic programming, and propositionallogic subject to standard and non-standard semantics. Weshow for each of those formalisms that one can easily specify its semantics using a few second-order sentences and thatone can effectively obtain a solver for that semantics usingour automated solver generation procedure.For instance, in the case of argumentation frameworks, weobtain 16 different solvers, each solving one of four inference tasks for one of four major argumentation semantics andshow that our solvers (slightly) outperform the best solverfrom the last system competition despite not being tuned forargumentation instances.

Cause-effect relations are an important part of human knowledge. In real life, humans often reason about complex causes linked to complex effects. By comparison, existing formalisms for representing knowledge about causal relations are quite limited in the kind of specifications of causes and effects they allow. In this paper, we present the new language C-Log, which offers a significantly more expressive representation of effects, including such features as the creation of new objects. We show how C-Log integrates with first-order logic, resulting in the language FO(C). We also compare FO(C) with several related languages and paradigms, including inductive definitions, disjunctive logic programming, business rules and extensions of Datalog.