Bart Bogaerts

Predicate logic as a modeling language: Modeling and solving some machine learning and data mining problems with IDP3

This paper provides a gentle introduction to problem solving with the IDP3 system. The core of IDP3 is a finite model generator that supports first order logic enriched with types, inductive definitions, aggregates and partial functions. It offers its users a modeling language that is a slight extension of predicate logic and allows them to solve a wide range of search problems. Apart from a small introductory example, applications are selected from problems that arose within machine learning and data mining research. These research areas have recently shown a strong interest in declarative modeling and constraint solving as opposed to algorithmic approaches. The paper illustrates that the IDP3 system can be a valuable tool for researchers with such an interest. The first problem is in the domain of stemmatology, a domain of philology concerned with the relationship between surviving variant versions of text. The second problem is about a somewhat related problem within biology where phylogenetic trees are used to represent the evolution of species. The third and final problem concerns the classical problem of learning a minimal automaton consistent with a given set of strings. For this last problem, we show that the performance of our solution comes very close to that of a state-of-the art solution. For each of these applications, we analyze the problem, illustrate the development of a logic-based model and explore how alternatives can affect the performance.

Model Expansion in the Presence of Function Symbols Using Constraint Programming

The traditional approach to Model Expansion (MX) is to reduce the theory to a propositional language and apply a search algorithm to the resulting theory. Function symbols are typically replaced by predicate symbols representing the graph of the function, an operation that blows up the reduced theory. In this paper, we present an improved approach to handle function symbols in a ground-and-solve methodology, building on ideas from Constraint Programming. We do so in the context of FO(.)IDP, the knowledge representation language that extends First-Order Logic (FO) with, among others, inductive definitions, arithmetic and aggregates. An MX algorithm is developed, consisting of (i) a grounding algorithm for FO(.)^IDP, parametrised by the function symbols allowed to occur in the reduced theory, and (ii) a search algorithm for unrestricted, ground FO(.)^IDP. The ideas are implemented in the IDP knowledge-base system and experimental evaluation shows that both more compact groundings and improved search performance are obtained.

Grounded fixpoints and their applications in knowledge representation

In various domains of logic, researchers have made use of a similar intuition: that facts (or models) can be derived from the ground up. They typically phrase this intuition by saying, e.g., that the facts should be grounded, or that they should not be unfounded, or that they should be supported by cycle-free arguments, et cetera. In this paper, we formalise this intuition in the context of algebraical fixpoint theory. We define when a lattice element x ? L is grounded for lattice operator O : L ? L . On the algebraical level, we investigate the relationship between grounded fixpoints and the various classes of fixpoints of approximation fixpoint theory, including supported, minimal, Kripke-Kleene, stable and well-founded fixpoints. On the logical level, we investigate groundedness in the context of logic programming, autoepistemic logic, default logic and argumentation frameworks. We explain what grounded points and fixpoints mean in these logics and show that this concept indeed formalises intuitions that existed in these fields. We investigate which existing semantics are grounded. We study the novel semantics for these logics that is induced by grounded fixpoints, which has some very appealing properties, not in the least its mathematical simplicity and generality. Our results unveil a remarkable uniformity in intuitions and mathematics in these fields.

Improved static symmetry breaking for SAT

An effective SAT preprocessing technique is the construction of symmetry breaking formulas: auxiliary clauses that guide a SAT solver away from needless exploration of symmetric subproblems. However, during the past decade, state-of-the-art SAT solvers rarely incorporated symmetry breaking. This suggests that the reduction of the search space does not outweigh the overhead incurred by detecting symmetry and constructing symmetry breaking formulas. We investigate three methods to construct more effective symmetry breaking formulas. The first method simply improves the encoding of symmetry breaking formulas. The second detects special symmetry subgroups, for which complete symmetry breaking formulas exist. The third infers binary symmetry breaking clauses for a symmetry group as a whole rather than longer clauses for individual symmetries. We implement these methods in a symmetry breaking preprocessor, and verify their effectiveness on both hand-picked problems as well as the 2014 SAT competition benchmark set. Our experiments indicate that our symmetry breaking preprocessor improves the current state-of-the-art in static symmetry breaking for SAT and has a sufficiently low overhead to improve the performance of modern SAT solvers on hard combinatorial instances.

Simulating dynamic systems using linear time calculus theories

Dynamic systems play a central role in fields such as planning, verification, and databases. Fragmented throughout these fields, we find a multitude of languages to formally specify dynamic systems and a multitude of systems to reason on such specifications. Often, such systems are bound to one specific language and one specific inference task. It is troublesome that performing several inference tasks on the same knowledge requires translations of your specification to other languages. In this paper we study whether it is possible to perform a broad set of well-studied inference tasks on one specification. More concretely, we extend IDP 3 with several inferences from fields concerned with dynamic specifications.

Symmetry Propagation: Improved Dynamic Symmetry Breaking in SAT

For constraint programming, many well performing dynamic symmetry breaking techniques have been devised. For propositional satisfiability solving, dynamic symmetry breaking is still either slower or less general than static symmetry breaking. This paper presents Symmetry Propagation, which is an improvement to Lightweight Dynamic Symmetry Breaking, a dynamic symmetry breaking approach from CP. Symmetry Propagation uses any given symmetry as a propagator, and as a result is a general symmetry breaking technique. Experiments with an implementation in the SAT solver Minisat show that on many benchmarks, Symmetry Propagation outperforms the state-of-the-art static symmetry breaking method Shatter.

Stable-unstable semantics: Beyond NP with normal logic programs

Standard answer set programming (ASP) targets at solving search problems from the first level of the polynomial time hierarchy (PH). Tackling search problems beyond NP using ASP is less straightforward. The class of disjunctive logic programs offers the most prominent way of reaching the second level of the PH, but encoding respective hard problems as disjunctive programs typically requires sophisticated techniques such as saturation or meta-interpretation. The application of such techniques easily leads to encodings that are inaccessible to non-experts. Furthermore, while disjunctive ASP solvers often rely on calls to a (co-)NP oracle, it may be difficult to detect from the input program where the oracle is being accessed. In other formalisms, such as Quantified Boolean Formulas (QBFs), the interface to the underlying oracle is more transparent as it is explicitly recorded in the quantifier prefix of a formula. On the other hand, ASP has advantages over QBFs from the modeling perspective. The rich high-level languages such as ASP-Core-2 offer a wide variety of primitives that enable concise and natural encodings of search problems. In this paper, we present a novel logic programming–based modeling paradigm that combines the best features of ASP and QBFs. We develop so-called combined logic programs in which oracles are directly cast as (normal) logic programs themselves. Recursive incarnations of this construction enable logic programming on arbitrarily high levels of the PH. We develop a proof-of-concept implementation for our new paradigm.

Modeling Machine Learning and Data Mining Problems with FO(

This paper reports on the use of the FO(·) language and the IDP framework for modeling and solving some machine learning and data mining tasks. The core component of a model in the IDP framework is an FO(·) theory consisting of formulas in first order logic and definitions; the latter are basically logic programs where clause bodies can have arbitrary first order formulas. Hence, it is a small step for a well-versed computer scientist to start modeling. We describe some models resulting from the collaboration between IDP experts and domain experts solving machine learning and data mining tasks. A first task is in the domain of stemmatology, a domain of philology concerned with the relationship between surviving variant versions of text. A second task is about a somewhat similar problem within biology where phylogenetic trees are used to represent the evolution of species. A third and final task is about learning a minimal automaton consistent with a given set of strings. For each task, we introduce the problem, present the IDP code and report on some experiments.

Symmetric Explanation Learning: Effective Dynamic Symmetry Handling for SAT

The presence of symmetry in Boolean satisfiability (SAT) problem instances often poses challenges to solvers. Currently, the most effective approach to handle symmetry is by static symmetry breaking, which generates asymmetric constraints to add to the instance. An alternative way is to handle symmetry dynamically during solving. As modern SAT solvers can be viewed as propositional proof generators, adding a symmetry rule in a solver’s proof system would be a straightforward technique to handle symmetry dynamically. However, none of these proposed symmetrical learning techniques are competitive to static symmetry breaking. In this paper, we present symmetric explanation learning, a form of symmetrical learning based on learning symmetric images of explanation clauses for unit propagations performed during search. A key idea is that these symmetric clauses are only learned when they would restrict the current search state, i.e., when they are unit or conflicting. We further provide a theoretical discussion on symmetric explanation learning and a working implementation in a state-of-the-art SAT solver. We also present extensive experimental results indicating that symmetric explanation learning is the first symmetrical learning scheme competitive with static symmetry breaking.

Safe Inductions: An Algebraic Study

In many knowledge representation formalisms, a constructive semantics is defined based on sequential applications of rules or of a semantic operator. These constructions often share the property that rule applications must be delayed until it is safe to do so: until it is known that the condition that triggers the rule will remain to hold. This intuition occurs for instance in the well-founded semantics of logic programs and in autoepistemic logic. In this paper, we formally define the safety criterion algebraically. We study properties of so-called safe inductions and apply our theory to logic programming and autoepistemic logic. For the latter, we show that safe inductions manage to capture the intended meaning of a class of theories on which all classical constructive semantics fail.