Broes De Cat

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.

Lazy model expansion: interleaving grounding with search

Finding satisfying assignments for the variables involved in a set of constraints can be cast as a (bounded) model generation problem: search for (bounded) models of a theory in some logic. The state-of-the-art approach for bounded model generation for rich knowledge representation languages like Answer Set Programming (ASP) and FO(ċ) and a CSP modeling language such as Zinc, is ground-and-solve: reduce the theory to a ground or propositional one and apply a search algorithm to the resulting theory. An important bottleneck is the blow-up of the size of the theory caused by the grounding phase. Lazily grounding the theory during search is a way to overcome this bottleneck. We present a theoretical framework and an implementation in the context of the FO(ċ) knowledge representation language. Instead of grounding all parts of a theory, justifications are derived for some parts of it. Given a partial assignment for the grounded part of the theory and valid justifications for the formulas of the non-grounded part, the justifications provide a recipe to construct a complete assignment that satisfies the non-grounded part. When a justification for a particular formula becomes invalid during search, a new one is derived; if that fails, the formula is split in a part to be grounded and a part that can be justified. Experimental results illustrate the power and generality of this approach.

Lazy Model Expansion by Incremental Grounding

Ground-and-solve methods used in state-of-the-art Answer Set Programming and model expansion systems proceed by rewriting the problem specification into a ground format and afterwards applying search. A disadvantage of such approaches is that the rewriting step blows up the original specification for large input domains and is unfeasible in case of infinite domains. In this paper we describe a lazy approach to model expansion in the context of first-order logic that can cope with large and infinite problem domains. The method interleaves grounding and search, incrementally extending the current partial grounding only when necessary. It often allows to solve the original problem without creating the full grounding and is hence more widely applicable than ground-and-solve. We report on an existing implementation within the IDP system and on experiments that show the promise of the method.

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.

Detection and exploitation of functional dependencies for model generation

Recent work in Answer Set Programming has integrated ideas from Constraint Programming. This has led to a new eld called ASP Modulo CSP (CASP), in which the ASP language is enriched with constraint atoms representing constraint satisfaction problems. These constraints have a more compact grounding and are handled by a new generation of search algorithms. However, the burden is on the modeler to exploit these new constructs in his declarative problem speci cations. Here, we explore how to remove this burden by automatically generating constraint atoms. We do so in the context of FO(·), a knowledge representation language that extends rst-order logic with, among others, inductive de nitions, arithmetic and aggregates. We uncover functional dependencies in declarative problem speci cations with a theorem prover and exploit them with a transformation that introduces functions. Experimental evaluation shows that we obtain more compact groundings and better search performance.

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.

Fo(fd): Extending classical logic with rule-based fixpoint definitions

We introduce fixpoint definitions, a rule-based reformulation of fixpoint constructs. The logic FO(FD), an extension of classical logic with fixpoint definitions, is defined. We illustrate the relation between FO(FD) and FO(ID), which is developed as an integration of two knowledge representation paradigms. The satisfiability problem for FO(FD) is investigated by first reducing FO(FD) to difference logic and then using solvers for difference logic. These reductions are evaluated in the computation of models for FO(FD) theories representing fairness conditions and we provide potential applications of FO(FD).

Towards computing revised models for FO theories

In many real-life computational search problems, one is not only interested in finding a solution, but also in maintaining a solution under varying circumstances. For example, in the area of network configuration, an initial configuration of a computer network needs to be obtained, but also a new configuration when one of the machines in the network breaks down. Currently, most such revision problems are solved manually, or with highly specialized software. A recent declarative approach to solve (hard) computational search problems involving a lot of domain knowledge, is by finite model generation. Here, the domain knowledge is specified as a logic theory T, and models of T correspond to solutions of the problem. In this paper, we extend this approach to solve revision problems. In particular, our method allows to use the same theory to describe the search problem and the revision problem, and applies techniques from current model generators to find revised solutions.