A Categorical View on Algebraic Lattices in Formal Concept Analysis

Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched with notions such as approximation by computation or representability. The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices. In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective. To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices. At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.Comment: 36 page

On cascade products of answer set programs

Describing complex objects by elementary ones is a common strategy in mathematics and science in general. In their seminal 1965 paper, Kenneth Krohn and John Rhodes showed that every finite deterministic automaton can be represented (or "emulated") by a cascade product of very simple automata. This led to an elegant algebraic theory of automata based on finite semigroups (Krohn-Rhodes Theory). Surprisingly, by relating logic programs and automata, we can show in this paper that the Krohn-Rhodes Theory is applicable in Answer Set Programming (ASP). More precisely, we recast the concept of a cascade product to ASP, and prove that every program can be represented by a product of very simple programs, the reset and standard programs. Roughly, this implies that the reset and standard programs are the basic building blocks of ASP with respect to the cascade product. In a broader sense, this paper is a first step towards an algebraic theory of products and networks of nonmonotonic reasoning systems based on Krohn-Rhodes Theory, aiming at important open issues in ASP and AI in general.Comment: Appears in Theory and Practice of Logic Programmin

Manufacturing a mathematical group: a study in heuristics

I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forwar

Memoization for Unary Logic Programming: Characterizing PTIME

We give a characterization of deterministic polynomial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. More precisely, we study the restriction of this framework to terms (and logic programs, rewriting rules) using only unary symbols. We prove it is complete for polynomial time computation, using an encoding of pushdown automata. We then introduce an algebraic counterpart of the memoization technique in order to show its PTIME soundness. We finally relate our approach and complexity results to complexity of logic programming. As an application of our techniques, we show a PTIME-completeness result for a class of logic programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201

Maude: specification and programming in rewriting logic

Maude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude's language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude