String-based Multi-adjoint Lattices for Tracing Fuzzy Logic Computations

Classically, most programming languages use in a predefined way thenotion of “string” as an standard data structure for a comfortable management of arbitrary sequences of characters. However, in this paper we assign a different role to this concept: here we are concerned with fuzzy logic programming, a somehow recent paradigm trying to introduce fuzzy logic into logic programming. In this setting, the mathematical concept of multi-adjoint lattice has been successfully exploited into the so-called Multi-adjoint Logic Programming approach, MALP in brief, for modeling flexible notions of truth-degrees beyond the simpler case of true and false. Our main goal points out not only our formal proof verifying that stringbased lattices accomplish with the so-called multi-adjoint property (as well as its Cartesian product with similar structures), but also its correspondence with interesting debugging tasks into the FLOPER system (from “Fuzzy LOgic Programming Environment for Research”) developed in our research group

A Fuzzy Logic Programming Environment for Managing Similarity and Truth Degrees

FASILL (acronym of "Fuzzy Aggregators and Similarity Into a Logic Language") is a fuzzy logic programming language with implicit/explicit truth degree annotations, a great variety of connectives and unification by similarity. FASILL integrates and extends features coming from MALP (Multi-Adjoint Logic Programming, a fuzzy logic language with explicitly annotated rules) and Bousi~Prolog (which uses a weak unification algorithm and is well suited for flexible query answering). Hence, it properly manages similarity and truth degrees in a single framework combining the expressive benefits of both languages. This paper presents the main features and implementations details of FASILL. Along the paper we describe its syntax and operational semantics and we give clues of the implementation of the lattice module and the similarity module, two of the main building blocks of the new programming environment which enriches the FLOPER system developed in our research group.Comment: In Proceedings PROLE 2014, arXiv:1501.0169

From IF to BI: a tale of dependence and separation

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Vaananen, and their compositional semantics due to Hodges. We show how Hodges' semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O'Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural role, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio

Completeness of Flat Coalgebraic Fixpoint Logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The mu-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL, and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus and the alternating-time mu-calculus (such as alternating-time temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer Science, Springer, 2010, pp. 524-53

Data types with symmetries and polynomial functors over groupoids

Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect is often prominent and justifies the terminology. For example, Tambara's theorem states that the category of finite polynomial functors is the Lawvere theory for commutative semirings. In this talk I will explain how an upgrade of the theory from sets to groupoids is useful to deal with data types with symmetries, and provides a common generalisation of and a clean unifying framework for quotient containers (cf. Abbott et al.), species and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan. The multi-variate setting also includes relations and spans, multispans, and stuff operators. An attractive feature of this theory is that with the correct homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits, etc. - the groupoid case looks exactly like the set case. After some standard examples, I will illustrate the notion of data-types-with-symmetries with examples from quantum field theory, where the symmetries of complicated tree structures of graphs play a crucial role, and can be handled elegantly using polynomial functors over groupoids. (These examples, although beyond species, are purely combinatorial and can be appreciated without background in quantum field theory.) Locally cartesian closed 2-categories provide semantics for 2-truncated intensional type theory. For a fullfledged type theory, locally cartesian closed \infty-categories seem to be needed. The theory of these is being developed by D.Gepner and the author as a setting for homotopical species, and several of the results exposed in this talk are just truncations of \infty-results obtained in joint work with Gepner. Details will appear elsewhere.Comment: This is the final version of my conference paper presented at the 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer Science. 16p