Maria A. Galán

Similarities between powersets of terms

Generalisation of the foundational basis for many-valued logic programming builds upon generalised terms in the form of powersets of terms. A categorical approach involving set and term functors as monads allows for a study of monad compositions that provide variable substitutions and compositions thereof. In this paper, substitutions and unifiers appear as constructs in Kleisli categories related to particular composed powerset term monads. Specifically, we show that a frequently used similarity-based approach to fuzzy unification is compatible with the categorical approach, and can be adequately extended in this setting; also some examples are included in order to illuminate the definitions.

Monads can be rough

Traditionally, rough sets build upon relations based on ordinary sets, i.e. relations on X as subsets of X×X. A starting point of this paper is the equivalent view on relations as mappings from X to the (ordinary) power set PX. Categorically, P is a set functor, and even more so, it can in fact be extended to a monad (P,η,μ). This is still not enough and we need to consider the partial order (PX,≤). Given this partial order, the ordinary power set monad can be extended to a partially ordered monad. The partially ordered ordinary power set monad turns out to contain sufficient structure in order to provide rough set operations. However, the motivation of this paper goes far beyond ordinary relations as we show how more general power sets, i.e. partially ordered monads built upon a wide range of set functors, can be used to provide what we call rough monads.

Partially Ordered Monads for Monadic Topologies, Rough Sets and Kleene Algebras

In this paper we will show that partially ordered monads contain sufficient structure for modelling monadic topologies, rough sets and Kleene algebras. Convergence represented by extension structures over partially ordered monads includes notions of regularity and compactness. A compactification theory can be developed. Rough sets [Z. Pawlak, Rough sets, Int. J. Computer and Information Sciences 5 (1982) 341356] are modelled in a generalized setting with set functors. Further, we show how partially ordered monads can be used in order to obtain monad based examples of Kleene algebras building upon a wide range of set functors far beyond just strings [S. C. Kleene, Representation of events in nerve nets and finite automata, In: Automata Studies (Eds. C. E. Shannon, J. McCarthy), Princeton University Press, 1956, 3-41] and relations [A. Tarski, On the calculus of relations, J. Symbolic Logic 6 (1941), 65-106].

Powersets of terms and composite monads

Composing various powerset functors with the term monad gives rise to the concept of generalized terms. This in turn provides a technique for handling many-valued sets of terms in a framework of variable substitutions, thus being the prerequisite for categorical unification in many-valued logic programming using an extended notion of terms. As constructions of monads involve complicated calculations with natural transformations, proofs are supported by a graphical approach that provides a useful tool for handling various conditions, such as those well known for distributive laws.

A categorical approach to unification of generalised terms

Unification of generalised terms in a many-valued setting involves considerations for equalities in the sense of similarity degrees between operators and thus similarities between terms. Further, allowing for substitutions of variables with powersets of terms requires ‘flattening’ operators for handling composition of variable substitutions. These techniques are available when using powerset functors composed with the term functor so that this composition of functors is extendable to a monad. In this paper we provide a framework for unification of such generalised terms.

A graphical approach to monad compositions

In this paper we show how composite expressions involving natural transformations can be pictorially represented in order to provide graphical proof support for providing monad compositions. Exampl ...

Paradigms for Non-classical Substitutions

We will present three paradigms for non-classical substitution. Firstly, we have the classical substitution of variables with terms. This is written in a strict categorical form supporting presentation of the other two paradigms. The second paradigm is substitutions of variables with many-valued sets of terms. These two paradigms are based on functors and monads over the category of sets. The third paradigm is the substitution of many-valued sets of variables with terms over many-valued sets of variables. The latter is based on functors and monads over the category of many-valued sets. This provides a transparency of the underlying categories and also makes a clear distinction between set-theoretic operation in the meta language and operations on sets and many-valued sets as found within respective underlying categories.

The Rough Powerset Monad

Rough sets provide a good environment to deal with vagueness and uncertainty situations. In this paper we show how monads can be used to generalize and interpret rough situations. In particular, the partially ordered ordinary power set monad turns out to contain sufficient structure in order to provide rough set operations.

Categorical innovations for rough sets

Categories arise in mathematics and appear frequently in computer science where algebraic and logical notions have powerful representations using categorical constructions. In this chapter we lean towards the functorial view involving natural transformations and monads. Functors extendable to monads, further incorporating order structure related to the underlying functor, turn out to be very useful when presenting rough sets beyond relational structures in the usual sense. Relations can be generalized with rough set operators largely maintaining power and properties. In this chapter we set forward our required categorical tools and we show how rough sets and indeed a theory of rough monads can be developed. These rough monads reveal some canonic structures, and are further shown to be useful in real applications as well. Information within pharmacological treatment can be structured by rough set approaches. In particular, situations involving management of drug interactions and medical diagnosis can be described and formalized using rough monads.

A logic with imprecise probabilities and an application to automated reasoning using rewriting techniques

A new probabilistic logic for handling imprecise probabilities is introduced, implemented in a rewriting system, and used to carry out an experiment. Each well-formed formula of the probabilistic logic is labeled with two values that represent possible minimum and maximum probabilities associated with the event related to the unlabeled formula. The aim of this logic is to facilitate application of rules to obtain an approximation of the probability interval associated with an event, without the necessity of knowing the precise probability of other events. The logic is described as a formal theory by means of its language, semantics and a proof theory. The soundness of the proof theory has been proven. Rewriting techniques are a powerful method for testing the behavior of a formal proof calculus through translation of calculus inference rules into rewrite rules. Implementation of the logic rules in a rewriting language such as Maude allows an automated reasoning system to be easily obtained, which can be consulted by applications. We developed a small game application and an experiment to test the application.