A substitution is a mapping from variables to terms. There is a consensus in the Logic Programming community that the work with substitutions is a source of errors due to their poor algebraic properties. This problem increases when we manipulate substitutions in the framework of similarity relations, where some classical concepts have more complex definitions and the intuition of the result of the operations involving them is easily lost. In this work we analyze some algebraic properties of substitutions in the framework of similarity relations. Specifically, we study the lattice structure of the set of idempotent substitutions with the definition of the weak parallel composition and weak parallel factorization operations, acting as the join and meet of the lattice. Also we relate these operations with the classical operation of composition of substitutions. The aim is to provide fuzzy logic programming researchers with reasoning tools with richer properties that facilitate formal proofs and, even, able to define compositional and parallel semantics.

Unification is a central concept in logic systems based on the resolution principle. As well, in knowledge representation, proximity relations (i.e., reflexive, symmetric, fuzzy binary relations) are useful for introducing semantics into a syntactic level by modeling the semantic closeness of different syntactic objects and managing vague or imprecise information. Proximity relations, in combination with the unification algorithm, make possible expressing certain forms of approximate reasoning in a logic programming framework. In this article, we use proximity relations in the context of a (fuzzy) logic programming system, called Bousi ~ Prolog, as a way of solving the limitations introduced by similarity relations (i.e., transitive proximity relations) to correctly represent fuzzy information. Recently, we introduced an accurate definition of proximity between expressions (terms or atomic formulas) and a new unification algorithm able to manage proximity relations properly. However, the so-called weak unification algorithm, which is an extension of Martelli and Montanari's unification algorithm supported by the new notion of proximity, does not have an efficient implementation. In this article, we present a method that facilitates such an efficient implementation, including an adaptation of the weak SLD resolution rule based on the new unification algorithm, and its integration and implementation into the fuzzy logic programming system Bousi ~ Prolog. A performance analysis to show its efficiency is also presented.

Abstract This paper introduces techniques to integrate WordNet into a Fuzzy Logic Programming system. Since WordNet relates words but does not give graded information on the relation between them, we have implemented standard similarity measures and new directives allowing the proximity equations linking two words to be generated with an approximation degree. Proximity equations are the key syntactic structures which, in addition to a weak unification algorithm, make a flexible query-answering process possible in this kind of programming language. This addition widens the scope of Fuzzy Logic Programming, allowing certain forms of lexical reasoning, and reinforcing Natural Language Processing (NLP) applications.

Abstract This paper explores the integration of hypothetical reasoning into an efficient implementation of the fuzzy logic language Bousi∼Prolog. To this end, we first analyse what would be expected from a logic inference system, equipped with what is called embedded implication, to model solving goals with respect to assumptions. We start with a propositional system and incrementally build more complex systems and implementations to satisfy the requirements imposed by a system like Bousi∼Prolog. Finally, we propose an inference system, operational semantics and the translation function to generate efficient Prolog programmes from Bousi∼Prolog programmes.

This paper presents a system implementing a novel addition to a fuzzy deductive database: hypothetical queries. Such queries allow users to dynamically make assumptions on a given database instance, either by adding or removing data, without changing the instance. Further, since a fuzzy database includes fuzzy relations, these relations can also be changed with assumptions. This ability for dynamic change seamlessly enables writing "what-if" applications such as decision-support systems. Here, the new language Fuzzy Hypothetical Datalog is presented, along with an operational semantics and stratified inference. It has been implemented in a working system DES readily available on-line.

The FASILL programming language (acronym of "Fuzzy Aggregators and Similarity Into a Logic Language") combines a weak unification algorithm, based on similarity relations, along with a rich repertoire of fuzzy connectives and aggregators, whose truth functions can be defined on a complete lattice. In this paper, after recalling the operational semantics of this language, we provide a notion of fuzzy correct answer for a program and a goal and we prove the soundness of FASILL for this operational semantics and the notion of correct answer introduced.

Unification is a central concept in deductive systems based on the resolution principle. Recently, we introduced a newweak unification algorithmbased on proximity relations (i.e., reflexive, symmetric, fuzzy binary relations). Proximity relations are able to manage vague or imprecise information and, in combination with the unification algorithm, allow certain forms of approximate reasoning in a logic programming framework. In this paper, we present a reformulation of the weak unification algorithm and an elaborated method to implement it efficiently.

This paper describes a proposal for a deductive database system with fuzzy Datalog as its query language. Concepts supporting the fuzzy logic programming system Bousi~Prolog are tailored to the needs of the deductive database system DES. We develop a version of fuzzy Datalog where programs and queries are compiled to the DES core Datalog language. Weak unification and weak SLD resolution are adapted for this setting, and extended to allow rules with truth degree annotations. We provide a public implementation in Prolog, which is open source, multiplatform, portable, and in-memory, featuring a graphical user interface. A distinctive feature of this system is that, unlike others, we have formally demonstrated that our implementation techniques fit the proposed operational semantics. We also study the efficiency of these implementation techniques through a series of detailed experiments. Moreover, a database example for a recommender system is used to illustrate some of the features of the system and its usefulness.

This article describes the implementation of the fuzzy deductive database system FuzzyDES, where concepts underlying the fuzzy logic programming system Bousi~Prolog are adapted and improved to be transferred to the DES deductive database system. We take advantage of the DES tabled-based implementation to propose new methods for rule compiling and t-closure computing, developing a terminating query answering system with graded rules. A description of the system, an example, and a link to a publicly-available, comprehensive system are provided.

This paper aims to propose a model of reasoning based on semantic relations among words and to incorporate it in the inference mechanism of a logic programming language. This model is integrated in a fuzzy logic programming framework and it is implemented into the Bousi~Prolog system by using WordNet. All this process is transparent to the programmer and the reasoning with words is automatic. The lexical semantics between symbols (words) provides us with the ability of reasoning with words and turns the knowledge representation in a more natural and less complex process.

Bousi~Prolog is a fuzzy logic programming language whose main objective is to make flexible the query answering process. Its operational mechanism is a extension of the SLD-resolution (called weak resolution) where the classical syntactic unification algorithm has been replaced by a fuzzy one. This paper presents a generic method for the unification of linguistic terms (i.e. fuzzy sets) which is also applicable to other programming languages with an operational semantics based on some kind of weak resolution mechanism. The basic idea is to compile the information provided by fuzzy sets, generating a binary fuzzy relation on the set of their associated linguistic labels. Subsequently, this fuzzy relation can be used in a standard, completely integrated way inside the unification mechanism of the Bousi~Prolog system, what allows us to handle linguistic labels on an equal basis with regard other syntactic symbols occurring in the source program. This is a novel approach because it is the first time that fuzzy sets are introduced into the core of a Prolog system by means of compilation techniques and combining fuzzy relations with weak unification. An important feature of this approach is its simplicity, since the inclusion is carried out in a very natural way without affecting the operational semantics of the Bousi~Prolog language and with very few syntactical modifications. All these reasons convert our approach in a good alternative to the techniques used by other fuzzy Prolog systems.

Bousi~Prolog is a fuzzy logic programming language with an operational semantics which is an adaptation of the SLD resolution principle, where classical unification has been replaced by a fuzzy unification algorithm based on proximity relations. Hence, it is a programming language well suited for dealing with uncertainty and approximate reasoning. There are several practical applications where Bousi~Prolog can be useful: flexible query answering; advanced pattern matching; information retrieval where textual information is selected or analyzed using an ontology; text cataloging and analysis; etc. In this paper we give a model-theoretic semantics for a pure subset of this language: we formalize the notion of least fuzzy Herbrand model as the declarative semantics for definite programs. We prove various important properties of these models. Finally we define an immediate consequences operator, which is proved monotonous and continuous, obtaining a fixpoint characterization of the least fuzzy Herbrand model.

Needed narrowing is a complete and optimal operational principle for modern declarative languages which integrate the best features of lazy functional and logic programming. We investigate the formal relation between needed narrowing and another (not so lazy) narrowing strategy which is the basis for popular implementations of lazy functional logic languages. We demonstrate that needed narrowing and lazy narrowing are computationally equivalent over the class of uniform programs. We also introduce a complete refinement of lazy narrowing, called uniform lazy narrowing, which is still equivalent to needed narrowing over the aforementioned class. Since actual implementations of functional logic languages are based on the transformation of the original program into a uniform one—which is then executed using a lazy narrowing strategy—our results can be thought of as a formal basis for the correctness of these implementations.

Partial evaluation is a method for program specialization based on fold/unfold transformations [8, 25]. Partial evaluation of pure functional programs uses mainly static values of given data to specialize the program [15, 44]. In logic programming, the so-called static/dynamic distinction is hardly present, whereas considerations of determinacy and choice points are far more important for control [12]. We discuss these issues in the context of a (lazy) functional logic language. We formalize a two-phase specialization method for a non-strict, first order, integrated language which makes use of lazy narrowing to specialize the program w.r. t. a goal. The basic algorithm (first phase) is formalized as an instance of the framework for the partial evaluation of functional logic programs of [2, 3], using lazy narrowing. However, the results inherited by [2, 3] mainly regard the termination of the PE method, while the (strong) soundness and completeness results must be restated for the lazy strategy. A post-processing renaming scheme (second phase) is necessary which we describe and illustrate on the well-known matching example. This phase is essential also for other non-lazy narrowing strategies, like innermost narrowing, and our method can be easily extended to these strategies. We show that our method preserves the lazy narrowing semantics and that the inclusion of simplification steps in narrowing derivations can improve control during specialization.