LNAI 9868 Oscar Luaces · José A Gámez Edurne Barrenechea · Alicia Troncoso Mikel Galar · Héctor Quintián Emilio Corchado (Eds.) Advances in Artificial Intelligence 17th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2016 Salamanca, Spain, September 14–16, 2016, Proceedings 123 Lecture Notes in Artificial Intelligence Subseries of Lecture Notes in Computer Science LNAI Series Editors Randy Goebel University of Alberta, Edmonton, Canada Yuzuru Tanaka Hokkaido University, Sapporo, Japan Wolfgang Wahlster DFKI and Saarland University, Saarbrücken, Germany LNAI Founding Series Editor Joerg Siekmann DFKI and Saarland University, Saarbrücken, Germany 9868 More information about this series at http://www.springer.com/series/1244 Oscar Luaces José A Gámez Edurne Barrenechea Alicia Troncoso Mikel Galar Héctor Quintián Emilio Corchado (Eds.) • • • Advances in Artificial Intelligence 17th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2016 Salamanca, Spain, September 14–16, 2016 Proceedings 123 Editors Oscar Luaces Artificial Intelligence Center University of Oviedo Gijón Spain José A Gámez University of Castilla-La Mancha Albacete Spain Edurne Barrenechea Public University of Navarre Pamplona Spain Mikel Galar Public University of Navarre Pamplona, Navarra Spain Héctor Quintián University of Salamanca Salamanca Spain Emilio Corchado University of Salamanca Salamanca Spain Alicia Troncoso Universidad Pablo de Olavide Sevilla Spain ISSN 0302-9743 ISSN 1611-3349 (electronic) Lecture Notes in Artificial Intelligence ISBN 978-3-319-44635-6 ISBN 978-3-319-44636-3 (eBook) DOI 10.1007/978-3-319-44636-3 Library of Congress Control Number: 2016938377 LNCS Sublibrary: SL7 – Artificial Intelligence © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Preface This volume contains a selection of the papers accepted for oral presentation at the 17th Conference of the Spanish Association for Artificial Intelligence (CAEPIA 2016), held in Salamanca (Spain), during September 14–16, 2016 This was the 17th biennial conference in the CAEPIA series, which was started in 1985 Previous events took place in Madrid, Alicante, Málaga, Murcia, Gijón, Donostia, Santiago de Compostela, Salamanca, Seville, La Laguna, Madrid, and Albacete This time, CAEPIA was coordinated with various symposia, each one corresponding to a main track in Artificial Intelligence (AI) research: 11th Symposium on Metaheuristics, Evolutive and Bioinspired Algorithms (MAEB); 6th Symposium of Fuzzy Logic and Soft Computing (LODISCO); 8th Symposium of Theory and Applications of Data Mining (TAMIDA); and the 3rd Symposium on Information Fusion and Ensembles (FINO) CAEPIA is a forum open to researchers worldwide, to present and discuss the latest scientific and technological advances in AI Its main aims are to facilitate the dissemination of new ideas and experiences, to strengthen the links among the different research groups, and to help spread new developments to society All perspectives — theory, methodology, and applications — are welcome Apart from the presentation of technical full papers, the scientific program of CAEPIA 2016 included an App contest, a Doctoral Consortium and, as a follow-up to the success achieved at previous CAEPIA conferences, a special session on outstanding recent papers (Key Works) already published in renowned journals or forums With the aim of maintaining CAEPIA as a high-quality conference, and following the model of current demanding AI conferences, the CAEPIA review process runs under the double-blind model The number of submissions received by CAEPIA and associated tracks was 166; however, only 47 submissions were selected to be published in the LNAI Springer volume These 47 papers were carefully peer-reviewed by three members of the CAEPIA Program Committee with the help of additional reviewers from each of the associated symposia The reviewers judged the overall quality of the submitted papers, together with their originality and novelty, technical correctness, awareness of related work, and quality of presentation The reviewers stated their confidence in the subject area in addition to detailed written comments On the basis of the reviews, the program chairs made the final decisions The six distinguished invited speakers at CAEPIA 2016 were Serafín Moral (University of Granada, Spain), Xin Yao (University of Birmingham, UK), Enrique Alba Torres (University of Málaga, Spain), Sancho Salcedo Sanz (University of Alcalá de Henares, Spain), Richard Benjamins (BI & DATA, Telefonica, Spain), and Alberto Bugarín Diz (University of Santiago de Compostela, Spain) They presented six very interesting topics on current AI research: “Algoritmos de Inferencia Aproximados para Modelos Gráficos Probabilísticos” (Moral), “Ensemble Approaches to Class Imbalance Learning” (Yao), “Sistemas Inteligentes para Ciudades Inteligentes” (Alba Torres), VI Preface “Nuevos Algoritmos para Optimización y Búsqueda Basados en Simulación de Arrecifes de Coral” (Salcedo), “Creating Value from Big Data” (Benjamins), and “A Bunch of Words Worth More than a Million Data: A Soft Computing View of Data-to-Text” (Bugarín) The Doctoral Consortium (DC) was specially designed for the interaction between PhD students and senior researchers AEPIA and the organization of CAEPIA recognized the best PhD work submitted to the DC with a prize, as well as the best student and conference paper presented at CAEPIA 2016 Furthermore, and with the aim of promoting the presence of women in AI research, as in previous editions, a prize was set at CAEPIA 2016: the Frances Allen award, which is devoted to the two best AI PhD Thesis presented by a woman during the last two years The editors would like to thank everyone who contributed to CAEPIA 2016 and associated events: authors, members of the Scientific Committees, additional reviewers, invited speakers, etc Final thanks go to the Organizing Committee, our local sponsors (BISITE and the University of Salamanca), the Springer team, and AEPIA for their support September 2016 Oscar Luaces José A Gámez Edurne Barrenechea Alicia Troncoso Mikel Galar Héctor Quintián Emilio Corchado Organization General Chairs Oscar Luaces Emilio Corchado University of Oviedo at Gijón, Spain Univesity of Salamanca, Spain Program Chairs Co-chair of MAEB Francisco Herrera José A Gámez University of Granada, Spain University of Castilla-La Mancha, Spain Co-chair of LODISCO Luis Martínez Edurne Barrenechea University of Jaen, Spain Public University of Navarre, Spain Co-chair of TAMIDA José Riquelme Alicia Troncoso University of Seville, Spain Universidad Pablo de Olivine, Spain Co-chair of FINO Emilio Corchado Mikel Galar Bruno Baruque University of Salamanca, Spain Public University of Navarre, Spain University of Burgos, Spain Program Committee Jesús S Aguilar-Ruiz Pedro Aguilera Aguilera Enrique Alba Rafael Alcala Jesus Alcala-Fdez Francisco Almeida Amparo Alonso-Betanzos Ada Álvarez Ramón Álvarez-Valdés Alessandro Antonucci University Pablo de Olavide, Spain University of Almería, Spain University of Málaga, Spain University of Granada, Spain University of Granada, Spain University of La Laguna, Spain University of A Cora, Spain Universidad Autónoma de Nuevo Ln, Spain University of Valencia, Spain IDSIA, Switzerland VIII Organization Lourdes Araujo Olatz Arbelaitz Marta Arias Ángel Arroyo Gualberto Asencio Jaume Bacardit Emili Balaguer-Ballester Edurne Barrenechea Senén Barro Bruno Baruque Iluminada Baturone Joaquín Bautista José Manuel Benítez Pablo Bermejo Concha Bielza Lozoya Christian Blum Fernando Bobillo Daniel Borrajo Julio Brito Alberto Bugarín Humberto Bustince Pedro Cabalar Rafael Caballero José M Cadenas Tomasa Calvo Jose Luis Calvo-Rolle David Camacho Vicente Campos Andrés Cano Cristóbal Carmona Pablo Carmona Andre Carvalho Jorge Casillas José Luis Casteleiro Roca Pedro A Castillo Francisco Chávez Francisco Chicano Carlos A Coello José Manuel Colmenar Ángel Corberán Emilio Corchado Juan Manuel Corchado Oscar Cordón Carlos Cotta Inés Couso Javier Cózar UNED, Spain University of Ps Vasco, Spain Polytechnic University of Catala, Spain University of Burgos, Spain University Pablo de Olavide, Spain Newcastle University, UK Bournemouth University, UK Public University of Navarra, Spain University of Santiago de Compostela, Spain University of Burgos, Spain Instituto de Microelectrónica de Sevilla-CSIC, Spain Polytechnic University of Cataluña, Spain University of Granada, Spain University of Castilla-La Mancha, Spain Polytechnic University of Madrid, Spain IKERBASQUE, Spain University of Zaragoza, Spain University Carlos III de Madrid, Spain University of la Laguna, Spain University of Santiago de Compostela, Spain Public University of Navarra, Spain University of A Coruña, Spain University of Málaga, Spain University of Murcia, Spain University of Alcalá, Spain University of A Cora, Spain Universidad Autónoma de Madrid, Spain University of Valencia, Spain University of Granada, Spain University of Burgos, Spain University of Extremadura, Spain University of Saõ Paulo, Brazil University of Granada, Spain University of Coruña, Spain University of Granada, Spain University of Extremadura, Spain University of Málaga, Spain CINVESTAV – IPN, Spain Universidad Rey Juan Carlos, Spain University of Valencia, Spain University of Salamanca, Spain University of Salamanca, Spain University of Granada, Spain University of Málaga, Spain University of Oviedo, Spain University of Castilla-La Mancha, Spain Organization Leticia Curiel Sergio Damas Rocío de Andrés Calle Luis M de Campos Cassio De Campos Luis de la Ossa José del Campo Juan J del Coz María José del Jesús Irene Díaz Julián Dorado Bernabé Dorronsoro Abraham Duarte Richard Duro Thomas Dyhre Nielsen José Egea Francisco Javier Elorza Sergio Escalera Anna Esparcia Francesc Esteva Javier Faulín Francisco Fernández Alberto Fernández Antonio J Fernández Elena Fernández Javier Fernandez Alberto Fernández Hilario Antonio Fernández-Caballero Juan M Fernández-Luna Francesc J Ferri Aníbal Ramón Figueiras-Vidal Maribel G Arenas Mikel Galar José Gámez Mario Garcia Nicolás García Salvador García Carlos García Martínez Nicolás García Pedrajas José Luis García-Lapresta Josep M Garrell Karina Gibert University of Burgos, Spain European Centre for Soft Computing, Spain University of Salamanca, Spain University of Granada, Spain Queen’s University Belfast, UK University of Castilla-La Mancha, Spain University of Málaga, Spain University of Oviedo, Spain University of Jaén, Spain University of Oviedo, Spain Universidad da Coruña, Spain University of Cádiz, Spain Universidad Rey Juan Carlos, Spain University of A Coruña, Spain Aalborg University, Denmark Polytechnic University of Cartagena, Spain Polytechnic University of Madrid, Spain University of Barcelona, Spain ITI – UPV, Spain Instituto de Investigación en Inteligencia Artificial-CSIC, Spain Public University of Navarra, Spain University of Extremadura, Spain University of Granada, Spain University of Málaga, Spain Polytechnic University of Cataluña, Spain Public University of Navarra, Spain University of Jaén, Spain University of Castilla-La Mancha, Spain University of Granada, Spain University of Valencia, Spain Universidad Carlos III de Madrid, Spain University of Granada, Spain Public University of Navarra, Spain University of Castilla-La Mancha, Spain Instituto Politécnico de Tijuana, Spain University of Córdoba, Spain University of Granada, Spain University of Córdoba, Spain University of Córdoba, Spain University of Valladolid, Spain Universitat Ramon Llull, Spain Polytechnic University of Cataluña, Spain IX 496 J.L Gonz´ alez S´ anchez et al Table Product measure of k-specificity SpP k P P P P P SpP k (µ0 ) Spk (µ1 ) Spk (µ2 ) Spk (µ3 ) Spk (µ4 ) Spk (µ5 ) k = 1.00 0.00 0.00 0.00 0.00 0.00 k = 0.00 1.00 0.00 0.00 0.00 0.00 k = 0.00 0.00 1.00 0.00 0.00 0.00 k = 0.00 0.00 0.00 1.00 0.00 0.00 k = 0.00 0.00 0.00 0.00 1.00 0.00 k = 0.00 0.00 0.00 0.00 0.00 1.00 Proof If µ is a crisp set with k elements, then aj = for every j: ≤ j ≤ k and aj = for every j: k + ≤ j ≤ n Let h be an integer such that ≤ h ≤ n F If h < k, since (1 − ak ) = 0, then SpF h = If h > k, since ah = 0, then Sph = P Therefore, if Spk is a measure of k-specifictiy, then Spk (µ) ≤ Spk (µ) for every crisp subset µ on the universe X 4.2 Comparing Measures of k-Specificity for Not Crisp Subsets Example Le be X = {e1 , e2 , e3 , e4 , e5 } be a crisp set with n = elements Consider the fuzzy sets: µ6 = {0.2e1 , 0.7|e2 , 0|e3 , 0.9|e4 , 0.5|e5 } µ7 = {0.8e1 , 0.3|e2 , 1|e3 , 0.1|e4 , 0.5|e5 } = − µ6 = µ6 Reordering µ6 we obtain {0.9|e4 , 0.7|e2 , 0.5|e5 , 0.2|e1 , 0|e3 }, so {aj }6 = {0.9, 0.7, 0.5, 0.2, 0}; similarly, {aj }7 = {1, 0.8, 0.5, 0.3, 0.1} F We can calculate the Uniform Linear SpL k , the Fractional Spk , and the Prodmeasures of k-specificity of µ and µ (Table 4): uct SpP k Table Measures of k-specifcity for a fuzzy set and his complementary fuzzy set L F F P P SpL k (µ6 ) Spk (µ7 ) Spk (µ6 ) Spk (µ7 ) Spk (µ6 ) Spk (µ7 ) k = 0.54 0.46 0.19 0.00 0.41 0.00 k = 0.55 0.57 0.35 0.37 0.53 0.50 k = 0.57 0.60 0.56 0.60 0.58 0.61 k = 0.60 0.57 0.64 0.65 0.61 0.58 k = 0.57 0.55 0.57 0.63 0.50 0.53 k = 0.46 0.54 0.46 0.54 0.00 0.41 Remark Note that the Uniform Linear and Product measures of k-specificity have the perfect separation property, and the Fractional measure of k-specificity does not Proposition The Fractional measure of k-specificity does not have the perfect separation property with the usual negation operator Some New Measures of k-Specificity 497 Proof See previous example F P Proposition The SpL k , Spk , and the Spk are not totally ordered for all fuzzy sets Proof See Fig Fig Comparing three k-specifity measures of fuzzy set µ6 Conclusions The Product and Fractional measures of k-specificity on fuzzy sets have been defined and some examples are provided They have been compared with the known Uniform Linear measure of k-specificity The Perfect Separation property has been defined and the lowest measure of k-specificity for crisp sets has been found References Garmendia, L., Gonz´ alez del Campo, R., Yager, R.R.: Recursively spreadable and reductible measures of specificity Inf Sci 326, 270–277 (2016) Gonz´ alez del Campo, R., Garmendia, L., Yager, R.R.: A measure of k-specificity of fuzzy sets and the use of maximum k-specificity In: IV Congreso Espa˜ nol de inform´ atica, CEDI 2013, pp 1134–1143 (2013) Yager, R.R.: Measuring tranquility and anxiety in decision making: an application of fuzzy sets Int J Gen Syst 8, 139–146 (1982) Yager, R.R.: Ordinal measures of specificity Int J Gen Syst 17, 57–72 (1990) Yager, R.R.: Measures of specificity In: Kaynak, O., Zadeh, L.A., Tă urkásen, B., Rudas, I.J (eds.) Computational Intelligence: Soft Computing and Fuzzy-Neuro Integration wiht Applications NATO ASI Series Series F: Computer and Systems Sciences, vol 162, pp 94–113 Springer, Heidelberg (1998) Yager, R.R.: Expansible measures of specificity Int J Gen Syst 41(3), 247–263 (2012) On a Three-Valued Logic to Reason with Prototypes and Counterexamples and a Similarity-Based Generalization Soma Dutta1,2 , Francesc Esteva3 , and Lluis Godo3(B) Vistula University, Warsaw, Poland somadutta9@gmail.com University of Warsaw, Warsaw, Poland IIIA-CSIC, Bellaterra, Spain {esteva,godo}@iiia.csic.es Abstract In this paper, the meaning of a vague concept α is assumed to be rendered through two (mutually exclusive) finite sets of prototypes and counterexamples In the remaining set of situations the concept is assumed to be applied only partially A logical model for this setting can be fit into the three-valued Lukasiewicz’s logic L3 set up by considering, besides the usual notion of logical consequence |= (based on the truth preservation), the logical consequence |=≤ based on the preservation of all truth-degrees as well Moreover, we go one step further by considering a relaxed notion of consequence to some degree a ∈ [0, 1], by allowing the prototypes (counterexamples) of the premise (conclusion) be a-similar to the prototypes (counterexamples) of the conclusion (premise) We present a semantical characterization as well as an axiomatization Introduction A vague, in the sense of gradual, property is characterized by the existence of borderline cases; that is, objects or situations for which the property only partially applies The aim of this paper is to investigate how a logic for vague concepts can be defined starting from the most basic description of a vague property or concept α in terms of a set of prototypical situations or examples [α+ ] ⊆ Ω, where α definitely applies, and a set of counterexamples [α− ] ⊆ Ω, where α does not apply for sure In this paper we will further assume to work with complete descriptions of this kind: that is, for each concept α, the remaining set of situations Ω \ ([α+ ] ∪ [α− ]) will be those where we know α only partially applies to Of course, to be in a consistent scenario, we will require that there is no situation where α both fully applies and does not apply to, in other words, the constraint [α+ ] ∩ [α− ] = ∅ is always satisfied In such a case, one lead to a three-valued framework, where for each situation w ∈ Ω, the degree app(w, α) to which α applies at w (or, equivalently, the truth degree of the assertion “w is c Springer International Publishing Switzerland 2016 O Luaces et al (Eds.): CAEPIA 2016, LNAI 9868, pp 498–508, 2016 DOI: 10.1007/978-3-319-44636-3 47 On a Three-Valued Logic to Reason with Prototypes and Counterexamples 499 α”) can be naturally defined as follows: ⎧ ⎨ 1, if w ∈ [α+ ] app(w, α) = 0, if w ∈ [α− ] ⎩ 1/2, otherwise We want to emphasize that in this 3-valued model, the third value 1/2 is not meant to represent ignorance about whether a concept applies or not to a situation, rather it is meant to represent that the concept only partially applies to a situation, or equivalently, that the situation is a borderline case for the concept (see [3] for a discussion on this topic) The paper is structured as follows After this short introduction, Sect is devoted to develop a logical approach to reason with vague concepts represented by examples and counterexamples based on the three-valued Lukasiewicz logic L3 In Sect we show how by introducing a similarity relation into the picture one can define three kinds of graded notions of approximate logical consequence among vague propositions and we characterize them Finally, in Sect we formally define a sort of graded modal logic to capture reasoning about the approximate consequences and prove completeness We end up with some conclusions Three-Valued Logics to Reason with Examples and Counterexamples In our framework, we assume that we have evaluations e such that for atomic concepts α, e(α) = ([α+ ], [α− ]), providing a disjoint pair of examples and counterexamples A first question is how this evaluation propagates to compound concepts We consider a language with four connectives: conjunction (∧), disjunction (∨), negation (¬) and implication (→) Given e(α) = ([α+ ], [α− ]) and e(β) = ([β + ], [β − ]), the rules for ∧, ∨ and ¬ seem clear as given follows: e(α ∧ β) = ([α+ ] ∩ [β + ], [α− ] ∪ [β − ]) e(α ∨ β) = ([α+ ] ∪ [β + ], [α− ] ∩ [β − ]) e(¬α) = ([α− ], [α+ ]) The case for → is not that straightforward as above Generalising the classical definition of material implication, one could take α → β := ¬α ∨ β, and hence e(¬α ∨ β) = ([α− ] ∪ [β + ], [α+ ] ∩ [β − ]) In that case, the framework turns out to be the one corresponding to the wellknown Kleene’s three-valued logic However, it is also well-known that in Kleene’s logic the interpretation of the intermediate value 1/2 is usually considered as ignorance This makes it natural to claim that if it is not known whether w is an example or counterexample of both α and β, it remains unknown whether it is an example or counterexample of α → β However, if 1/2 is assumed to denote a borderline case, it is perfectly natural to consider, in that case, that w is an example of α → β This small change in the framework amounts to move from Kleene’s three-valued logic to Lukasiewicz’s three-valued logic In such a case, we have 500 S Dutta et al e(α → β) = ([α− ] ∪ [β + ] ∪ ([α∼ ] ∩ [β ∼ ]), [α+ ] ∩ [β − ]), where we use the notation [γ ∼ ] = Ω \ ([γ + ] ∪ [γ − ]) Let us formalize this framework from a three-valued logic point of view To so, let V ar denote a (finite) set of atomic concepts, or propositional variables, from which compound concepts (or formulas) are built using the connectives ∧, ∨, → and ¬ We will denote the set of formulas by F m3 (V ar), in short F m3 Further, let Ω be the set of all possible situations, that we will identify with the set of all evaluations v of atomic concepts V ar into the truth set {0, 1/2, 1}, that is Ω = {0, 1/2, 1}V ar , with the following intended meaning: v(α) = means that v is an example of α (resp v is a model of α in logical terms), v(α) = means that v is a counterexample of α (resp v is a counter-model of α), and v(α) = 1/2 means that v is a borderline situation for α, i.e it is neither an example nor a counterexample According to the previous discussion, truthevaluations v will be extended to compound concepts according to the semantics of 3-valued Lukasiewicz logic L3 , defined by following truth-tables: ∧ 1/2 1/2 0 0 1/2 1/2 1/2 ∨ 1/2 1/2 1/2 1/2 1/2 1 1 → 1/2 1/2 1 1 1/2 1 1/2 ¬ 1/2 1/2 These truth-tables can also be given by means of the following truth-functions: for all x, y ∈ {0, 1/2, 1}, x∧y = min(x, y), x∨y = max(x, y), x → y = min(1, 1− x + y) and ¬x = − x Notation For any concept ϕ we will denote by [ϕ] the 3-valued (fuzzy) set of models of ϕ, i.e [ϕ] : Ω → {0, 1/2, 1} defined as [ϕ](w) = w(ϕ) We will write [ϕ] ≤ [ψ] when [ϕ](w) ≤ [ψ](w) for all w ∈ Ω In L3 , a strong conjunction and a strong disjunction connectives can be defined from → and ¬ as follows: ϕ ⊗ ψ := ¬(ϕ → ¬ψ) and ϕ ⊕ ψ := ¬ϕ → ψ.1 Actually, for each concept ϕ ∈ F m3 , the connective ⊗ allows one to define three related Boolean concepts: ϕ+ := ϕ ⊗ ϕ, ϕ− := (¬ϕ) ⊗ (¬ϕ) = (¬ϕ)+ , ϕ∼ := ¬ϕ+ ∧ ¬ϕ− , with the following semantics: w(ϕ+ ) = if w(ϕ) = 1; w(ϕ+ ) = otherwise; w(ϕ∼ ) = if w(ϕ) = 1/2; w(ϕ∼ ) = otherwise; w(ϕ− ) = if w(ϕ) = 0; w(ϕ− ) = otherwise; and therefore [ϕ+ ], [ϕ− ], [ϕ∼ ] capture respectively the (classical) sets of examples, counterexamples and borderline cases of ϕ Actually, one could take → and ¬ as the only primitive connectives since ∧ and ∨ can be defined from → and ¬ as well: ϕ ∧ ψ = ϕ ⊗ (ϕ → ψ) and ϕ ∨ ψ = (ϕ → ψ) → ψ On a Three-Valued Logic to Reason with Prototypes and Counterexamples 501 The usual notion of logical consequence in 3-valued Lukasiewicz logic is defined as follows: for any set of formulas Γ ∪ {ϕ}, Γ |= ψ if for any evaluation v, v(ϕ) = for all ϕ ∈ Γ, then v(ψ) = It is well known that this consequence relation can be axiomatized by the following axioms and rule (see e.g [2]): ϕ → (ψ → ϕ), (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)), (¬ϕ → ¬ψ) → (ψ → ϕ), (ϕ ∨ ψ) → (ψ ∨ ϕ), ϕ ⊕ ϕ ↔ ϕ ⊕ ϕ ⊕ ϕ, ϕ, ϕ → ψ (MP) The rule of modus ponens: ψ (L1) (L2) (L3) (L4) (L5) This axiomatic system, denoted L3 , is strongly complete with respect to the above semantics; that is, for a set of formulas Γ ∪ {ϕ}, Γ |= ϕ iff Γ ϕ, where , the notion of proof for L3 , is defined from the above axioms and rule in the usual way Remark: In the sequel we will restrict ourselves on considerations about logical consequences from finite set of premises In such a case, if Γ = {ϕ1 , , ϕn } then it holds that Γ |= ψ iff ϕ1 ∧ ∧ ϕn |= ψ, and hence it will be enough to consider premises consisting of a single formula Lemma For all formulas ϕ, ψ, it holds that ϕ |= ψ iff [ϕ+ ] ⊆ [ψ + ] This makes clear that |= is indeed the consequence relation that preserves the examples of concepts Similarly we can also consider the consequence relation that preserves counterexamples Namely, one can contrapositively define a falsity-preserving consequence as: ϕ |=C ψ if ¬ψ |= ¬ϕ, that is, if for any evaluation v, v(ψ) = implies v(ϕ) = Unlike classical logic, in 3-valued Lukasiewicz logic it is not the case that ϕ |= ψ iff ¬ψ |= ¬ϕ As we have seen that the former amounts to require [ϕ+ ] ⊆ [ψ + ], while the latter, as shown next, amounts to require [ψ − ] ⊆ [ϕ− ] Clearly these conditions, in general, are not equivalent, except when ϕ and ψ not have borderline cases, that is, when [ϕ+ ] ∪ [ϕ− ] = [ψ + ] ∪ [ψ − ] = Ω Lemma For all formulas ϕ, ψ, it holds that ϕ |=C ψ iff [ψ − ] ⊆ [ϕ− ] Equivalently, ϕ |=C ψ holds iff for any evaluation v ∈ Ω, v(ϕ) ≥ 1/2 implies v(ψ) ≥ 1/2, or in other words, [ϕ+ ] ∪ [ϕ∼ ] ⊆ [ψ + ] ∪ [ψ ∼ ] Now we define the consequence relation that preserves both examples and counterexamples in the natural way Definition ϕ |=≤ ψ if ϕ |= ψ and ϕ |=C ψ, that is, if [ϕ+ ] ⊆ [ψ + ] and [ψ − ] ⊆ [ϕ− ] 502 S Dutta et al Note that, for instance, ϕ |= ϕ+ holds, while ϕ |=≤ ϕ+ Indeed, while the examples of ϕ and ϕ+ are the same, the counterexamples of ϕ+ include not only the counterexamples but also those borderline cases of ϕ From the above observations, we have these equivalent characterizations of |=≤ Lemma For all formulas ϕ, ψ, the following conditions are equivalent: – – – – ϕ |=≤ ψ, |= ϕ → ψ, [ϕ] ≤ [ψ], [ϕ → ψ] = Ω These characterizations justify the use of the superscript ≤ in the symbol of consequence relation And indeed, the consequence relation |=≤ is known in the literature as the degree-preserving companion of |=, as opposed to the truthpreserving consequence |=, that preserves the truth-value ‘1’ [1] |=≤ can also be axiomatized by taking as axioms those of L3 and the following two inference rules: (Adj) : ϕ, ψ ϕ∧ψ (M P r) : ϕ, ϕ→ψ ψ The resulting logic is denoted by L≤ , and its notion of proof is denoted by ≤ Notice that (MPr) is a weakened version of modus ponens, called restricted modus ponens, since ϕ → ψ has to be a theorem of L3 for the rule to be applicable As a summary of this section, we can claim that L≤ (or its semantical counterpart |=≤ ) provides a more suitable logical framework to reason about concepts described by examples and counterexamples than the usual three-valued Lukasiewicz logic L3 A Similarity-Based Refined Framework In the previous section we have discussed a logic for reasoning about vague concepts described in fact as 3-valued fuzzy sets A more fine grained representation, moving from 3-valued to [0, 1]-valued fuzzy sets, can be introduced by assuming the availability of a (fuzzy) similarity relation S : Ω × Ω → [0, 1] among situations Indeed, for instance, assume that all examples of ϕ are examples of ψ, but some counterexamples of ψ are not counterexamples of ϕ Hence, we cannot derive that ψ follows from ϕ according to |=≤ However, if these counterexamples of ψ greatly resemble to counterexamples of ϕ, it seems reasonable to claim that ψ follows approximately from ϕ Actually, starting from Ruspini’s seminal work [7], a similar approach has already been investigated in the literature in order to extend the notion of entailment in classical logic in different frameworks and using formalisms, see e.g [6] Here we will follow this line and propose a graded generalization of the |=≤ in On a Three-Valued Logic to Reason with Prototypes and Counterexamples 503 the presence of similarity relation S on the set of 3-valued Lukasiewicz interpretations Ω, that allows to draw approximate conclusions Since, by definition ϕ |=≤ ψ if both ϕ |= ψ and ϕ |=C ψ, that is, if [ϕ+ ] ⊆ [ψ + ] and [ψ − ] ⊆ [ϕ− ], it seems natural to define that ψ is an approximate consequence of ϕ to some degree a ∈ [0, 1] when every example of ϕ is similar (at least to the degree a) to some example of ψ, as well as every counterexample of ψ is similar (to at least to the degree a) to some counterexample of ϕ In other words, this means that to relax |=≤ we propose to relax both |= and |=C This idea is formalized next, where we assume that a ∗-similarity relation S : Ω × Ω → [0, 1] be given, satisfying the properties: – S(w, w ) = iff w = w , – S(w, w ) = S(w , w), – S(w, w ) ∗ S(w , w ) ≤ S(w, w ), where ∗ is a t-norm operation Moreover, for any subset A ⊂ Ω and value a ∈ [0, 1] we define its a-neighborhood as Aa = {w ∈ Ω | there exists w ∈ A such that S(w, w ) ≥ a} Definition For any pair of formulas ϕ, ψ and for each degree a ∈ [0, 1], we ≤ define the consequence relations |=a , |=C a and |=a as follows: (i) ϕ |=a ψ if for every w ∈ Ω such that w(ϕ) = there exists w ∈ Ω with S(w, w ) ≥ a and w (ψ) = In other words, ϕ |=a ψ if [ϕ+ ] ⊆ [ψ + ]a (ii) ϕ |=C a ψ if for every w ∈ Ω such that w(ψ) = there exists w ∈ Ω with − − a S(w, w ) ≥ a and w (ϕ) = In other words, ϕ |=C a ψ if [ψ ] ⊆ [ϕ ] ≤ C + + a (iii) ϕ |=a ψ if both ϕ |=a ψ and ϕ |=a ψ i.e if both [ϕ ] ⊆ [ψ ] and [ψ − ] ⊆ [ϕ− ]a Taking into account that for any formula χ it holds [(¬χ)+ ] = [χ− ], it is clear ≤ a that |=C a (and thus |=a as well) can be expressed in terms of |= Lemma For any formulas ϕ and ψ, the following conditions hold: – ϕ |=C a ψ iff ¬ψ |=a ¬ϕ – ϕ |=≤ a ψ iff ϕ |=a ψ and ¬ψ |=a ¬ϕ The consequence relations |=a are very similar to the so-called approximate graded entailment relations defined in [4] and further studied in [6,9,10] The main difference is that in [4] the authors consider classical propositions while in this paper we consider three-valued Lukasiewicz propositions Nevertheless we can prove very similar characterizing properties for the |=a ’s In the following theorem, for each evaluation w ∈ Ω, w denotes the following proposition: w=( p∈X:w(p)=1 p+ ) ∧ ( p∈X:w(p)=1/2 p∼ ) ∧ ( p− ) p∈X:w(p)=0 So, w is a (Boolean) formula which encapsulates the complete description provided by w Moreover, for every w ∈ Ω, w (w) = if w = w and w (w) = otherwise 504 S Dutta et al Theorem The following properties hold for the family {|=a : a ∈ [0, 1]} of graded entailment relations on F m3 × F m3 induced by a ∗-similarity relation S on Ω: (i) Nestedness: if ϕ |=a ψ and b ≤ a, then ϕ |=b ψ (ii) |=1 coincides with |=, while |= |=a if a < Moreover, if ψ |= ⊥, then ϕ |=0 ψ for any ϕ (iii) Positive-preservation: ϕ |=a ψ iff ϕ+ |=a ψ + (iv) ∗-Transitivity: if ϕ |=a ψ and ψ |=b χ then ϕ |=a∗b χ (v) Left-OR: ϕ ∨ ψ |=a χ iff ϕ |=a χ and ψ |=a χ (vi) Restricted Right-OR: for all w ∈ Ω, w |=a ϕ ∨ ψ iff w |=a ϕ or w |=a ψ (vii) Restricted symmetry: for all w, w ∈ Ω, w |=a w iff w |=a w (viii) Consistency preservation: if ϕ |= ⊥ then ϕ |=a ⊥ only if a = (ix) Continuity from below: If ϕ |=a ψ for all a < b, then ϕ |=b ψ Conversely, for any family of graded entailment relations { a : a ∈ [0, 1]} on F m3 × F m3 satisfying the above properties, there exists a *-similarity relation S such that a = |=a for each a ∈ [0, 1] Proof (Sketch) The proof follows the same steps than the one of [4, Theorem 1] in the case of a classical propositional setting The key points to take into account here are: – it is easy to check that, for any formula ϕ ∈ F m3 , ϕ+ is logically equivalent in L3 to the disjunction w∈Ω:w(ϕ)=1 w – (ϕ ∨ ψ)+ is logically equivalent to ϕ+ ∨ ψ + – for every w, w ∈ Ω, w |=a w iff S(w, w ) ≥ a For the converse direction, the latter property is used to define the corresponding similarity S for a family of consequence relations { a : a ∈ [0, 1]} satisfying (i)– (ix) as S(w, w ) = sup{a ∈ [0, 1] | w a w } Taking into account Lemma 4, a sort of dual characterization for |=C a , that we omit, can easily be derived from the above one for |=a On the other hand, the above properties also indirectly characterize |=≤ a in the sense that, in our ≤ finite setting, |=a (and thus |=C a as well) can be derived from |=a as well as the following lemma shows Lemma For any ϕ, ψ ∈ F m3 , we have that ϕ |=a ψ iff for every w ∈ Ω such that w(ϕ) = there exists w ∈ Ω such that w(ψ) = and w |=≤ a w Proof It directly follows from properties (iv) and (v) of Theorem 1, by checking that, for every w ∈ Ω, w |=≤ a w iff w |=a w However, admittedly, the resulting characterization of |=≤ a we would obtain using this lemma is not very elegant On a Three-Valued Logic to Reason with Prototypes and Counterexamples 505 A Logic to Reason About Graded Consequences |=a, ≤ |=C a and |=a In this section we will define a Boolean (meta) logic LAC3 to reason about ≤ the graded entailments |=a , |=C a and |=a The idea is to consider expressions C corresponding to ϕ |=a ψ, ϕ |=a ψ and ϕ |=≤ a ψ as the concerned objects of our logic, and then to use Theorem to devise a complete axiomatics to capture the intended meaning of such expressions To avoid unnecessary complications, we will make the following assumption: all ∗-similarity relations S will take values in a finite set G of [0, 1], containing and 1, and ∗ will be a given finite t-norm operation on G, that is, (G, ∗) will be a finite totally ordered semi-group In this way, we keep our language finitary and avoid the use of an infinitary inference rule to cope with Property (ix) of Theorem Our logic will be a two-tired logic, where at a first level we will have formulas and semantics of the 3-valued Lukasiewicz logic L3 and at the second level we will have propositional classical logic CPC We start by defining the syntax of LAC3, with two languages: – Language L0 : built from a finite set of propositional variables V ar = {p, q, r, } and using L3 connectives ¬, ∧, ∨, → Other derived connectives are ⊕ and ⊗, defined as in Sect We will use and ⊥ as abbreviations for p → p and ¬(p → p) respectively, and ϕ+ and ϕ− as abbreviations of ϕ ⊗ ϕ and (¬ϕ)+ respectively – Language L1 : atomic formulas of L1 are only of the form φ P a ψ, where φ, ψ are L0 -formulas and a ∈ G, and compound L1 -formulas are built from atomic ones with the usual Boolean connectives ¬, ∧, ∨, →.2 P Moreover, we will be using φ C a ψ as abbreviations of ¬ψ a ψ and φ a ¬φ P C and (φ a ψ) ∧ (φ a ψ) respectively The semantics is given by similarity Kripke models M = (W, S, e) where W is a finite set of worlds, S : W × W → G is a ∗-similarity relation, and e : W × V ar → {0, 12 , 1} is a 3-valued evaluation of propositional variables in every world, which is extended to arbitrary L0 -formulas using L3 truth-functions For every formula ϕ ∈ L0 , we define: [ϕ]M : W → {0, 1/2, 1} such that w → e(w, ϕ), [ϕ+ ]M = {w ∈ W | e(w, ϕ) = 1}, and [ϕ− ]M = {w ∈ W | e(w, ϕ) = 0} Each similarity Kripke model M = (W, S, e) induces a function eM : L1 → {0, 1}, which is a (Boolean) truth evaluation for L1 -formulas defined as follows: – for atomic L1 -formulas: P ψ) = if [φ+ ]M ⊆ ([ψ + ]M )a , i.e., if minw∈[φ+ ]M eM (φ a maxw ∈[ψ+ ]M S(w, w ) ≥ a; eM (φ P a ψ) = otherwise – for compound formulas, use the usual Boolean truth functions Although we are using symbols ∧, ∨, ¬, → for both formulas of L0 and L1 , it will be clear from the context when they refer to L3 or when they refer to Boolean connectives 506 S Dutta et al P Note that, by definition, eM (φ C a ψ) = iff eM (¬φ a ¬ψ) = 1, and eM (φ C ψ) = iff eM (φ P ψ) = and e (φ ψ) = M a a In the next lemma we list some useful properties of eM a Lemma The following conditions hold: – – – – eM (φ eM (φ eM (φ eM ((φ ψ) = iff [ψ − ]M ⊆ ([φ− ]M )a + + a − − a a ψ) = iff [φ ]M ⊆ ([ψ ]M ) and [ψ ]M ⊆ ([φ ]M ) ψ) = iff [φ]M ≤ [ψ]M ψ) ∧ (ψ φ)) = iff [ϕ]M = [ψ]M , iff [φ ↔ ψ] = W C a Now we define the notion of logical consequence in LAC3 for L1 -formulas Definition Let T ∪ {Φ} be a set of L1 -formulas We say that Φ logically follows from T , written T |=LAC3 Φ, if for every similarity Kripke model M = (W, S, e), if eM (Ψ ) = for every Ψ ∈ T , then eM (Φ) = as well Finally we propose the following axiomatization of LAC3 Definition The following are the axioms for LAC3: (A1) (A2) (A3) (A4) (A5) (A6) (A7) (A8) (A9) (A10) (A11) (A12) Axioms of CPC for L1 -formulas φ 1P ψ, where φ, ψ are such that φ |= ψ P ¬( → ⊥) P (φ a ψ) → (φ bP ψ), where a ≤ b (φ 1P ψ) → (φ+ ∧ ¬ψ + 1P ⊥) ¬(ψ 1P ⊥) → (φ 0P ψ) (φ aP ⊥) → (φ 1P ⊥) ¬(w 1P ⊥) ∧ (w aP w ) → (w aP w), for w, w ∈ Ω (φ aP χ) ∧ (ψ aP χ) → (φ ∨ ψ aP χ) (w aP φ ∨ ψ) → (w aP φ) ∨ (w cP ψ) P χ) (φ aP ψ) ∧ (ψ bP χ) → (φ a∗b P + P + ψ ) (φ a ψ) ↔ (φ a The only rule of LAC3 is modus ponens The notion of proof defined from the above axioms and rule will be denoted LAC3 Finally, we have the following soundness and completeness theorem for LAC3 Theorem For any set T ∪ {Φ} of L1 -formulas, it holds that T |=LAC3 Φ if, and only if, T LAC3 Φ Proof One direction is soundness, and it basically follows from Theorem As for the converse direction, assume T LAC3 Φ The idea is to consider the graded expressions φ aP ψ as propositional (Boolean) variables that are ruled by the axioms together with the laws of classical propositional logic CPC Let Γ be the set of all possible instantiations of axioms (A1)–(A12) Then it implies that Φ does not follow from T ∪ Γ using CPC reasoning, i.e T ∪ Γ CP C Φ By completeness of CPC, there exists a Boolean interpretation v such that v(Ψ ) = On a Three-Valued Logic to Reason with Prototypes and Counterexamples 507 for all Ψ ∈ T ∪ Γ and v(Φ) = Now we will build a ∗-similarity Kripke model M such that eM (Ψ ) = for all Ψ ∈ T and eM (Φ) = To that we take Ω and define S : Ω × Ω → G by S (w, w ) = max{a ∈ G | v(w P a w ) = 1} By axioms (A2), (A8) and (A11), S is a ∗-similarity Note that, by definition and Axiom (A4), S(w, w ) ≥ a iff v(w P a w ) = Finally we consider the model M = (Ω, S, e), where for each w ∈ Ω and p ∈ V ar, e(w, p) = w(p) What remains is to check that eM (Ψ ) = v(Ψ ) for every LAC3-formula Ψ It suffices to show that, for every φ, ψ ∈ L0 and a ∈ G, we have eM (φ aP ψ) = v(φ aP ψ), that is, to prove that v(φ P a ψ) = iff max w∈[φ+ ]M w ∈[ψ + ]M S(w, w ) ≥ a First of all, recall that for every φ, L3 proves the equivalence φ+ ↔ ∨w∈Ω:w(ϕ)=1 w, and by axioms (A12), (A9) and (A10), we have that LAC3 proves φ P a ψ ↔ w P a w w∈Ω:w(φ)=1 w ∈Ω:w (ψ)=1 Therefore, v(φ aP ψ) = iff for all w in Ω such that w(φ) = 1, there exists w such that w (ψ) = and v(w P a w ) = But, as we have previously w ) = holds iff S(w, w ) ≥ a In other words, we actually observed, v(w P a have v(φ aP ψ) = iff minw∈[φ+ ]M maxw ∈[ψ+ ]M S(w, w ) ≥ a This concludes the proof Conclusions and Future Work We have presented an approach towards considering graded entailments between vague concepts (or propositions) based on the similarity between both the prototypes and counterexamples of the antecedent and the consequent This approach is a natural generalization of the Lukasiewicz’s three-valued consequence (|=≤ ) that preserves truth-degrees The provided axiomatization is for the operators P a , based on both a , which are based on prototypes only, while the operators prototypes and counterexamples, can be naturally obtained as a derived operators in the system To derive a complete axiomatic system directly for the operators a is an issue under current investigation Besides, we leave other interesting issues for further research First, in this paper, we have assumed app(ω, α) to be a C three-valued concept, and to define |=≤ a from |=a and |=a we have used a conjunctive aggregation of the two aspects of similarity, similarity among prototypes and similarity among counterexamples Another approach could be to let app(ω, α) (1 − to admit itself a finer distinction by defining app∗ (ω, α) = S(w, [α]+ ) S(w, [α− ])) with S(w, [α+ ]) = maxω ∈[α+ ] S(ω, ω ) and analogusly for S(w, [α− ]) Then the extent to which α entails β can be defined based on the relationship of app∗ (ω, α) and app∗ (ω, β) considering all possible situations ω This direction 508 S Dutta et al seems to have lots of challenges as might not be as simple as a conjunctive operation; also different notions of consequence can be worth exploring in the line of [5,6,8] Acknowledgments The authors are thankful to the anonymous reviewers for their helpful comments Esteva and Godo acknowledge partial support of the project TIN2015-71799-C2-1-P (MINECO/FEDER) References Bou, F., Esteva, F., Font, J.M., Gil, A., Godo, L., Torrens, A., Verd´ u, V.: Logics preserving degrees of truth from varieties of residuated lattices J Logic Comput 19(6), 1031–1069 (2009) Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of ManyValued Reasoning Trends in Logic, vol Kluwer, Dordrecht (1999) Ciucci, D., Dubois, D., Lawry, J.: Borderline vs unknown: comparing three-valued representations of imperfect information Int J Approx Reason 55(9), 1866–1889 (2014) Dubois, D., Prade, H., Esteva, F., Garcia, P., Godo, L.: A logical approach to interpolation based on similarity relations Int J Approx Reason 17, 1–36 (1997) Dutta, S., Bedregal, B.R.C., Chakraborty, M.K.: Some instances of graded consequence in the context of interval-valued semantics 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Quintián Emilio Corchado (Eds.) • • • Advances in Artificial Intelligence 17th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2016 Salamanca, Spain, September 14–16, 2016. .. selection of the papers accepted for oral presentation at the 17th Conference of the Spanish Association for Artificial Intelligence (CAEPIA 2016) , held in Salamanca (Spain), during September 14–16, 2016. .. of maintaining CAEPIA as a high-quality conference, and following the model of current demanding AI conferences, the CAEPIA review process runs under the double-blind model The number of submissions