Volume 6 Volume Journal of applied logics (Print) ISSN 2631-9810 Journal of applied logics (Online) ISSN 2631-9829 Issue 1 January 2019 Journal of Contents Articles Applied Logics Covariant-Contravariant Refi nement Modal Logic The IfCoLog Journal of Logics and their Applications Huili Xing, Zhaohui Zhu and Jinjin Zhang 1 Did the Neo-Babylonians Construct a Symbolic Logic for Volume 6 Issue 1 January 2019 Legal Proceedings? Andrew Schumann 31 Weight Predication on Missing Links in Social Networks. A Cross-Entropy-Based Approach Wilhelm Rödder, Andreas Dellnitz, Ivan Gartner and Sebastian Litzinger 83 AlphaGo’s Decision Making Woosuk Park, Sungyong Kim, Keunhyoung Luke Kim Journal of Applied Logics and Jeounghoon Kim 105 Strengthening Gossip Protocols using Protocol-Dependent Knowledge Hans van Ditmarsch, Malvin Gattinger, Louwe B. Kuijer and Pere Pardo 157 The IfCoLog Journal of Logics and their Ap plications Sponsored by Published by Available online at www.collegepublications.co.uk/journals/ifcolog/ Free open access Journal of Applied Logics - IfCoLog Journal of Logics and their Applications Volume 6, Number 1 January 2019 Disclaimer Statements of fact and opinion in the articles in Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (JALs-FLAP) are those of the respective authors and contributors and not of the JALs-FLAP. Neither College Publications nor the JALs-FLAP make any representation, express or implied, in respect of the accuracy of the material in this journal and cannot accept any legal responsibility or liability for any errors or omissions that may be made. The reader should make his/her own evaluation as to the appropriateness or otherwise of any experimental technique described. c Individual authors and College Publications 2019 All rights reserved. ISBN 978-1-84890-299-2 ISSN (E) 2631-9829 ISSN (P) 2631-9810 College Publications Scientific Director: Dov Gabbay Managing Director: Jane Spurr http://www.collegepublications.co.uk All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise without prior permission, in writing, from the publisher. Editorial Board Editors-in-Chief Dov M. Gabbay and Jörg Siekmann Henri Prade Marcello D’Agostino Melvin Fitting David Pym Natasha Alechina Michael Gabbay Ruy de Queiroz Sandra Alves Murdoch Gabbay Ram Ramanujam Arnon Avron Thomas F. Gordon Chrtian Retoré Jan Broersen Wesley H. Holliday Ulrike Sattler Martin Caminada Sara Kalvala Jörg Siekmann Balder ten Cate Shalom Lappin Jane Spurr Agata Ciabttoni Beishui Liao Kaile Su Robin Cooper David Makinson Leon van der Torre Luis Farinas del Cerro George Metcalfe Yde Venema Esther David Claudia Nalon Rineke Verbrugge Didier Dubois Valeria de Paiva Heinrich Wansing PM Dung Jeff Paris Jef Wijsen Amy Felty David Pearce John Woods David Fernandez Duque Brigitte Pientka Michael Wooldridge Jan van Eijck Elaine Pimentel Anna Zamansky iii iv Scope and Submissions This journal considers submission in all areas of pure and applied logic, including: pure logical systems dynamic logic proof theory quantum logic constructive logic algebraic logic categorical logic logic and cognition modal and temporal logic probabilistic logic model theory logic and networks recursion theory neuro-logical systems type theory complexity nominal theory argumentation theory nonclassical logics logic and computation nonmonotonic logic logic and language numerical and uncertainty reasoning logic engineering logic and AI knowledge-based systems foundations of logic programming automated reasoning belief revision knowledge representation systems of knowledge and belief logic in hardware and VLSI logics and semantics of programming natural language specification and verification concurrent computation agent theory planning databases This journal will also consider papers on the application of logic in other subject areas: philosophy, cognitive science, physics etc. provided they have some formal content. Submissions should be sent to Jane Spurr ([email protected]) as a pdf file, preferably compiled in LATEX using the IFCoLog class file. v vi Contents ARTICLES Covariant-Contravariant Refinement Modal Logic ................ 1 Huili Xing, Zhaohui Zhu and Jinjin Zhang Did the Neo-Babylonians Construct a Symbolic Logic for Legal Proceedings? 31 Andrew Schumann Weight Predication on Missing Links in Social Networks. A Cross-Entropy-Based Approach ......................... 83 Wilhelm Rödder, Andreas Dellnitz, Ivan Gartner and Sebastian Litzinger AlphaGo’s Decision Making .............................. 105 Woosuk Park, Sungyong Kim, Keunhyoung Luke Kim and Jeounghoon Kim Strengthening Gossip Protocols using Protocol-Dependent Knowledge ... 157 Hans van Ditmarsch, Malvin Gattinger, Louwe B. Kuijer and Pere Pardo vii viii 0 Covariant-Contravariant Refinement Modal Logic Huili Xing College of Computer Science and Technology Nanjing University of Aeronautics and Astronautics, China Zhaohui Zhu College of Computer Science and Technology Nanjing University of Aeronautics and Astronautics, China [email protected] Jinjin Zhang College of Information Science Nanjing Audit University, China Abstract The notion of covariant-contravariant refinement (CC-refinement, for short) is a generalization of the notions of bisimulation and refinement. This paper introduces CC-refinement modal logic (CCRML) obtained from the modal sys- tem K by adding CC-refinement quantifiers, and provides a sound and complete axiom system for CCRML. Keywords: modal logic, covariant-contravariant refinement, axiom system 1 Introduction Recently, Laura Bozzelli et al. presented and explored refinement modal logic (RML) [8, 9], which provides a more abstract perspective of future event logic [20, 21] The authors are grateful to the anonymous referees for their valuable suggestions, which have helped us to improve the presentation of the paper. In particular, the decidability of CCRML is pointed out by one of referees. This work received financial support of the National Natural Science Foundation of China (No. 61602249) and Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX17_0288). Vol. 6 No. 1 2019 Journal of Applied Logics — IfCoLog Journal of Logics and their Applications Huili Xing and Zhaohui Zhu and Jinjin Zhang and arbitrary public announcement logic [5]. RML is obtained from the modal sys- tem K by adding a refinement operator . The semantics of the refinement op- ∃B erator is given in terms of the notion of B-refinement. Such notion captures ∃B refinement preorders between Kripke models and the operator acts as a quanti- ∃B fier over the set of all B-refinements of a given pointed model. In addition to the refinement operator , a kind of non-standard propositional quantifier so-called ∃B bisimulation quantifier appeared earlier in the literature (see, e.g., [11, 12, 16]), ∃X where X is a proposition letter. A formula α is true in a pointed model M if ∃X s there is a pointed modele N satisfying α and an (Atom X )-bisimulation linking t − { } M and N . Here an (Atom X )-bisimulatione is a relation between two models s t − { } in which related states satisfy the same proposition letters in Atom X and have − { } matching transition possibilities. It has been shown that refinement quantification is bisimulation quantification plus relativization [8]. In addition to refinement and bisimulation, there exist a number of different compatible relations between Kripke models. This paper will focus on the notion of covariant-contravariant refinement (CC-refinement, for short). The notion of CC- refinement (simulation), introduced in [13], is a behavioural preorder over labelled transitions systems (LTSs). The standard definition of simulation (see, e.g., [7, 22]) is related to reactive systems whose actions are passive, that is, their execution must be triggered by the environment. Any correct implementation of a reactive system is required to simulate the actions of its specification, in other words, the former should respond to the user’s requests at least as well as the latter. Clearly, the standard notion of simulation provides a reasonable mathematical description of the refine- ment relations between reactive systems. However, for the systems with generative (or, active) actions, e.g., input/output (I/O) automata, such notion isn’t adequate because, for these generative actions (e.g., output), it is the system that produces the output and the environment is obliged to accept the output produced by the system. One of the motivations behind introducing the notion of CC-refinement is to provide an adequate behavioural preorder for the systems referring to the gener- ative actions. The definition of CC-refinement depends on partitioning all actions into three sorts: covariant, contravariant and bivariant actions. The covariant ac- tions denote the passive actions of a system, whose execution is under the control of the environment. The transitions labelled with these actions in a given specification should be simulated by any correct implementation, which is exactly the requirement suggested by the standard notion of simulation. The contravariant actions represent the generative actions being under the control of a system itself. The transitions of these actions in an implementation must be simulated by its specification, in other words, the transitions induced by the generative actions must be allowed by its spec- ification. The bivariant actions