Lambda-calculus and formal language theory Sylvain Salvati To cite this version: Sylvain Salvati. Lambda-calculus and formal language theory. Computer Science [cs]. Université de Bordeaux, 2015. tel-01253426 HAL Id: tel-01253426 https://hal.archives-ouvertes.fr/tel-01253426 Submitted on 10 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÉ DE BORDEAUX ÉCOLE DOCTORALE DE MATHÉMATIQUE ET INFORMATIQUE DE BORDEAUX Habilitation à diriger les recherches Soutenue publiquement le 10 décembre 2015 Sylvain Salvati Lambda-calculus and formal language theory Jury: Rapporteurs: Thomas Ehrhard - Directeur de recherche CNRS / PPS Giorgio Satta - Professeur Università Padua, Italie Sophie Tison - Professeur Université de Lille / CRIStAL Examinateurs: Bruno Courcelle - Professeur émérite Université de Bordeaux / LaBRI Philippe de Groote - Directeur de recherche INRIA / Loria Jérôme Leroux - Directeur de recherche CNRS / LaBRI Contents Contents iii 1 Introduction 1 1.1 Research context . 1 1.2 Research orientation and main contributions . 2 1.3 Organization of the document . 5 2 Preliminaries and recognizability 7 2.1 Simply typed λ-calculus . 8 2.2 Special constants . 10 2.3 Some syntactic restrictions on simply typed λ-calculus . 12 2.4 Models of λ-calculi . 14 2.5 Recognizability in simply typed λ-calculus . 21 2.6 Conclusion and perspective . 26 3 Abstract Categorial Grammars 29 3.1 Abstract Categorial Grammars . 30 3.2 Expressiveness . 34 3.3 Parsing Algorithms . 39 3.4 OI grammars . 42 3.5 Conclusion and perspectives . 47 4 Mildly Context Sensitive Languages 51 4.1 On mild-context sensitivity and its limitations . 53 4.2 Multiple Context Free Grammars . 59 4.3 The language MIX . 62 4.4 Iteration properties for MCFGs . 70 4.5 Classifying Mildly Context Sensitive Formalisms . 72 4.6 Conclusion and perspectives . 73 5 Logic and Verification 77 5.1 Schematology . 79 5.2 Parity automata . 81 5.3 Wreath product and weak parity automata . 87 5.4 λY -calculus and abstract machines . 90 iii 5.5 A λY -model for parity automata . 96 5.6 Adequacy of the model via parity games . 102 5.7 Conclusion and perspectives . 109 6 Conclusion 113 Bibliography . 116 Personal bibliography . 134 iv Abstract Formal and symbolic approaches have offered computer science many application fields. The rich and fruitful connection between logic, au- tomata and algebra is one such approach. It has been used to model natural languages as well as in program verification. In the mathemat- ics of language it is able to model phenomena ranging from syntax to phonology while in verification it gives model checking algorithms to a wide family of programs. This thesis extends this approach to simply typed lambda-calculus by providing a natural extension of recognizability to programs that are representable by simply typed terms. This notion is then applied to both the mathematics of language and program verification. In the case of the mathematics of language, it is used to generalize parsing algorithms and to propose high-level methods to describe languages. Concerning program verification, it is used to describe methods for verifying the behavioral properties of higher-order programs. In both cases, the link that is drawn between finite state methods and denotational semantics provide the means to mix powerful tools coming from the two worlds. v Acknowledgments First of all, I would like to thank the members of the jury for the attention the have paid to my work. I am grateful to Burno Courcelle for having accepted to be the president of the jury. When I was writing this document, I realized how much his work has influenced mine in many respects. I thank the reviewers for having read this document in details. Moreover, as this document reports on work that I did in two different fields, I must thank them for the patience they had to go through chapters which had little to do with their daily research. I thank Thomas Ehrhard for having done this with computational linguistics and also for the discussions I could have with him during the past years on denotational semantics. Giorgio Satta had the difficult task to accommodate the peculiarities of the French habilitation and I am very happy that he accepted to do this. The third reviewer is Sophie Tison who has a high profile in automata theory. I have remarked that the community working on automata does not like λ-calculus so much, I am thankful to her for having accepted to have overcome this reluctance. I am very pleased that Jérôme Leroux accepted to be part of the jury. First because, I really like his work and also, because, now, as the head of the team Méthodes formelles, he represents the stimulating environment in which my work has grown. I am glad that Philippe de Groote is in this jury. I started research with him and, still, in the rare occasions that we find to discuss research topics or technical problems, I am always impressed by the elegance and clarity of his ideas. I want to thank the PoSET team, David Janin and Myriam de Sainte- Catherine with whom I am trying to find the best tune. When I arrived at LaBRI, I was immediately welcome by the team Méthodes formelles. Not only did it provide me with a strong scientific environment, but also could I have very stimulating discussions with its members. Among these people, I address special thanks to Géraud Sénizergues whose vivid enthusiasm and joyful pleasure in research is strongly contagious; to Igor Walukiewicz who introduced me to the fascinating world of infinite objects; and to Anca Muscholl for her support and encouragement. Research is a matter of ideas and inspiration. It seems impossible to do anything of interest without the insights and the ideas of other researchers. vii This is true for me and I must say that I have had the chance to collaborate with very talented researchers to whom my work owes most. I want to thank Makoto Kanazawa who has provided me with a postdoc when I needed one, for the common work and his patient listening to my so often wrong intuitions. I thank also Greg Kobele for the discussions on linguistics and the formal work. He is the only person I know who is combining at such a high degree an expertise in linguistic and in mathematics. Ryo Yoshinaka did his PhD nearly at the same time as I did and I remember cheerfully the moment when I could work with him. I want to thank my PhD students Pierre and Jérôme and my master students for having trusted me as a superviser. We still have not produced research together, but I hope this will happen before he retires, so I thank Pierre Casteran for teaching me CoQ and sharing with me his smart ideas about coding in the calculus of constructions. Finally I would like to thank my wife for her constant support and for taking care of everything at home when I am away. A final word for my sons, Arthur and Timothée, to whom I want to say again how much I care for them. viii Chapter 1 Introduction 1.1 Research context This document is a synthesis of the research that I carried out in the past ten years and is part of my file to obtain the habilitation à diriger les recherches.I review the work I have done since I obtained my PhD in 2005. I did this work from the end of 2005 until the beginning of 2007 as a postdoc at the National Institute of Informatics in Tokyo under the supervision of Makoto Kanazawa. I then joined INRIA Bordeaux Sud-Ouest as a researcher in the project Signes. My work has taken place in LaBRI and I have been strongly influenced by its scientific environment. While I have been in the team Signes, I have been working on formalisms for natural language, aiming at understanding their algorithmic properties, their limitations and their relations. In 2011, when the team Signes was ended, I was part of the ANR project FREC, on Frontiers of RECognizability. This project allowed me to apply methods I had developed for grammatical formalisms to verification problems and infinitary systems. In this period of time I have had the opportunity to supervise two PhD students: • Pierre Bourreau who defended his thesis in 2012. He worked on general- izing parsing methods for abstract categorial grammars based on datalog programs. • Jérôme Kirman who is expected to defend his thesis by the end of the year. He worked on high-level modeling of natural language based on logics related to finite state automata. In the context of FREC, I have supervised the postdoctoral studies of Lau- rent Braud on higher-order schemes. I also supervised a number of master students on various topics related to formal language theory. 1 1.2 Research orientation and main contributions My research activities find applications in two different fields: mathematics of language and verification. In my work I do not really make a distinction between those two seemingly different fields. Indeed, ideas or tools I use for solving a problem in one field may then find some applications in the other. This is mainly due to the technical apparatus that I use: λ-calculus and formal language theory.