語用論(Pragmatics)を考える

Coq is a proof assistant based on a higher-order logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.ens-lyon.fr:/pub/LIP/COQ/V6.1

https://hal.inria.fr/inria-00069968/document

The Coq proof assistant : reference manual, version 6.1

The π-Calculus: A theory of mobile processes by Davide Sangiorgi and David Walker Formal methods have formed the foundation of Computer Science since its inception. Although, initially these formal methods dealt with processes and systems on an individual basis, the paradigm has shifted with the dawn of the age of computer networks. When dealing with systems with interconnected, communicating, dependent, cooperative, and competitive components, the older outlook of analyzing and developing singular systems—and the tools that went with it—were hardly suitable. This led to the development of theories and tools that would support the new paradigm. On the tools end, the development has been widespread and satisfactory: programming languages, development frameworks, databases, and even end-user software products such as word processors, have gained network-awareness. However on the theoretical end, the development has been far less satisfactory. The major work was done by Robin Milner, Joachim Parrow, and David Walker who developed the formalism known as π-calculus in 1989. π-calculus is a process calculus that treats communication between its components as the basic form of computation. It has been quite successful as a foundation of several other calculi in the field and as Milner puts it, it has become common to express ideas about interactions and mobility in variants of the calculus. Introduction The current book serves as a comprehensive reference to π-calculus. Besides Milner's own book on the subject, this is the only other book-length publication on the topic. In many ways, it is much more comprehensive than Milner's: a much wider area of topics are dealt with and in more detail as well. Contents The book is split into seven part. The first part presents the basic theory of π-calculus. However, basic does not mean concise: every topic is discussed in great detail. The section on bisimulation is particularly intensive and provides several insights about the motivation for the theory. Part two discusses several variants of the original calculus. By varying several characteristics of the calculus, such as whether a process can communicate with more than processes at a time, we can obtain these variants. A number of interesting properties of the language are studied by the other when discussing these variants. As can be understood from the title, the calculus' contribution to analyzing mobile processes is a major topic, and it is dealt with extensively starting from part three. The basics are introduced by the way of a sophisticated typing system whose application in speciying complex processes is presented in part four. Part five looks at higher-order π-calculus in which composed systems are considered as first-class citizens. Part six is one of the best in the book and discusses the relation between π-calculus and lambda-calculus, which is an older and more basic calculus. Finally part seven shows how π-calculus can be employed in studying practical, modern software engineering concepts such as object-oriented programming. Impressions One of my disappointments with this book is in how often the reader is left perplexed with some of the theoretical developments, specially in part three. π-calculus is a complicated topic, even for the graduate student audience to which this book is directed, and the author would have done much better by reducing the number of topics and instead focusing on more lucid and detailed explanations. There are several experimental digressions throughout the book, which although interesting, take away from some of the momentum of sequential study. For example, topics such as comparison and encoding of one language to another could be easily moved to a separate section in order to make the content more suitable for self-study. Another issue is the little effort towards making the connection from the theoretical to the practical. The main reason why formal methods have not been adopted in mainstream software development pracitces is that often it is unclear to developers how formalisms can contribute towards the software engineering process. The book would have served its purpose much better if it had spent part of eah chapter discussing the practical application of that chapter's content. For example, congruence checking and bisimulation can be incredbily exciting topics for programmers to learn if they can see practical applications of such powerful techniques. Beyond the above criticism, the book is absolutely indispensible to students and researchers in the field of formal methods. Currently it serves as the primary reference for anyone who wishes to learn the various aspects of π-calculus in detail. Raheel Ahmad

The Ï-Calculus: A theory of mobile processes

Modal and temporal logics are finding new and varied applications in Computer Science in fields as diverse as Artificial Intelligence [Marek et al.,1991], Models for Concurrency [Stirling, 1992] and Hardware Verification [Nakamura et al.,1987]. Often the eventual use of these logics boils down to the task of deducing whether a certain formula of a logic is a logical consequence of a set of other formula of the same logic. The method of semantic tableaux is now well established in the field of Automated Deduction [Oppacher and Suen, 1988; Baumgartner et al., 1995; Beckert and Possega, 1995] as a viable alternative to the more traditional methods based on resolution [Chang and Lee, 1973]. In this chapter we give a systematic and unified introduction to tableau methods for automating deduction in modal and temporal logics. We concentrate on the propositional fragments restricted to a two-valued (classical) basis and assume some prior knowledge of modal and temporal logic, but give a brief overview of the associated Kripke semantics to keep the chapter self-contained.

Tableau Methods for Modal and Temporal Logics

Many fields of science investigate states and processes as resources. Chemistry, thermodynamics, Shannon's theory of communication channels, and the theory of quantum entanglement are prominent examples. Questions addressed by these theories include: Which resources can be converted into which others? At what rate can many copies of one resource be converted into many copies of another? Can a catalyst enable a conversion? How to quantify a resource? We propose a general mathematical definition of resource theory. We prove general theorems about how resource theories can be constructed from theories of processes with a subclass of processes that are freely implementable. These define the means by which costly states and processes can be interconverted. We outline how various existing resource theories fit into our framework, which is a first step in a project of identifying universal features and principles of resource theories. We develop a few general results concerning resource convertibility.

A mathematical theory of resources

Extended Curry-Howard Correspondence for a Basic Constructive Modal Logic

The paper studies the properties of the subnets of proof-nets. Very simple proofs are obtained of known results on proof-nets for MLL ? , Multiplicative Linear Logic without propositional constants.

Subnets of proof-nets in MLL -

We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite well-behaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour.

/pdf/categorical-proof-theory-of-classical-propositional-calculus-23tlk7dj9s.pdf

Categorical proof theory of classical propositional calculus

This paper studies the properties of the subnets of a proof-net
 for first-order Multiplicative Linear Logic without propositional constants ( MLL − ), extended with the rule of Mix: from [vdash ]Γ and [vdash ]Δ infer [vdash ]Γ, Δ. Asperti's correctness criterion and its interpretation in terms of concurrent processes are extended to the first-order case. The notions of kingdom and empire of a formula are extended from MLL − to MLL − + MIX . A new proof of the sequentialization theorem is given. As a corollary,
 a system of proof-nets is given for De Paiva and Hyland's Full Intuitionistic Linear Logic with Mix; this result gives a general method for translating Abramsky-style term assignments into proof-nets, and vice versa .

Subnets of proof-nets in multiplicative linear logic with MIX

The logic for pragmatics extends classical logic in order to characterize the logical properties of the operators of illocutionary force such as that of assertion and obligation and of the pragmatic connectives which are given an intuitionistic interpretation. Here we consider the cases of assertions and conjectures: the assertion that a mathematical proposition α is true is justified by the capacity to present an actual proof of α, while the conjecture that α is true is justified by the absence of a refutation of α. We give sequent calculi of type G3i and G3im inspired by Girard's LU, with subsystems characterizing intuitionistic reasoning and some forms of classical reasoning with such operators. Extending Godel, McKinsey, Tarski and Kripke's translations of intuitionistic logic into S4, we show that our sequent calculi are sound and complete with respect to Kripke's semantics for S4.

Gianluigi Bellin

Papers

Extended Curry-Howard Correspondence for a Basic Constructive Modal Logic

Subnets of proof-nets in MLL -

Categorical proof theory of classical propositional calculus

Subnets of proof-nets in multiplicative linear logic with MIX

Towards a Logic for Pragmatics. Assertions and Conjectures