This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.

http://types10.mimuw.edu.pl/content/slides/day1/HenkBarendregt.pdf

Lambda Calculus with Types

We present an order-extensional, order (or inequationally) fully abstract model for Scott's language pcf. The approach we have taken is very concrete and in nature goes back to S. C Kleene (1978, in “General Recursion Theory II, Proceedings of the 1977 Oslo Symposium,” North-Holland, Amsterdam) and R. O. Gandy (1993, “Dialogues, Blass Games, Sequentiality for Objects of Finite Type,” unpublished manuscript) in one tradition, and to G. Kahn and G. D. Plotkin (1993, Theoret. Comput. Sci.121, 187?278) and G. Berry and P.-L. Curien (1982, Theoret. Comput. Sci.20, 265?321) in another. Our model of computation is based on a kind of game in which each play consists of a dialogue of questions and answers between two players who observe the following principles of civil conversation: 1.Justification. A question is asked only if the dialogue at that point warrants it. An answer is proffered only if a question expecting it has already been asked. 2.Priority. Questions pending in a dialogue are answered on a last-asked-first-answered basis. This is equivalent to Gandy's no-dangling-question-mark condition. We analyze pcf-style computations directly in terms of partial strategies based on the information available to each player when he or she is about to move. Our players are required to play an innocent strategy: they play on the basis of their view which is that part of the history that interests them currently. Views are continually updated as the play unfolds. Hence our games are neither history-sensitive nor history-free. Rather they are view-dependent. These considerations give expression to what seems to us to be the nub of pcf-style higher-type sequentiality in a (dialogue) game-semantical setting.

/pdf/on-full-abstraction-for-pcf-3zs5sgq4zx.pdf

On Full Abstraction for PCF

The finite π-calculus has an explicit set-theoretic functor-category model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plotkin and Power. Thus we obtain a novel algebraic description for models of the π-calculus, and validate an existing construction as the universal such model.

The algebraic operations are intuitive, covering name creation, communication of names over channels, and nondeterminism; the equations then combine these features in a modular fashion. We work in an enriched setting, over a “possible worlds” category of sets indexed by available names. This expands significantly on the classical notion of algebraic theories, and in particular allows us to use nonstandard arities that vary as processes evolve.

Based on our algebraic theory we describe a category of models for the π-calculus, and show that they all preserve bisimulation congruence. We develop a direct construction of free models in this category; and generalise previous results to prove that all free-algebra models are fully abstract.

/pdf/free-algebra-models-for-the-p-calculus-hwrnnbew6o.pdf

Free-algebra models for the π-calculus

This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logics is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading.

/pdf/an-introduction-to-substructural-logics-46gq1uirpk.pdf

An Introduction to Substructural Logics

https://homepages.inf.ed.ac.uk/gdp/publications/Comb_Effects_Jour.pdf

Combining effects: sum and tensor

In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky [1]) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these and consider their computational content.

/pdf/a-term-calculus-for-intuitionistic-linear-logic-v39izvmy4p.pdf

A Term Calculus for Intuitionistic Linear Logic

Glueing and orthogonality for models of linear logic

https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf

The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads

An equality relation on the terms of the A-calculus is an equivalence relation closed under the (syntactical) operations of application and A-abstraction. We may distinguish between syntactic and semantic ways of introducing equality relations, /^-equality is introduced syntactically; it is the least equality relation satisfying the equations for aand ^-conversion. For a more subtle way of introducing equality relations syntactically, consider the relations =f and =h of §5 of this paper. To give a semantic characterization of an equality relation, we simply take the relation ' has the same value in £>', where D is some model for the A-calculus. Of course, no equality relation is of interest to the intended interpretation of the A-calculus, unless it extends /^-equality. An equality relation is inconsistent if and only if it sets all terms equal; otherwise it is consistent. It is maximal consistent if and only if it is consistent and has no consistent proper extensions. In this paper we consider a class of continuous lattice models for the A-calculus, and a particular model, the Graph model. The same equality is induced by all the continuous lattice models; we shall refer to them as the Scott models (see [3], where they were first constructed). For the history of the Graph model see [4]. We shall give, in this paper, syntactic characterizations of the equality induced by the Scott models, and by the Graph model; and we shall show that the equality induced by the Scott models is the unique maximal consistent equality relation, extending the relation = H, which was proved consistent in [1]. We use x, y,z, w ... for variables, and M,N, P ... for terms of the A-calculus (with subscripts as necessary). D will refer to whatever model or models are under consideration. The content of our Theorem 5.4 (a) has been discovered independently by C. P. Wadsworth.

Martin Hyland

Papers

Combining effects: sum and tensor

A Term Calculus for Intuitionistic Linear Logic

Glueing and orthogonality for models of linear logic

The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads

A Syntactic Characterization of the Equality in Some Models for the Lambda Calculus