Abstracts of the lectures at the Winter Workshop in Logics, Types and Rewriting

Cedar Room, Heriot-Watt University, Edinburgh

1-3 February 2000

 ULTRA groupUseful Logics, Types, Rewriting, and Applications UKII UK Institute of Informatics, UK

The list of abstracts of the lectures is as follows:

Henk Barendregt Catholic University of Nijmegen

• Proof-assistants for mathematics
• Ample attention has been given to the question what formal systems a proof-assistant should support and also to the meta-theory of those systems. Somewhat neglected is the question how proof-assistants can be user-friendly for developing mathematics. Some requirements will be presented.

Stefano Berardi University of Turin

• A notion of computation for Analysis
• We investigate which notion of computation would be suitable to validate a constructive interpretation of classical second order choice. We propose a notion of computation extending Coquand-Ong game model of simple typed lambda calculus, by allowing possible transfinite computations when functional argument are discontinous.

N.G. de Bruijn Eindhoven University of Technology:

• The Design of Automath
• Around 1967 the Automath system was designed in a relatively short time, in a world that was not yet ready for the idea of automatic checking. A number of decisions had to be taken in order to get to a simple flexible system, able to cater for various kinds of foundation, and good enough to handle mathematics on a very large scale. One of the novelties was that the system did not just want to check proofs, but complete theories, or rather complete books. As a consequence, definitions became essential parts of the books, completely similar to proved statements; another consequence was that there was no longer a difference in treating logic and mathematics. Everything fitted nicely in the setting of a general form of type theory. An important decision was to stay out of artificial intelligence, apart from the fundamental task of automated type checking. Another decision was to resist the temptation to go after time saving devices (at the price of complicating the system). In particular it was decided that induction or recursion were not to be implemented in the system, since they could easily be treated as subjects in the books.

Alan Bundy University of Edinburgh

• What is a Proof?
• To those brought up in a logic-based tradition there seems to be a simnple and clear definition of proof. But this is largely a 20th century invention; earlier proofs had a different nature. Even in recent decades there has been fierce debate about the nature of proofs -- a debate fueled by efforts to automate the proof process. We will present and contrast some different notions of proof and the rival arguments between them.

Gilles Dowek INRIA

• Axioms vs. Rewrite rules : from completeness to cut elimination
• Orienting axioms as rewrite rules is a traditionnal way to cut off search space in automated theorem proving. We try to show in this talk that such an orientation is not just a way to implement a selective use of axioms but that it may cut off search space more dramatically. In particular, the completeness of proof search may be lost in some cases and new proof theoretical tools are needed when it is not.

Martin Hofmann University of Edinburgh

• A type system for bounded space and functional in-place update
• We show how linear typing can be used to obtain functional programs which modify heap-allocated data structures in place.
We present this both as a design pattern'' for writing C-code in a functional style and as a compilation process from linearly typed first-order functional programs into malloc()-free C code.
The main technical result is the correctness of this compilation.
The crucial innovation over previous linear typing schemes consists of the introduction of a resource type <> which controls the number of constructor symbols such as cons in recursive definitions and ensures linear space while restricting expressive power surprisingly little.
While the space efficiency brought about by the new typing scheme and the compilation into C can also be realised by with state-of-the-art optimising compilers for functional languages such as Ocaml, the present method provides guaranteed bounds on heap space which will be of use for applications such as languages for embedded systems or proof carrying code.
We discuss various extensions, in particular an extension with FIFO queues admitting constant time catenation and enqueuing, and an extension of the type system to fully-fledged intuitionistic linear logic.

Helene Kirchner University of Nancy

• Algebraic Specifications, Higher-Order Types and Set-Theoretic Models
• In most algebraic specification frameworks, the type system is restricted to sorts, subsorts, and first-order function types. This is in marked contrast to the so-called model-oriented frameworks, which provide higher-order types, interpreted set-theoretically as Cartesian products, function spaces, and power-sets. This talk presents a simple framework for enriching algebraic specifications with higher-order types and set-theoretic models, while keeping to the tractable logic of Horn clauses. It may be regarded as the basis for a Horn-clause approximation to the Z framework, and has the advantage of being amenable to prototyping and automated reasoning. Standard set-theoretic models are considered, and conditions are given for the existence of initial reducts of such models. Algebraic specifications for various set-theoretic concepts are considered.

Jan-Willem Klop Free University of Amsterdam

• Proof systems for cyclic term graphs
• Cyclic term graphs are important in functional programming and have also other applications such as recursive types. We discuss and compare two complete proof systems, one being the usual 'folklore' system based on a fixed point rule, and one based on a coinductive proof rule due to Brandt and Henglein.

Gordon Plotkin University of Edinburgh

• Another Meta-Language for Programming with Bound Names Modulo Renaming
• Pitts and Gabbay have recently described a language for metaprogramming for languages with binders. (See: http://www.cl.cam.ac.uk/users/amp12/papers/index.html#metlpb).
It is based on their Fraenkel-Mostowski (FM) set model. Inspired by this work, we present a very similar language based instead on the Fiore-Plotkin-Turi presheaf model (over the category of finite cardinals).
The main difference is in the approach to the type theory. For Gabbay and Pitts the judgments are that a value has a certain type and that it does not depend on certain atoms. We rather take a positive approach and the judgment is that the atoms depended on are included in a certain set (cf. analogous statements about free variables in languages with binders). This positive judgment is then part of the structural logic of the type theory and is internalised by the abstraction functor, yielding a straightforward semantics (just as commas in contexts are internalised by products).
It may be that the two approaches are equivalent, apart from a certain linearity restriction inherent in the FM model. We have at any rate shown that one is not restricted to the FM model for the semantics of variable-binding metalanguages, and it remains unclear which model is to be preferred.

Simona Ronchi Della Rocca University of Turin

• Intuitionistic Logics and Intersection Type assignment System
• The intersection type assignment system for $\lambda$-calculus is an extension of the Curry's type assignment system, where types are built using, besides the arrow constructor for functional types, an {\em intersection} constructor, allowing to assign to the same variable more than one type. The so obtained system is very powerful, it has interesting properties (it types all and only the solvable terms) and, once equipped with a suitable notion of subtyping, it can supply useful tools for programs analysis. Here we study the logical counterpart of the intersection type assignment system, in order to look at its properties from a purely logical point of view. The related logics can be obtained from the intuitionistic logics with implication and conjunction, by allowing the parallel derivation of a finite number of formulas, and by restricting the introduction of the conjunction just to formulas which have been proved in parallel. We define a kind of {\em Curry-Howard isomorphism} between terms of $\lambda$-calculus and such a logic: in the spirit of Girard's proof nets, we associate to every term a {\em proof schema}, i.e., a tree whose nodes are labelled by names of the logical rules. Then a term can be typed if and only if its associated proof schema can be transformed in a true proof by:
• fixing the degree of parallelism of every rule in it;
• modifying its structure by introducing some conjunction introduction rules;
• decorating the resulting proof schema by logical formulas.
Then the principal type property of the type assignment system, which has been proved in the past through a difficult expansion-unification property of types, becomes now an easy consequence of some very elementary properties of the logical system. Moreover the logical system supplies useful tools for designing both a type inference semi-algorithm and its decidable restrictions.

Jonathan Seldin Lethbridge University

• On Lists and Other Abstract Data Types in the Calculus of Constructions
• The representation of the inductively defined abstract data type for lists was left incomplete in \cite[\S 9]{seldin:PTCC}. Here that representation is completed, and it is proved that all extra axioms needed are consistent. Among the innovations of this paper is a definition of \textsf{cdr}, whose definition was left for future work in \cite[\S 9]{seldin:PTCC}. The results are then extended to other abstract data types, those of \cite{berardi:EDTP}. The method used to define \textsf{cdr} for lists is extended to obtain the definition of an inverse for each argument of each constructor of an abstract data type. These inverses are used to prove the injective property for the constructors. Also, Dedekind's method of defining the natural numbers is used to define a predicate associated with each abstract data type, and the use of this predicate makes it unnecessary to postulate the induction principle. The only axioms left to be proved are those asserting the disjointness of the co-domains of different constructors, and it is shown that those axioms can be proved consistent.

\begin{thebibliography}{}

\bibitem[Berardi 1993]{berardi:EDTP} Stefano Berardi. (1993) \newblock Encoding of data types in pure construction calculus: a semantic justification. \newblock In G\'{e}rard Huet and Gordon Plotkin, editors, {\em Logical Environments}, pages 30--60. Cambridge University Press.

\bibitem[Pfenning and Paulin-Mohring 1989]{pfenning/paulin-mohring:IDTC} Frank Pfenning and Christine Paulin-Mohring. (1989) \newblock Inductively defined types in the calculus of constructions. \newblock In M.~Main, A.~Melton, M.~Mislove, and D.~Schmidt, editors, {\em Mathematical Foundations of Programming Semantics: 5th International Conference, Tulane University, New Orleans, Louisiana, USA, March 29--April 1, 1989, Proceedings}, volume 442 of {\em Lecture Notes in Computer Science}, pages 209--228. Springer-Verlag.

\bibitem[Seldin 1997]{seldin:PTCC} Jonathan~P. Seldin. (1997) \newblock On the proof theory of {C}oquand's calculus of constructions. \newblock {\em Annals of Pure and Applied Logic}, 83:23--101.

\end{thebibliography}

Joe Wells Heriot-Watt University

• A Paradigm Shift in Program Analysis and Transformation via Intersection and Union Types
• Implementations of modern higher-order typed programming languages (e.g., Haskell, Java, ML) increasingly rely on advanced type systems to ensure that desirable properties (e.g., safety, security, freedom from deadlock) are preserved or enforced by compilation and also to guide representation and optimization decisions. For programming flexibility (e.g., code reuse) and more precise program analysis, modern type systems must support some form of type polymorphism, and generally do so with universal and/or existential quantifiers in types. The most well-known example is Milner's type system, used at the core of Haskell and ML. Mbr>
While there is a rich theory of universal/existential quantifiers, there are two important difficulties in practice. (1) Systems with these quantifiers generally do not have a strong principality property, i.e., there is no most general analysis for each program fragment. This makes it difficult to perform type analysis compositionally (i.e., where program fragments are analyzed independently) and sometimes even makes type analysis impossible. (2) These quantifiers hide information, making it hard to extend type analysis to flow analysis and to perform type/flow-directed program transformations.

To address these difficulties, I have designed type systems which support type polymorphism with intersection and union types instead of universal/existential types. In collaboration with my colleagues in the Church Project, I have developed analysis and transformation algorithms using these systems. Overcoming difficulty (1), we have used intersection type systems with strong principality properties to develop new type analysis algorithms which are compositional and also accept many safe programs rejected as ill-typed by earlier algorithms. Overcoming difficulty (2), we have shown how to use a system with intersection/union types to express polyvariant flow analyses and to guide a new style of program transformation and optimization: type/flow-directed compilation. This new technology is the foundation for an experimental compiler we are developing. I will discuss how our new approach to compilation will contribute to efficient, reliable, and modular compilation.

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