# Abstracts of talks at Festival Workshop in Foundations and Computations

## Heriot-Watt University, Edinburgh

### Sunday 16 July-Tuesday 18 July 2000

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

The list of abstracts is as follows:

• Laura Crosilla (Leeds, UK)
• A Weak Constructive Set Theory witsh Inaccessible sets
• See this extended abstract

• Gilles Dowek (INRIA-Rocquencourt, FR):
• In some theories (e.g. set theory, the Stratified Foundations, ...) the usual notion of proof is extended with folding and unfolding steps, a sequence of such steps is called a folding-unfolding cut and a proof containing such a cut can be reduced. We show that this notion of folding-unfolding cut is an instance of the more general notion of "cut modulo" and that cut elimination modulo for these theories implies cut elimination with folding-unfolding cuts. Cut elimination for a theory modulo can be proved by constructing a so called pre-model. We show that the notion of normalization model, introduced by Crabbe' to prove cut elimination for the Stratified Foundations, is more or less an instance of the notion of pre-model. The techniques developed by Crabbe' to construct a normalization model for the Stratified Foundations can be adapted to construct a pre-model of this theory. This permits to give a simpler and more modular cut elimination proof for the Stratified Foundation. We discuss at last the generality of this pre-model construction method.

• Jan van Eijck(University of Amsterdam, NL):
• Dynamic First Order Logic
• Dynamic First Order Logic results from interpreting quantification over a variable v as change of valuation over the v position, conjunction as sequential composition, disjunction as nondeterministic choice, and negation as (negated) test for continuation. We present a tableau style calculus for DFOL with explicit (simultaneous) substitution, prove its soundness and completeness, and point out its relevance for programming with dynamic first order logic, by interpreting global variable declarations as dynamic existential quantifiers and local variable declarations as dynamic existential quantifiers in the immediate scope of a double negation to block off the dynamic effects.

• Massimo Felici (University of Edinburgh, UK):
• A Tableaux system for a fragment of the hyperset theory
• Non well founded phenomena arises in a variety of ways in informatics. This talk develops a tableaus system for the hyperset theory which is based on non well founded set theory. See this extended abstract.

• Jacques Fleuriot (University of Edinburgh, UK):
• Automating Newton's calculus in Isabelle
• In this talk, I shall give an overview of the construction and use of the hyperreals in the theorem prover Isabelle within the framework of higher order logic (HOL):
The theory, which includes infinitesimals and infinite numbers, is based on the hyperreal number system developed by Abraham Robinson in his Nonstandard Analysis (NSA). I shall first outline the ultrapower : construction of the hyperreals in Isabelle. This was carried out strictly through the use of definitions to ensure that the foundations of NSA in the prover are sound. The mechanization has required constructing the various number systems leading to the reals, and then going one step further to define the hyperreals using a mechanized theory of filters and ultrafilters.
I shall next describe the use of the new types of numbers and new relations on them to formalize familiar concepts from analysis such as sequences, limits, continuous functions, derivatives, power series, and transcendental functions. I will point out the merits of the nonstandard approach with respect to the practice of analysis and mechanical theorem proving. In particular, I will argue that the overall result is an intuitive, yet rigorous, development of real analysis, with a relatively high degree of proof automation in many cases.

• Therese Hardin (Paris 6 and INRIA-Rocquencourt, FR):
• The calculus of contexts of lambda sigma
• The notion of context, i.e. a lambda-term with a hole, is rather informally defined in teaching books on lambda-calculi, where it allows a kind of operational view of the congruence properties of the reduction. A more formal treatment of contexts was implicit in the pionner works on higher-order unification, which is unification of contexts. Recently, the notion of context has been used to represent modules, environments, etc. This talk recalls first the coding of contexts in lambda-sigma as first-order terms with first-order term variables representing the holes. Then, some comparisons with other approachs are made.

• Roger Hindley (University of Swansea, UK):
• The birth of lambda-calculus and combinatory logic
• This talk will be about the early days of the concepts of function-abstraction and application, leading to lambda-calculus and combinatory logic. Both these systems achieved very satisfying successes in their early years before 1940, which were not surpassed until their application to computer science in the 1970's. The talk will be based on an article being written with Felice Cardone for a forthcoming history of modern logic to be edited by D. van Dalen and J. Dawson. It will include a test which will distinguish between computer scientists who are genuine and those who are just logicians looking for money.

• Yoko Motohama (Torino University, Italy):
• Compositional Characterizations of lambda terms using intersection types
• See this extended abstract

• Cesar Munoz (ICASE, NASA, USA) :
• Aircraft Trajectory Modeling and Analysis: A challenge to Formal Methods
• The Airborne Information for Lateral Spacing (AILS) program at NASA aims at giving pilots the information necessary to make independent approaches to parallel runways with spacing down to 2500 feet in Instrument Meteorological Conditions. The AILS concept consists of accurate traffic information visible on the navigation display and an alerting algorithm which warns the crew when one of the aircraft involved in a parallel landing is diverting from its intended flight path. In this talk we present a formal model of aircraft approaches to parallel runways. Based on this model, we analyze the alerting algorithm with the objective of verifying its correctness. The formalization is conducted in the general verification system PVS.

• Alan Mycroft (Cambridge University and AT&T Labs, UK):
• Type Based Decompilation
• We describe a system which decompiles (reverse engineers) C programs f rom target machine code by type-inference techniques. This extends recent trends in the converse process of compiling high-level languages whereby type information is preserved during compilation. The algorith ms remain independent of the particular architecture by virtue of treating target instructions as register-tr ansfer specifications. Target code expressed in such RTL form is then transformed into SSA form (undoing register c olouring etc.); this then generates a set of type constraints. Iteration and recursion over data-structure s causes synthesis of appropriate recursive C structs; this is triggered by and resolves occurs-check constraint violation. Other constraint violations are resolved by C's casts and unions. In the limit we use heuristics to select between equally suitable C code---a good GUI would clearly facilitate its professional u se.

• Gopalan Nadathur (University of Minnesota, USA):
• Correspondences between Classical, Intuitionistic and Uniform Provability and their Impact on Proof Search
• Based on an analysis of the inference rules used, it is possible to characterize the situations in which classical and intuitionistic provability coincide. This observation is useful, for instance, in sanctioning the use of classical principles in determining intuitionistic derivability for formulas satisfying certain syntactic restrictions. It is similarly possible to relate classical and intuitionistic derivability to uniform provability, a constrained form of intuitionistic provability that embodies a special notion of goal-directedness. Uniform provability has been employed in describing abstract logic programming languages and this analysis permits the richest such languages within the relevant derivation systems to be identified. Even when the other provability notions deviate from uniform derivability, it is often possible to reduce them to uniform provability through the addition of the negation of the formula to be proved to the assumption set. Further, this enhancement of the assumptions can be factored into certain derived rules in a sequent calculus setting. An interesting aspect of these observations is that they lead directly to a proof procedure for classical logic that incorporates into a single framework previously proposed mechanisms for dealing with disjunctive information in assumptions and for handling hypotheticals. In this talk we will explore the correspondences alluded to above between the different derivability notions and we will also sketch their relevance to proof search.

• Catherine Piliere(INRIA, FR)
• Guarded Exception Handling: Some results
• See this extended abstract

• Sarah Rees (Newcastle, UK):
• Formal languages applied to group theory
• For a group defined abstractly as the set of all products (strings) of a finite set of symbols, subject to a finite set of equations holding between products, some questions about the group may be theoretical undecidable. When they are decidable, an analysis of their complexity may be of great interest. Examples of questions of this type include the word problem (Is a given product of symbols equal to the identity element?'), the conjugacy problem (Are two given products u,v' related by an equation ux=xv' for some x?) and the isomorphism problem (Are two given groups G and H isomorphic, i.e. abstractly the same, but differently described?'). The formal language hierarchy provides one way of describing the complexity of a decision problem. For the word problem, the regular languages define the complexity for a very natural family of groups, those with finitely many elements, and the context-free languages a similarly easily defined family, the virtually free groups. These facts are well known, but much less is known about groups whose word problem is defined by language families further up the hierarchy. Regular languages are also naturally associated with another problem for groups, that of finding a good normal form, i.e. a set of strings which represents the group's elements of the groups without too much repetition and displays well the algebra relating the elements. Groups within the family of automatic groups can be combed' by regular languages. A defining geometric property of a combing can also be expressed in terms of regular languages. Various non-automatic groups can be combed by languages further up the language hierarchy, such as indexed families, real-time families, but less is known about these more general combings. It is known that groups with regular synchronous combings have deterministic context-sensitive word problem, while any group with a combing has non deterministic context-sensitive word problem. This talk will survey what is known about the language complexity of the word problem and of combings, and about connections between these two problems.

• Extensional Set Equality in the Calculus of Constructions
• I developed a significant part of set theory in the calculus of constructions. This followed a suggestion of Huet.
Berardi (personal communication) has informed me that set theory has not been implemented this way in Coq because contradictions follow from the axiom of extensionality. In Coq set theory has been implemented with a new type for sets. None of the consistency results of \cite{seldin:PTCC} apply to this implementation. It is hardly surprising that contradictions follow from the axiom of extensionality, for the equality which is Leibniz' equality, is intensional rather than extensional. The language of the calculus of constructions is rich enough to provide predicates that can distinguish different equivalent propositions.
In ordinary axiomatic set theory, set equality behaves like Leibniz equality in that equals may always be replaced by equals. This is clearly untrue in the calculus of constructions with $\mathsf{SetQ}$. However, there is a class of properties with respect to which equals under $\mathsf{SetQ}$ can be replaced by equals. This is the class of \emph{extensional properties}. The class of extensional properties can be shown to parallel the definition of well formed formulas in first-order axiomatic set theory.
As pointed out in \cite[Remark 17]{seldin:PTCC}, while we can prove most of the axioms of constructive set theory, we cannot prove the axioms of power set and $\in$-induction. Furthermore, we cannot expect to get all of set theory this way, because the consistency of this implementation follows from the strong normalization theorem for the calculus of constructions. If a complete set theory satisfying these axioms is desired, then the implementation given here will not do. But the need for a different (and, presumably, stronger implementation does not follow from any problem with the axiom of extensionality. See this extended abstract for further details.
• Eleni Spiliopoulou (Bristol, UK):
• The Brisk machine: the next step in the execution of functional languages
• Here is the abstract.

• Iain Stewart (University of Leicester, UK):
• A programming approach to descriptive complexity theory
• Finite model theory is the study of what can and cannot be said about classes of finite structures using different logics. A substantial sub-area of finite model theory is descriptive complexity theory; essentially, the study of the expressibility of logics on classes of finite structures in relation to computational complexity. Perhaps the seminal result in descriptive complexity theory is Fagin's Theorem which states that a problem is in the complexity class NP if, and only if, it can be defined in existential second-order logic. Most logics studied within descriptive complexity theory are traditional in the sense that they are extensions of first-order logic by, for example, inductive operators, second-order constructs or (sequences of) Lindstroem quantifiers. However, one can take a much more computational view and consider program schemes and the (classes of) finite structures they accept. Program schemes are much closer to the notion of a high-level program than are logical formulae but are still amenable to logical analysis. Program schemes were extensively studied in the seventies and eighties but have since declined somewhat. In this talk, I shall introduce the rudiments of descriptive complexity theory and explain how the study of program schemes can yield new finite-model theoretic results and techniques.

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