Jan M. Smith

Inductively generated formal topologies

Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined.

An Interpretation of Martin-Lof's Type Theory in a Type-Free Theory of Propositions

We present a formal theory of propositions and combinator terms, and in this theory we give an interpretation of Martin-Lofs type theory. The construction of the interpretation is inspired by the semantics for type theory, but it can also be viewed as a formalized realizability interpretation.

The Independence of Peano's Fourth Axiom from Martin-Lof's Type Theory Without Universes

In Hilbert and Ackermann [2] there is a simple proof of the consistency of first order predicate logic by reducing it to propositional logic. Intuitively, the proof is based on interpreting predicate logic in a domain with only one element. Tarski [7] and Gentzen [1] have extended this method to simple type theory by starting with an individual domain consisting of a single element and then interpreting a higher type by the set of truth valued functions on the previous type. I will use the method of Hilbert and Ackermann on Martin-Lofs type theory without universes to show that ¬Eq( A, a, b ) cannot be derived without universes for any type A and any objects a and b of type A . In particular, this proves the conjecture in Martin-Lof [5] that Peanos fourth axiom (∀ x ϵ N )¬ Eq( N , 0, succ( x )) cannot be proved in type theory without universes. If by consistency we mean that there is no closed term of the empty type, then the construction will also give a consistency proof by finitary methods of Martin-Lofs type theory without universes. So, without universes, the logic obtained by interpreting propositions as types is surprisingly weak. This is in sharp contrast with type theory as a computational system, since, for instance, the proof that every object of a type can be computed to normal form cannot be formalized in first order arithmetic.

Propositions and specifications of programs in Martin-Löf's type theory

The constructive meaning of mathematical propositions makes it possible to identify specifications for computer programs with propositions. In Martin-Löfs type theory, constructing a program satisfying a specification corresponds to proving the proposition expressing the specification. These ideas are explained and some examples of specifications are given.

Support for teaching formal methods

This report describes a growth path for the area referred to as formal methods within the computing education community. We define the term formal methods and situate it within our field by highlighting its role in Computing Curricula 1991, Computing Curricula 2001, and the SoftWare Engineering Body Of Knowledge (SWEBOK). The working group proposes an enhancement to an existing web resource, which is a rich collection of materials and links related to formal methods. The new resource is designed to provide a bridge between the general computing education community and the formal methods community. The goal is to allow the latter to provide useful support for the former for the ultimate benefit of all of our students. Eventually, the working group aspires to see the concepts of formal methods integrated seamlessly into the computing curriculum so that it is not necessary to separate them in our discussions.

Formal Topologies on the Set of First-Order Formulae

The completeness proof for first-order logic by Rasiowa and Sikorski [13] is a simplification of Henkins proof [7] in that it avoids the addition of infinitely many new individual constants. Instead they show that each consistent set of formulae can be extended to a maximally consistent set, satisfying the following existence property: if it contains (∃ x ) ϕ it also contains some substitution ϕ ( y / x ) of a variable y for x . In Fefermans review [5] of [13], an improvement, due to Tarski, is given by which the proof gets a simple algebraic form. Sambin [16] used the same method in the setting of formal topology [15], thereby obtaining a constructive completeness proof. This proof is elementary and can be seen as a constructive and predicative version of the one in Fefermans review. It is a typical, and simple, example where the use of formal topology gives constructive sense to the existence of a generic object, satisfying some forcing conditions; in this case an ultrafilter satisfying the existence property. In order to get a formal topology on the set of first-order formulae, Sambin used the Dedekind-MacNeille completion to define a covering relation ⊲ DM . This method, by which an arbitrary poset can be extended to a complete poset, was introduced by MacNeille [9] and is a generalization of the construction of real numbers from rationals by Dedekind cuts. It is also possible to define an inductive cover, ⊲ I , on the set of formulae, which can also be used to give canonical models, see Coquand and Smith [3].

Program derivation in type theory: A partitioning problem

Abstract Martin-Lofs type theory is a theory in which one can write both specifications and programs. By interpreting propositions as types, predicate logic is available when formulating a specification. The rules of type theory are formulated as tactics which makes a “top down” construction of programs possible. These ideas are illustrated by a formal derivation of a program for a partitioning problem.

Optimized Encodings of Fragments of Type Theory in First Order Logic

The paper presents sound and complete translations of several fragments of Martin-Lofs monomorphic type theory to first order predicate calculus. The translations are optimised for the purpose of automated theorem proving in the mentioned fragments. The implementation of the theorem prover Gandalf and several experimental results are described.

Evolution and Logic

Hume’s analysis of causality opens up for an Darwinistic understanding of causality. It is argued that basic logic can be given a similar analysis as causality and hence also can be seen as a result of biological evolution. Philosophically, there is a close connection to Kant’s forms of intuition.

Types for Proofs and Programs

A type theoretic programming language is introduced that is based on lambda calculus with coproducts, products and inductive types, and additionally allows the definition of recursive functions in the way that is common in most functional programming languages. A formal system is presented that checks whether such a definition is structurally recursive and a soundness theorem is shown for this system. Thus all functions passing this check are ensured to terminate on all inputs. For the moment only non-mutual recursive functions are considered.