LTT: a type-theoretic framework for foundational pluralism

1. LTT: a type-theoretic frameworkfor foundational pluralism Zhaohui Luo Dept of Computer Science Royal Holloway, Univ of London

2. Type theory and applications • Proof assistants based on TTs • Agda (Sweden/Japan) and NuPRL (USA) • implementing Martin-Löf’s type theory • Coq (France), Lego/Plastic (UK) • implementing CIC (Calculus of Inductive Constructions) and ECC/UTT (Unifying Theory of dependent Types) • Application examples • Computer science • Program verification (eg, analysis of security protocols) • Dependently-typed programming • Formalisation of mathematics • Four-colour Theorem in Coq

3. Foundational pluralism • Two extreme positions in FOM • Neo-platonism (eg, set-theoretic foundation: Gödel/Maddy) • Revisionists (eg, intuitionism: Brouwer/Martin-Löf) • A pragmatic position – “pluralism” • Various maths based on different logical foundations • “Foundational pluralism” • Support in type theory and the associated tech? • Theorem proving technology based on TTs is not just for constructive reasoning! • Eg, Classical logic as well as intuitionistic logic

4. Consider the “combinations” of the following and their “negations”: (C) Classical logic (I) Impredicative definitions We would have • (CI) Ordinary (classical, impredicative) math Classical set theory/simple type theory, HOL/Isabelle • (C°I°) Predicative constructive math Martin-Löf’s TT, Agda/NuPRL • (C°I) Impredicative constructive math CIC/ECC/UTT, Coq/Lego/Plastic • (CI°) Predicative classical math Weyl, Feferman, Simpson, … Uniform foundational framework for formalisation to support pluralism?

5. Set-theoretic reasoning in type theory? • Current type theories • Strong in type-theoretic reasoning (eg, inductive types) • Not so strong in set-theoretic reasoning • Note: Types are NOT sets! (cf, non-inductive sets) • “a : A” – judgemental, meta-level • “s  S” – propositional • How should set-theoretic reasoning be supported? • Traditional (untyped) ZF set theory? (cf, Isabelle/ZF) • Combining with type-theoretic reasoning? • Two roles of (the usual notion of) sets: • Domain/range of functions: f : AB • Separation/selection from a domain: { x : A | P(x) } Now, types for (i) and typed sets for (ii)!

6. Type-theoretic framework LTT: structure LTT = LF + Logic-enriched TTs + Typed Sets • LF – Logical framework (cf, Edin LF, Martin-Löf’s LF, PAL+, …) • Logic-enriched type theories [Aczel/Gambino02,06] • Typed sets: sets with base types (see later) LTT = Logics + Types Logic Types • Logics – specified in LF \ / • Types – inductive types + types of sets \ / LF (Luo 2007, LNCS 4435.)

7. Key components of LTT (I): types & props • Types and propositions: • Two worlds: objects in the “real world” and their properties (cf, ECC/UTTLTT) • Types • Eg, inductive types like N, x:A.B, List(A), Tree(A), … • Eg, types of sets like Set(A) • Propositions: • Describing properties of objects (x:A.P(x) with type A) • Classical laws may be introduced • eg, double negation: DN[P,p] : Prf(P), if P : Prop and p : Prf(¬¬P). • Induction rules • Linking the world of logical propositions and that of types • Enabling proofs of properties about objects of types

8. Example of inductive types: natural numbers • Formation and introduction • N : Type • 0 : N • succ(n) : N, for n : N • Elimination over types and computation: • ElimT(C,c,f,n) : C(n), for C(n) : Type where n : N • Plus computational rules for ElimT: eg, ElimT(C,c,f,0) = c ElimT(C,c,f,succ(n)) = f(n,ElimT(C,c,f,n)) • Induction over propositions: • ElimP(P,c,f,n) : P(n), for P(n) : Prop where n : N • Key to prove logical properties of natural numbers

9. Key components of LTT (II): typed sets • Typed sets • Set(A) : Type for A : Type • { x:A | P(x) } : Set(A) • t  { x:A | P(x) } means P(t) (cf, de Bruijn’s use of this terminology) • Impredicativity and predicativity • Impredicative sets (LTTi) • A can be any type (e.g., Set(B)) • P(x) can be any proposition • eg, P(x) = s:Set(N). sS & xs, for S : Set(Set(N)) • Predicative sets (LTTp) • Universes of small types and small propositions • A must be small (in particular, A is not Set(…)) • P(x) must be small (not allowing quantifications over sets)

10. Implementations and case studies • Plastic (Callaghan/Luo 2001) • Proof assistant • Plastic implements LF and inductive types (UTT) • Implemention of LTT in Plastic (Callaghan) • Simple extension of Plastic • Case studies • Formalisation of Weyl’s predicative mathematics (Adams/Luo 2007, LNCS 4502) • Analysis of security protocols (Luo 2007, LNCS 4435) • OO-modelling and verification (in progress)

11. Formalisation of Weyl’s predicative math • H. Weyl. The Continuum (Das Kontinuum), 1918. • Historical development (paradox etc.) • Predicative development of the real number system • The notion of category • Classical logic • Weyl/Feferman/Simpson’s work on predicativity • Predicativity (E.g., { x | φ(x) } with φ being “arithmetical” – without quantification over sets) • Formalisation of Weyl’s book in Plastic • In LTT, use classical logic and predicative sets • Weyl’s categories as types • “Exact match” (and further research …)

12. OO-modelling in intensional type theory • Functional model • Cf, work by Abadi, Bruce, Cardelli, Kamin, Pierce, … OO  F ?? • Features such as bounded quantification (BQ) are problematic. • Modelling OO-features in intensional type theory (ITT) • LTT with classical logic • LTT is “intensional”: no problematic features such as BQ or extensional features such as -like equalities. • A model in an intensional TT will lead to, eg, verification of programs with OO-features (eg, in Coq) OO  ITT  Coq/Plastic/… • Work in progress (S. Han) • OO-model and verification of OO-programs in Coq • Future Work • Eg, automatic translation (OO-programs  models, properties  propositions, … )