1. Formal Reasoning with Different Logical Foundations Zhaohui Luo Dept of Computer Science Royal Holloway, Univ of London

2. 2 Mathematical pluralism Some positions in foundations of math: Neo-platonism (eg, set-theoretic foundation: Gödel/Manddy) Revisionists (eg, intuitionism: Brouwer/Martin-Löf) Pragmatic position – “pluralism” Incorporating different approaches Classical v.s. Constructive/intuitionistic Impredicative v.s. Predicative Set-theoretic v.s. Type-theoretic Support to such a position in theorem proving? A uniform foundational framework?

3. 3 TT-based Theorem Proving Technology Proof assistants based on TT mainly intuitionistic logic special features (e.g., predicativity/impredicativity) set-theoretic reasoning? Proof assistants based on LFs Edinburgh LF? Twelf? Plastic? Isabelle?

4. 4 Framework Approach: LTT Type-theoretic framework LTT 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) Alternatively, LTT = Logics + Types Logics – specified in LF Types – inductive types + types of sets

5. 5 Key components of LTT: types and props Types and propositions: Type and El(A): kinds of types and objects of type A  Eg, inductive types like N,  x:A.B, List(A), Tree(A), …  Eg, types of sets like Set(A) Prop and Prf(P): kinds of propositions and proofs of proposition P  Eg,  x:A.P(x) : Prop, where A : Type and P : (A)Prop.  Eg, DN[P,p] : Prf(P), if P : Prop and p : Prf(¬¬P). Induction rule Linking the world of logical propositions and that of types Enabling proofs about objects of types

6. 6 Example: natural numbers Formation and introduction N : Type 0 : N succ[n] : N [n : N] Elimination over types and computation: Elim T [C,c,f,n] : C[n], for C[n] : Type [n : N] Plus computational rules for Elim T : eg, Elim T [C,c,f,succ(n)] = f[n,Elim T [C,c,f,n]] Induction over propositions: Elim P [P,c,f,n] : P[n], for P[n] : Prop [n : N] Key to prove logical properties of objects

7. 7 Key components of LTT: typed sets Typed sets Set(A) : Type for A : Type { x:A | P(x) } : Set(A)  t  { x:A | P(x) } means P(t) Impredicativity and predicativity Impredicative sets  A can be any type (e.g., Set(B))  P(x) can be any proposition (e.g.,  s:Set(N). s  S & x  s) Predicative sets  Universes of small types and small propositions  A must be small (in particular, A is not Set(…))  P must be small (not allowing quantifications over sets)

8. 8 Case studies and future work Case studies (Simple) Implementation of LTT in Plastic (Callaghan) Formalisation of Weyl’s predicative math (Adams & Luo) Analysis of security protocols Future work Comparative studies with other systems (eg, ACA 0 ) Comparative studies in practical reasoning (eg, set- theoretical reasoning) Meta-theoretic research … …

9. 9 References Z. Luo. A type-theoretic framework for formal reasoning with different logical foundations. ASIAN’06, LNCS 4435. 2007. R. Adams and Z. Luo. Weyl's predicative classical mathematics as a logic-enriched type theory. TYPES’06, LNCS 4502. 2007. Available from http://www.cs.rhul.ac.uk/home/zhaohui/type.html