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### Formal Reasoning with Different Logical Foundations

Zhaohui Luo

Dept of Computer Science

Royal Holloway, Univ of London

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?

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?

Framework Approach: LTT

- Type-theoretic frameworkLTT

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

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

Example: natural numbers

- Formation and introduction
- N : Type
- 0 : N
- succ[n] : N [n : N]
- Elimination over types and computation:
- ElimT[C,c,f,n] : C[n], for C[n] : Type [n : N]
- Plus computational rules for ElimT: eg,

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 [n : N]
- Key to prove logical properties of objects

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). sS & xs)
- 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)

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, ACA0)
- Comparative studies in practical reasoning (eg, set-theoretical reasoning)
- Meta-theoretic research
- … …

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 fromhttp://www.cs.rhul.ac.uk/home/zhaohui/type.html

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