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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)

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Formal reasoning with different logical foundations

Formal Reasoning with Different Logical Foundations

Zhaohui Luo

Dept of Computer Science

Royal Holloway, Univ of London


Mathematical pluralism
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
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
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
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
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
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)


Case studies and future work
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
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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