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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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