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Formal Reasoning with Different Logical Foundations

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Revisionists (eg, intuitionism: Brouwer/Martin-L f) Pragmatic position 'pluralism' ... LF Logical framework (cf, Edin LF, Martin-L f's LF, PAL ... – PowerPoint PPT presentation

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Title: Formal Reasoning with Different Logical Foundations


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

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/intuitionis
    tic
  • Impredicative v.s. Predicative
  • Set-theoretic v.s. Type-theoretic
  • Support to such a position in theorem proving? A
    uniform foundational framework?

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

4
Framework Approach LTT
  • Type-theoretic framework LTT
  • LTT LF Logic-enriched TTs Typed Sets
  • LF Logical framework (cf, Edin LF, Martin-Löfs
    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
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, ?xA.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, ?xA.P(x) Prop, where A Type and P
    (A)Prop.
  • Eg, DNP,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
Example natural numbers
  • Formation and introduction
  • N Type
  • 0 N
  • succn N n N
  • Elimination over types and computation
  • ElimTC,c,f,n Cn, for Cn Type n N
  • Plus computational rules for ElimT eg,
  • ElimTC,c,f,succ(n) fn,ElimTC,c,f,n
  • Induction over propositions
  • ElimPP,c,f,n Pn, for Pn Prop n N
  • Key to prove logical properties of objects

7
Key components of LTT typed sets
  • Typed sets
  • Set(A) Type for A Type
  • xA P(x) Set(A)
  • t ? xA 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., ?sSet(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
Case studies and future work
  • Case studies
  • (Simple) Implementation of LTT in Plastic
    (Callaghan)
  • Formalisation of Weyls 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

9
References
  • Z. Luo. A type-theoretic framework for formal
    reasoning with different logical foundations.
    ASIAN06, LNCS 4435. 2007.
  • R. Adams and Z. Luo. Weyl's predicative
    classical mathematics as a logic-enriched type
    theory. TYPES06, LNCS 4502. 2007.
  • Available from http//www.cs.rhul.ac.uk/home/zha
    ohui/type.html
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