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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. Some positions in foundations of math: Neo-platonism (eg, set-theoretic foundation: Gödel/Manddy)

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Formal Reasoning with Different Logical Foundations

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  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/intuitionistic 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/impredicativity) • set-theoretic reasoning? • Proof assistants based on LFs • Edinburgh LF? Twelf? • Plastic? • Isabelle?

  4. 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

  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. 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

  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. 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 • … …

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

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