Formal Reasoning with Different Logical Foundations

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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

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). 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
• (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