λ-Calculus & Intuitionistic Logic Project Aims Cartesian Closed Categories Correspondence between CCC & λ unit, →, × 1© CHUANGJIE XU 2010

2  Study the foundations of the Curry-Howard Isomorphism  Probe into one of its extensions: The three-way-correspondence between intuitionistic logic, typed lambda calculus and cartesian closed categories

3© CHUANGJIE XU 2010  A family of prototype programming languages  A λ-term M can be an atom (term-variable), an application, or an abstraction, i.e.  Three types of equivalences: α -conversion: β -conversion: η -conversion:

4© CHUANGJIE XU 2010  A typed interpretation of the lambda calculus with one type constructor → that builds function types  Type: can be an atom (type-variable) or a composite type ( σ →τ, where σ both τ and are types)  Type-assignment: any term M : σ where M is a λ-term and σ is a type  The syntax of λ → is essentially that of the λ-calculus itself

5© CHUANGJIE XU 2010  Typing rules for λ → : If x is a variable and σ is a type, then x: σ has type σ ; ( →E ): If M has type σ →τ and N has type σ, then application MN has type τ ; ( →I ): If term M has type τ and variable x has type σ, then abstraction λx: σ. M has type σ →τ.

6© CHUANGJIE XU 2010  Judgments about statements: Existence of a proof or construction of that statement  The set Φ of formulas in intuitionist propositional logic:

7© CHUANGJIE XU 2010 Systems of LogicComputational Calculi FormulasTypes ProofsTerms

8© CHUANGJIE XU 2010  A typed interpretation of the lambda calculus with one type constructor → that builds function types Intuitionist Propositional Logic VS λ null, unit, →, ⨉, +

9© CHUANGJIE XU 2010  A typed interpretation of the lambda calculus with one type constructor → that builds function types