Normalization by Evaluation and the Foundations of Constructive - PowerPoint PPT Presentation

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 Normalization by Evaluation and the Foundations of Constructive Mathematics 1972 - 2009 Peter Dybjer Chalmers tekniska högskola, Göteborg, Sweden Workshop on Normalization by Evaluation August 15, 2009 University of California at Los Angeles FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 The question Is normalization by evaluation an intrinsic part of a foundational framework for constructive mathematics? FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 The question Is normalization by evaluation an intrinsic part of a foundational framework for constructive mathematics? in 1972 - 73 FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 The question Is normalization by evaluation an intrinsic part of a foundational framework for constructive mathematics? in 1972 - 73 progress 1979 - 2009 FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 The question Is normalization by evaluation an intrinsic part of a foundational framework for constructive mathematics? in 1972 - 73 progress 1979 - 2009 an open problem FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 It’s 1972. State of the art Normalization proofs with "computablility" predicates: FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 It’s 1972. State of the art Normalization proofs with "computablility" predicates: Tait 1967 - simply typed lambda calculus Girard 1971 - system F FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 It’s 1972. State of the art Normalization proofs with "computablility" predicates: Tait 1967 - simply typed lambda calculus Girard 1971 - system F Formulas as types: FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 It’s 1972. State of the art Normalization proofs with "computablility" predicates: Tait 1967 - simply typed lambda calculus Girard 1971 - system F Formulas as types: Howard 1968, intuitionistic predicate logic with equality de Bruijn 1968, Automath Scott 1970, Constructive Validity FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 It’s 1972. State of the art Normalization proofs with "computablility" predicates: Tait 1967 - simply typed lambda calculus Girard 1971 - system F Formulas as types: Howard 1968, intuitionistic predicate logic with equality de Bruijn 1968, Automath Scott 1970, Constructive Validity Intuitionistic type theory with normalization proofs (unpublished): FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 It’s 1972. State of the art Normalization proofs with "computablility" predicates: Tait 1967 - simply typed lambda calculus Girard 1971 - system F Formulas as types: Howard 1968, intuitionistic predicate logic with equality de Bruijn 1968, Automath Scott 1970, Constructive Validity Intuitionistic type theory with normalization proofs (unpublished): Martin-Löf 1971, A theory of types. Dependent type theory with type : type. Girard’s paradox. Martin-Löf 1972, An intuitionistic theory of types. Predicative theory with a universe of small types. Published in 1997! FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 About models for intuitionistic type theories ... (p82) In the study of models of intuitionistic theories, one has the choice between classical and intuitionistic abstractions on the metalevel. ... An obstacle to the formulation of a general intuitionistic notion of model has been the lack of a sufficiently welldeveloped intuitionistic notion of set. Using the type-theoretic abstractions described in [17], I intend in the following to formulate an intuitionistic notion of model. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 ... and the notion of definitional equality (p82-83) The transistion to intuitionistic abstractions on the metalevel is both essential and nontrivial. Essential, because in what seems to me to be the most fruitful notion of model, the interpretation of the convertibility relation conv is standard, that is, it is interpreted as definitional equality = def in the model, and definitional equality is a notion which is unmentionable within the classical set theoretic framework. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 Term model of the positive implicational calculus (p 87) (a) Typ = def the type of pairs ( A , φ ) , where A is a type symbol and φ a species of closed terms with type symbol A . (b) Obj (( A , φ )) = def (Σ a ∈ Term ( A )) φ ( a ) . (c) F (( A , φ ) , ( B , ψ )) = def ( A → B , the species of all closed terms b with type symbol A → B such that ( ∀ x ∈ Obj (( A , φ )))( ∃ y ∈ Obj (( B , ψ )))( b ( p ( x )) red p ( y )) (d) Ap ( b , a ) = def p ( q ( b , a )) (e) ... K = def ( K , ( λ x )(( K ( p ( x )) , ( λ y )( x , the proof that K ( p ( x ) , p ( y )) red p ( x )) , the proof that K ( p ( x )) red K ( p ( x ))))) FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 You have just witnessed the birth of nbe! ... in the term model, we achieve that if a conv b, then the normal forms of a and b as well as the proofs which show that they are computable (hereditarily normalizable) are definitionally equal. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 Local formalizability, p 99 The proof of normalization for my intuitionistic type theory (see [17]) becomes locally formalizable in itself. When the dubious rule of lambda conversion was allowed, I could not carry out the proof of normalization for every specific term in the theory itself, contrary to what one would expect from one’s experience with other full scale formal theories. I was only able to prove C A and C B to be extensionally equal, whereas one would like to have C A = def C B . Here C A and C B are the computability predicates associated with the type symbols A and B, respectively. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 An Intuitionistic Theory of Types: Predicative Part Logic Colloquium 1973 in Bristol. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 An Intuitionistic Theory of Types: Predicative Part Logic Colloquium 1973 in Bristol. First published version of Martin-Löf’s intuitionistic type theory. (Super) combinator version, no bound variables. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 An Intuitionistic Theory of Types: Predicative Part Logic Colloquium 1973 in Bristol. First published version of Martin-Löf’s intuitionistic type theory. (Super) combinator version, no bound variables. Normalization "nbe-style". Not only proved that normal forms exist, but they were explicitly given ("computed"). FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 An Intuitionistic Theory of Types: Predicative Part Logic Colloquium 1973 in Bristol. First published version of Martin-Löf’s intuitionistic type theory. (Super) combinator version, no bound variables. Normalization "nbe-style". Not only proved that normal forms exist, but they were explicitly given ("computed"). Several meta-theoretic results proved as corollaries of nbe 3.7 Church-Rosser (Hancock) 3.8 Decidability of convertibility 3.13 Decidability of the ∈ -relation (type-checking algorithm) FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 The model of closed normal terms, the normalization theorem (for closed terms) and its consequences In the present theory, however, the definiton of the notion of convertibility and the proof that an arbitrary term is convertible can no longer be separated, because the type symbols and the terms are generated simultaneously. Instead we shall show by induction on the length of a closed derivation, if it ends with a ∈ A, how to define a ′ and a ′′ , where a ′ is a closed normal term with type symbol A ′ , called the normal form of a, such that a red a ′ , and a ′′ is a proof of A ′′ ( a ′ ) , which it is sometimes more natural to think of as an object of type A ′′ ( a ′ ) , and if it ends with a conv b, that a ′ = def b ′ and a ′′ = def b ′′ FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 Why did not the story end in 1973? (Super) combinator version of intuitionistic type theory was considered too weak to be useful. Local formalizability of nbe-proof (and its corollaries) is claimed but not shown. No consistency proof relative to set theory. FLOPS 2008

1972 1973 1979 1984 1986 1991 1994 2004 2008 2009 1979 revolution - Constructive Mathematics and Computer Programming Meaning explanations! Lazy evaluation of closed terms to constructor form. Extensional type theory - a stronger theory Normalization of open expressions not part of meaning theory. Normalization and decidability of judgement do not hold. Nbe forgotten (implementations like NuPRL and GTT did not employ normalization). Against metatheory! FLOPS 2008