PLC type system. if (x : t) 2 G. G ` x : t. G, x : t 1 ` M : t 2 G ` lx : t 1 (M) : t 1 t 2. if x /2 dom(g)

Transcription

1 PLC type system (var) G ` x : t if (x : t) 2 G (fn) G, x : t 1 ` M : t 2 G ` lx : t 1 (M) : t 1 t 2 if x /2 dom(g) (app) G ` M : t 1 t 2 G ` M 0 : t 1 G ` MM 0 : t 2 (gen) G ` M : t G ` La (M) : 8a (t) if a /2 ftv(g) (spec) G ` M : 8a (t 1 ) G ` M t 2 : t 1 [t 2 /a]

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3 Functions on types In PLC, La (M) is an anonymous notation for the function F mapping each type t to the value of M[t/a] (of some particular type).

4 Functions on types In PLC, La (M) is an anonymous notation for the function F mapping each type t to the value of M[t/a] (of some particular type). F t denotes the result of applying such a function to a type.

5 Functions on types In PLC, La (M) is an anonymous notation for the function F mapping each type t to the value of M[t/a] (of some particular type). F t denotes the result of applying such a function to a type. Computation in PLC involves beta-reduction for such functions on types (La (M)) t! M[t/a]

6 Functions on types In PLC, La (M) is an anonymous notation for the function F mapping each type t to the value of M[t/a] (of some particular type). F t denotes the result of applying such a function to a type. Computation in PLC involves beta-reduction for such functions on types (La (M)) t! M[t/a] as well as the usual form of beta-reduction from l-calculus (lx : t (M 1 )) M 2! M 1 [M 2 /x]

7 Beta-reduction of PLC expressions M beta-reduces to M 0 in one step, M! M 0 means M 0 can be obtained from M (up to alpha-conversion, of course) by replacing a subexpression which is a redex by its corresponding reduct. The redex-reduct pairs are of two forms: (lx : t (M 1 )) M 2! M 1 [M 2 /x] (La (M)) t! M[t/a]

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15 Beta-reduction of PLC expressions M beta-reduces to M 0 in one step, M! M 0 means M 0 can be obtained from M (up to alpha-conversion, of course) by replacing a subexpression which is a redex by its corresponding reduct. The redex-reduct pairs are of two forms: (lx : t (M 1 )) M 2! M 1 [M 2 /x] (La (M)) t! M[t/a] M! M 0 indicates a chain of finitely many beta-reductions.

16 Beta-reduction of PLC expressions M beta-reduces to M 0 in one step, M! M 0 means M 0 can be obtained from M (up to alpha-conversion, of course) by replacing a subexpression which is a redex by its corresponding reduct. The redex-reduct pairs are of two forms: (lx : t (M 1 )) M 2! M 1 [M 2 /x] (La (M)) t! M[t/a] M! M 0 indicates a chain of finitely many beta-reductions. ( possibly zero which just means M and M 0 are alpha-convertible).

17 Beta-reduction of PLC expressions M beta-reduces to M 0 in one step, M! M 0 means M 0 can be obtained from M (up to alpha-conversion, of course) by replacing a subexpression which is a redex by its corresponding reduct. The redex-reduct pairs are of two forms: (lx : t (M 1 )) M 2! M 1 [M 2 /x] (La (M)) t! M[t/a] M! M 0 indicates a chain of finitely many beta-reductions. ( possibly zero which just means M and M 0 are alpha-convertible). M is in beta-normal form if it contains no redexes.

18 Properties of PLC beta-reduction on typeable expressions Suppose G ` M : t is provable in the PLC type system. Then the following properties hold: Subject Reduction. provable typing. If M! M 0,thenG ` M 0 : t is also a

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26 Properties of PLC beta-reduction on typeable expressions Suppose G ` M : t is provable in the PLC type system. Then the following properties hold: Subject Reduction. provable typing. If M! M 0,thenG ` M 0 : t is also a Church Rosser Property. If M! M 1 and M! M 2,then there is M 0 with M 1! M 0 and M 2! M 0.

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28 Properties of PLC beta-reduction on typeable expressions Suppose G ` M : t is provable in the PLC type system. Then the following properties hold: Subject Reduction. provable typing. If M! M 0,thenG ` M 0 : t is also a Church Rosser Property. If M! M 1 and M! M 2,then there is M 0 with M 1! M 0 and M 2! M 0. Strong Normalisation Property. There is no infinite chain M! M 1! M 2!... of beta-reductions starting from M.

29 Properties of PLC beta-reduction on typeable expressions Suppose G ` M : t is provable in the PLC type system. Then the following properties hold: Subject Reduction. provable typing. If M! M 0,thenG ` M 0 : t is also a Church Rosser Property. If M! M 1 and M! M 2,then there is M 0 with M 1! M 0 and M 2! M 0. Strong Normalisation Property. There is no infinite chain M! M 1! M 2!... of beta-reductions starting from M.

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33 PLC beta-conversion, = b By definition, M = b M 0 holds if there is a finite chain

34 PLC beta-conversion, = b By definition, M = b M 0 holds if there is a finite chain M M 0 where each is either! or, i.e. a beta-reduction in one direction or the other.

35 PLC beta-conversion, = b By definition, M = b M 0 holds if there is a finite chain M M 0 where each is either! or, i.e. a beta-reduction in one direction or the other. (A chain of length zero is allowed in which case M and M 0 are equal, up to alpha-conversion, of course.)

36 PLC beta-conversion, = b By definition, M = b M 0 holds if there is a finite chain M M 0 where each is either! or, i.e. a beta-reduction in one direction or the other. (A chain of length zero is allowed in which case M and M 0 are equal, up to alpha-conversion, of course.) Church Rosser + Strong Normalisation properties imply that, for typeable PLC expressions, M = b M 0 holds if and only if there is some beta-normal form N with M! N M 0

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38 Polymorphic booleans bool, 8a (a (a a))

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40 Polymorphic booleans bool, 8a (a (a a)) True, La (lx 1 : a, x 2 : a (x 1 )) False, La (lx 1 : a, x 2 : a (x 2 ))

41 Polymorphic booleans bool, 8a (a (a a)) True, La (lx 1 : a, x 2 : a (x 1 )) False, La (lx 1 : a, x 2 : a (x 2 )) if, La (lb : bool, x 1 : a, x 2 : a (b a x 1 x 2 ))

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