Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η β reduction without rule ξ Randy Pollack and Masahiko Sato Version of May 20, 2016
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Overview A concrete representation of lambda terms. Locally nameless: indexes for bound positions, names for free variables. Canonical: α conversion is syntactic identity. Abstraction, lam x M , is a defined function. Using the defined abstraction, the language looks like conventional notation. We can define various reduction relations without rule ξ . Only works for some relations. Apparently fails for η .
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Preterms and well formedness We use names for free variables, and indexes for bound variables. Let i , j , m , n , p , q , range over natural numbers. Fix a countable set of names , ranged over by x , y , z . The raw syntax of preterms (ranged over by M , N , P , Q ) is pt ::= X n x | J n j | ⌈ M , N ⌉ n In preterm syntax, n is the height of the preterm, written hgt M . Well formedness (written W M ) is defined inductively by j < n W P W Q n ≤ hgt P n ≤ hgt Q W X n x W J n j W ⌈ P , Q ⌉ n Well formed terms are called terms .
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Intended Meaning of (Well Formed) Terms j < n W P W Q n ≤ hgt P n ≤ hgt Q W X n x W J n j W ⌈ P , Q ⌉ n X n x represents λ 1 . . . λ n x . J n j represents λ 1 . . . λ n j . Require j < n for well formedness; otherwise j would be unbound. Consider W P (representing λ 1 . . . λ p T P ) and W Q (representing λ 1 . . . λ q T Q ). By induction, every index in P or Q is bound. ⌈ P , Q ⌉ 0 represents (( λ 1 . . . λ p T P ) ( λ 1 . . . λ q T Q )) . ⌈ P , Q ⌉ 1 represents λ 1 (( λ 2 . . . λ p T P ) ( λ 2 . . . λ q T Q )) . ⌈ P , Q ⌉ 2 represents λ 1 λ 2 (( λ 3 . . . λ p T P ) ( λ 3 . . . λ q T Q )) . . . . Thus well formedness of ⌈ P , Q ⌉ n requires n ≤ min ( hgt P , hgt Q ) .
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Abstraction defined as a function on preterms lam x ( X n y ) := if x = y then J n + 1 0 else X n + 1 y lam x ( J n j ) := J n + 1 ( j + 1 ) lam x ⌈ M , N ⌉ n := ⌈ lam x M , lam x N ⌉ n + 1 Abstraction preserves well formedness and raises height by one. W M = ⇒ W ( lam x M ) hgt ( lam x M ) = hgt M + 1 Conversely, every term with height a successor is an abstraction. W M ∧ hgt M = n + 1 = ⇒ ∃ P x . M = lam x P We use A , B as metavariables over abstractions.
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Examples Notations: write lam xy M for lam x lam y M . x for X 0 x . Some combinators: (assuming x � = y , x � = z , y � = z ) I = λ x . x = lam x x = J 1 0 = λ x y . x = lam xy x = J 2 0 K false = λ x y . y = lam xy y = J 2 1 S = λ x y z . ( x z ) ( y z ) = lam xyz ⌈⌈ x , z ⌉ 0 , ⌈ y , z ⌉ 0 ⌉ 0 = ⌈⌈ J 3 0 , J 3 2 ⌉ 3 , ⌈ J 3 1 , J 3 2 ⌉ 3 ⌉ 3 The isomorphism between terms and informal lambda terms: t 1 ∼ M 1 t 2 ∼ M 2 t ∼ M x ∼ X 0 x ( t 1 t 2 ) ∼ ⌈ M 1 , M 2 ⌉ 0 λ x . t ∼ lam x M
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Lifting To define instantiation we first introduce a lifting function ( X n y ) ↑ := X n + 1 y ( J n j ) ↑ := J n + 1 ( j + 1 ) ( ⌈ M , N ⌉ n ) ↑ := ⌈ ( M ) ↑ , ( N ) ↑ ⌉ n + 1 which we iterate as: ( M ) ↑ 0 := M ( M ) ↑ m + 1 := (( M ) ↑ m ) ↑ Lifting preserves well formedness and raises height by one. ⇒ W ( M ) ↑ ∧ hgt ( M ) ↑ = hgt M + 1 W M =
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Instantiation Instantiation is a binary function, M [ N ] . If hgt M = 0 ( M is under no binders), M [ N ] = M . Otherwise M [ N ] fills any holes J n + 1 0 in M and adjusts the rest of the term: X n + 1 y [ N ] := X n y J n + 1 0 [ N ] := ( N ) ↑ n J n + 1 ( j + 1 )[ N ] := J n j ⌈ M , P ⌉ n + 1 [ N ] := ⌈ M [ N ] , P [ N ] ⌉ n Instantiation is not substitution. Instantiation preserves well formedness: W M ∧ W N = ⇒ W ( M [ N ]) ∧ ( hgt M ) − 1 ≤ hgt M [ N ]
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Substitution Substitution is defined in terms of instantiation: M [ x ← P ] := ( lam x M )[ P ] All the expected properties hold. Usual substitution lemma: x � = y ∧ x / ∈ FV ( P ) ∧ W ( M , P , N ) = ⇒ M [ x ← N ][ y ← P ] = M [ y ← P ][ x ← N [ y ← P ]]
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η β reduction as usual Using abstraction we have a natural definition of β reduction: W M W N ( β ) β ⌈ lam x M , N ⌉ 0 → M [ x ← N ] β β → M ′ → N ′ M W N W M N β β ⌈ M , N ⌉ 0 → ⌈ M ′ , N ⌉ 0 ⌈ M , N ⌉ 0 → ⌈ M , N ′ ⌉ 0 β M → N ( ξ ) β lam x M → lam x N Any preterm that participates in this relation is well-formed. Correct β reduction w.r.t. the meaning of terms given above, Still contains rule ξ
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Properties of usual β reduction As usual, rule ξ is invertible: β β lam x M → lam x N = ⇒ M → N β reduction does not lower height: β M → N = ⇒ hgt M ≤ hgt N
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Generalized lifting To eliminate rule ξ from our presentation of β reduction, we define generalized lifting . ( X n y ) i ⇑ := X n + 1 y � ( j < i ) J n + 1 j ( J n j ) i ⇑ := J n + 1 ( j + 1 ) ( j ≥ i ) ( ⌈ M , N ⌉ n ) i ⇑ := ⌈ ( M ) i ⇑ , ( N ) i ⇑ ⌉ n + 1 Preserves well formedness and raises height by one. Many useful properties of generalized lifting are used, e.g. Injectivity: W ( M , N ) ∧ ( M ) i ⇑ = ( N ) i ⇑ = ⇒ M = N . We iterate generalized lifting: ( M ) i ⇑ 0 := M ( M ) i ⇑ m + 1 := (( M ) i ⇑ m ) i ⇑
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Generalized instantiation Generalized instantiation , M [ N ] i , leaves terms M of height 0 unchanged, and updates abstractions: X n + 1 y [ M ] i := X n y J n + 1 i [ M ] i := ( M ) i ⇑ n − i � J n j ( j < i ) J n + 1 j [ M ] i := J n ( j − 1 ) ( j > i ) ⌈ P , Q ⌉ n + 1 [ M ] i := ⌈ P [ M ] i , Q [ M ] i ⌉ n A [ P ] 0 = A [ P ] ⇒ n ≤ hgt ( A [ P ] n ) n < hgt A ∧ n ≤ hgt P = ⇒ W ( A [ P ] n ) n < hgt A ∧ n ≤ hgt P ∧ W A ∧ W P =
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η β without rule ξ Claim the relation • > • defined without a ξ rule: W A n < hgt A W N n ≤ hgt N ( β ) ⌈ A , N ⌉ n > A [ N ] n M > M ′ n ≤ hgt M W N n ≤ hgt N ⌈ M , N ⌉ n > ⌈ M ′ , N ⌉ n N > N ′ n ≤ hgt N W M n ≤ hgt M ⌈ M , N ⌉ n > ⌈ M , N ′ ⌉ n β is equivalent to the relation • → • given above (and thus to the usual notion of β reduction). β Proof that M > N = ⇒ M → N : by induction on the relation M > N . β Both congruence rule cases use invertibility of rule ξ for relation → . The converse direction is straightforward. �
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Tait–Martin-Löf parallel reduction: Usual presentation Parallel reduction (non-deterministic): p p → M ′ → N ′ M N ( β ) p p → M ′ [ x ← N ′ ] X 0 x → X 0 x ⌈ lam x M , N ⌉ 0 p p p → M ′ → N ′ M → N M N ( ξ ) p p → lam x N ⌈ M , N ⌉ 0 → ⌈ M ′ , N ′ ⌉ 0 lam x M Correct w.r.t. usual presentation. Overlap between rule ( β ) and application congruence. Complete development (deterministic, à la Takahashi): Remove overlap, forcing every β step to be taken: cd cd → M ′ → N ′ M N M not an abstraction cd ⌈ M , N ⌉ 0 → ⌈ M ′ , N ′ ⌉ 0
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Parallel reduction without rule ξ Parallel reduction: j < n X n y ≫ X n y J n j ≫ J n j M ≫ M ′ N ≫ N ′ n ≤ hgt M n ≤ hgt N ⌈ M , N ⌉ n ≫ ⌈ M ′ , N ′ ⌉ n n < hgt A A ≫ B n ≤ hgt M M ≫ N ⌈ A , M ⌉ n ≫ B [ N ] n Complete development (remove overlap): M ≫ M ′ N ≫ N ′ n = hgt M n ≤ hgt N ⌈ M , N ⌉ n ≫ ⌈ M ′ , N ′ ⌉ n
Syntax Operations on preterms β reduction Generalized instantiation Reduction without ξ η Church–Rosser theorem With parallel reduction and complete development, we can carry out Takahashi’s proof of Church–Rosser. Although there is no rule ξ , this proof is no easier than usual.
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