Logische Methoden des Software Engineerings Combinators and combinatory logic Jakob Rehof & Andrej Dudenhefner LS XIV – Software Engineering

Logische Methoden des Software Engineerings

Combinators and combinatory logic

Jakob Rehof & Andrej Dudenhefner LS XIV – Software Engineering

TU Dortmund WS 2018/19

WS 2018/19

Combinators

LetCbe a set ofcombinator symbols, ranged over byX, Y, Z. The setΞC of combinatory expressions, ranged over byF, G, H are defined inductively by:

X∈ C ⇒X∈ΞC, x∈ V ⇒x∈ΞC, F, G∈ΞC⇒(F G)∈ΞC

Consider the set SKI={S,K,I}of combinator symbols (referred to as a combinatory base) and define the notion ofweak reductionwonΞSKIby setting, for all

X, Y, Z∈ΞSKI:

IF w F

KF G w F

SF GH w (F H)(GH)

Let→wandw be the reduction relations onΞSKIinduced byw, by closingw

under contextsC::= []|(CF)|(F C), (F∈ΞSKI).

Combinators

LetCbe a set ofcombinator symbols, ranged over byX, Y, Z. The setΞC of combinatory expressions, ranged over byF, G, H are defined inductively by:

X∈ C ⇒X∈ΞC, x∈ V ⇒x∈ΞC, F, G∈ΞC⇒(F G)∈ΞC

Consider the set SKI={S,K,I}of combinator symbols (referred to as a combinatory base) and define the notion ofweak reductionwonΞSKIby setting, for all

X, Y, Z∈ΞSKI:

IF w F

KF G w F

SF GH w (F H)(GH)

Let→wandw be the reduction relations onΞSKIinduced byw, by closingw

under contextsC::= []|(CF)|(F C), (F∈ΞSKI).

Combinators

LetCbe a set ofcombinator symbols, ranged over byX, Y, Z. The setΞC of combinatory expressions, ranged over byF, G, H are defined inductively by:

X∈ C ⇒X∈ΞC, x∈ V ⇒x∈ΞC, F, G∈ΞC⇒(F G)∈ΞC

Consider the set SKI={S,K,I}of combinator symbols (referred to as a combinatory base) and define the notion ofweak reductionwonΞSKIby setting, for all

X, Y, Z∈ΞSKI:

IF w F

KF G w F

SF GH w (F H)(GH)

Let→w and^w be the reduction relations onΞSKIinduced byw, by closingw

under contextsC::= []|(CF)|(F C), (F∈ΞSKI).

Combinatory bases

By choosing different sets B ⊆ C of combinators (such as

B = SKI = {S, K, I}) we can study different combinatory calculi, since in each case we can consider a B-calculus generated from the combinators in B. In such cases, we refer to the set B as a combinatory base.

Exercise 1

Show that the combinator I can be coded in terms of S and K. Hint: Notice that (KF)(KF )

F .

In other words, the base SKI = {S, K, I} is redundant. Or, in yet other

words, the base SK = {S, K} is complete with respect to SKI-calculus.

For this reason, one also talks about SK-calculus.

Combinatory bases

By choosing different sets B ⊆ C of combinators (such as

B = SKI = {S, K, I}) we can study different combinatory calculi, since in each case we can consider a B-calculus generated from the combinators in B. In such cases, we refer to the set B as a combinatory base.

Exercise 1

Show that the combinator I can be coded in terms of S and K. Hint:

Notice that (KF)(KF )

F .

In other words, the base SKI = {S, K, I} is redundant. Or, in yet other words, the base SK = {S, K} is complete with respect to SKI-calculus.

For this reason, one also talks about SK-calculus.

Combinatory bases

Sch¨ onfinkel used, in addition to S and K, the combinators B and C with the definitions

BF GH

F (GH ) CF GH

F HG

But they are not strictly needed, for we can take B ≡ S(KS)K

C ≡ S(S(K(S(KS)K))S)(KK)

Combinatory bases

Sch¨ onfinkel used, in addition to S and K, the combinators B and C with the definitions

BF GH

F (GH ) CF GH

F HG But they are not strictly needed, for we can take

B ≡ S(KS)K

C ≡ S(S(K(S(KS)K))S)(KK)

Combinatory bases

Exercise 2 (One point basis) Define the combinator X by the rule

(XF )

((F S)K)

Show that (XX)

SK(KK).

Show that X(X(XX))

K.

Show that X(X(X(XX)))

S.

Conclude that {X} is complete with respect to SKI-calculus.

Ξ

7→ Λ

Define the map( )Λ: ΞSKI→Λby induction on expressions inΞSKI: (x)Λ ≡ x, forx∈ V

(I)Λ ≡ λx.x (K)Λ ≡ λxy.x (S)Λ ≡ λxyz.(xz)(yz) (F G)Λ ≡ (F)Λ(G)Λ

Proposition 1

IfF wG, then(F)Λβ (G)Λ.

Exercise 3

Prove Proposition 1 by induction on the length ofF ^wG.

Λ 7→ Ξ

Define, for eachx∈ V, the “bracket abstraction” map [x]: ΞSKI→ΞSKIby induction on expressions inΞSKI:

[x]x ≡ I

[x]F ≡ KF, ifx6∈FV(F) [x](F G) ≡ S([x]F)([x]G), otherwise

Proposition 2 (Combinatory completeness)

1 ∀F∈ΞSKI.∀G∈ΞSKI.([x]F)G^wF[x:=G]

2 ∀F∈ΞSKI.([x]F)Λβ λx.(F)Λ

3 ∀x∈ V.∀F ∈ΞSKI.∃H∈ΞSKI.∀G∈ΞSKI. HG^wF[x:=G]

Exercise 4

Prove Proposition 2.

Λ 7→ Ξ

Define, for eachx∈ V, the “bracket abstraction” map [x]: ΞSKI→ΞSKIby induction on expressions inΞSKI:

[x]x ≡ I

[x]F ≡ KF, ifx6∈FV(F) [x](F G) ≡ S([x]F)([x]G), otherwise

Proposition 2 (Combinatory completeness)

1 ∀F∈ΞSKI.∀G∈ΞSKI.([x]F)G^wF[x:=G]

2 ∀F∈ΞSKI.([x]F)Λβ λx.(F)Λ

3 ∀x∈ V.∀F ∈ΞSKI.∃H∈ΞSKI.∀G∈ΞSKI. HG^wF[x:=G]

Exercise 4

Prove Proposition 2.

Λ 7→ Ξ

Define the map( )Ξ: Λ→ΞSKIby induction onλ-terms:

(x)Ξ ≡ x, forx∈ V (M N)Ξ ≡ (M)Ξ(N)Ξ

(λx.M)Ξ ≡ [x](M)Ξ

Proposition 3

For allM∈Λ, one has((M)Ξ)Λ^βM.

Exercise 5

Prove Proposition 3.

Combinatory logic SKI

Γ, x:τ `_skix:τ(var)

Γ`_skiI:τ →τ(I)

Γ`_skiK:τ →σ→τ(K)

Γ`<sub>ski</sub>S: (τ→σ→ρ)→(τ →σ)→τ →ρ(S) Γ`_skiF:τ→σ Γ`_skiG:τ

Γ`_ski(F G) :σ (→E)

Notice that variablesxhave fixed,monomorphic types, whereas combinatorsS,K,I have infinitely many types (their types areschematicorpolymorphic).

Combinatory logic SKI

Lemma 1 (Deduction theorem for SKI)

IfΓ, x:σ`<sub>ski</sub>F :τ, thenΓ`_ski[x]F :σ→τ.

Proposition 4

1 IfΓ`<sub>ski</sub>F:τ, thenΓ`Λ(F)Λ:τ.

2 IfΓ`ΛM:τ, thenΓ`_ski(M)Ξ:τ.

Exercise 6

Prove Proposition 4. Hint: The first statement is proven by induction on the derivation ofΓ`<sub>ski</sub>F :τ. The second statement is proven by induction on the derivation of Γ`ΛM :τ using Lemma 1.