Calculus and Combinatory Logic

Type-Based Synthesis and the Inhabitation Problem

Lecture 1: Typed λ-Calculus and Combinatory Logic

Jakob Rehof TU Dortmund University, Germany

The Second IPM Advanced School on Computing Theory and Practice of Programming Languages 27-31 August 2017 Tehran, Iran

August 30, 2017

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 1 / 38 λ-Calculus

Untyped, pure λ-calculus

H.P. Barendregt: The Lambda Calculus. Its Syntax and Semantics, Studies in Logic and the Foundations of Mathematics, 2nd Edition. Elsevier Science Publishers 1984. [Bar84] Hindley and Seldin: Lambda-Calculus and Combinators, an Introduction, Cambridge University Press 2008. [HS08]

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 2 / 38 Assuming computational expressions such as x + 1 the operation of λ-abstraction constructs the expression λx.x + 1. An abstraction λx.M can be read as: “the function of x returning M(x)”. Functional expressions are anonymous. Mathematicians sometimes use notation like x 7→ x + 1, and that corresponds to λx.x + 1. In addition to the operation of abstraction there is a (dual) operation of application: If M and N are λ-expressions, then the application of M to N, written (MN), is also a λ-expression. Computation arises when abstractions are applied to arguments, for example ((λx.x + 1) 2). An expression of the form ((λx.M) N) is called a redex. Computation means substitution of arguments for the λ-abstracted variable of the function operator in a redex. For example:

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

λ-Calculus in One Slide

λ-calculus is a minimalistic formalism for deﬁning functional expressions and computations with such (a minimalistic functional programming language):

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 An abstraction λx.M can be read as: “the function of x returning M(x)”. Functional expressions are anonymous. Mathematicians sometimes use notation like x 7→ x + 1, and that corresponds to λx.x + 1. In addition to the operation of abstraction there is a (dual) operation of application: If M and N are λ-expressions, then the application of M to N, written (MN), is also a λ-expression. Computation arises when abstractions are applied to arguments, for example ((λx.x + 1) 2). An expression of the form ((λx.M) N) is called a redex. Computation means substitution of arguments for the λ-abstracted variable of the function operator in a redex. For example:

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

λ-Calculus in One Slide

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 Functional expressions are anonymous. Mathematicians sometimes use notation like x 7→ x + 1, and that corresponds to λx.x + 1. In addition to the operation of abstraction there is a (dual) operation of application: If M and N are λ-expressions, then the application of M to N, written (MN), is also a λ-expression. Computation arises when abstractions are applied to arguments, for example ((λx.x + 1) 2). An expression of the form ((λx.M) N) is called a redex. Computation means substitution of arguments for the λ-abstracted variable of the function operator in a redex. For example:

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

λ-Calculus in One Slide

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 In addition to the operation of abstraction there is a (dual) operation of application: If M and N are λ-expressions, then the application of M to N, written (MN), is also a λ-expression. Computation arises when abstractions are applied to arguments, for example ((λx.x + 1) 2). An expression of the form ((λx.M) N) is called a redex. Computation means substitution of arguments for the λ-abstracted variable of the function operator in a redex. For example:

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

λ-Calculus in One Slide

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 Computation arises when abstractions are applied to arguments, for example ((λx.x + 1) 2). An expression of the form ((λx.M) N) is called a redex. Computation means substitution of arguments for the λ-abstracted variable of the function operator in a redex. For example:

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

λ-Calculus in One Slide

λ-calculus is a minimalistic formalism for deﬁning functional expressions and computations with such (a minimalistic functional programming language): Assuming computational expressions such as x + 1 the operation of λ-abstraction constructs the expression λx.x + 1. An abstraction λx.M can be read as: “the function of x returning M(x)”. Functional expressions are anonymous. Mathematicians sometimes use notation like x 7→ x + 1, and that corresponds to λx.x + 1. In addition to the operation of abstraction there is a (dual) operation of application: If M and N are λ-expressions, then the application of M to N, written (MN), is also a λ-expression.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 Computation means substitution of arguments for the λ-abstracted variable of the function operator in a redex. For example:

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

λ-Calculus in One Slide

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 λ-Calculus in One Slide

((λx.x + 1) 2) →β (x + 1)[x := 2] ≡ 2 + 1

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 3 / 38 λ-terms

Let V denote a denumerable set of λ-variables, v1, v2, v3,....

Let x, y, z, . . . denote a denumerable set of metavariables ranging over V.

Deﬁnition 1

The set Λ of λ-terms, ranged over by M, N, P, Q, . . ., is deﬁned inductively by:

x ∈ V ⇒ x ∈ Λ (variables) M,N ∈ Λ ⇒ (MN) ∈ Λ (application) M ∈ Λ, x ∈ V ⇒ (λxM) ∈ Λ (abstraction)

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 4 / 38 λ-terms

Notation 1

We may abbreviate (((FM1)M2) ··· Mn) by

FM1M2 ··· Mn for n ≥ 0.

We may abbreviate (λx1(λx2(··· (λxnM) ··· ))) by

λx1.λx2. ··· λxn.M for n ≥ 0.

And further, λx1.λx2. ··· λxn.M by λx1x2. ··· xn.M.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 5 / 38 λ-terms

Deﬁnition 2 (Free and bound variables)

The set of free variables of M, denoted FV(M), is deﬁned by induction on M: FV(x) = {x} FV(MN) = FV(M) ∪ FV(N) FV(λx.M) = FV(M) \{x} A variable x ∈ FV(M) is called bound in λx.M (by that λ). A term M ∈ Λ is called closed, if FV(M) = ∅.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 6 / 38 λ-terms

Deﬁnition 3 (Syntactic identity)

We write M ≡ N to denote that M and N are the same term under renaming of bound variables (α-conversion). For example,

(λx.x)z ≡ (λx.x)z (λx.x)z ≡ (λy.y)z (λx.x)x ≡ (λy.y)x (λx.x)z 6≡ (λx.y)z (provided x 6≡ y)

Note that x ≡ y is possible, unless we have stipulated that x 6≡ y.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 7 / 38 λ-terms

Deﬁnition 4 (Substitution)

Let M[x := N] denote the result of substituting N for the free occurrences of x in M, deﬁned by induction on M:

x[x := N] ≡ N y[x := N] ≡ y, if x 6≡ y (PQ)[x := N] ≡ (P [x := N]Q[x := N]) (λy.P )[x := N] ≡ λy.P [x := N], if x 6≡ y (λx.P )[x := N] ≡ λx.P

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 8 / 38 λ-terms

Lemma 5 (Substitution lemma)

Assume x 6≡ y and x 6∈ FV(P ).Then

M[x := N][y := P ] ≡ M[y := P ][x := N[y := P ]]

Exercise 1

Prove Lemma 5 by induction on M.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 9 / 38 Notion of reduction

Deﬁnition 6

A notion of reduction on a set D is a binary relation £ ⊆ D × D.

Deﬁnition 7 (Notion of β-reduction)

The notion of β-reduction is the relation £β ⊆ Λ × Λ deﬁned by

(λx.M)N £β M[x := N]

for all M,N ∈ Λ. An expression of the form (λx.M)N is called a redex.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 10 / 38 Contexts

Contexts C over Λ are expressions of the form

C ::= [] | (CM) | (MC) | λx.C

We think of [] as a hole and of C as a term with a hole in it.

If C is a context and M a term then C[M] is the term which arises from exchanging [] with M (“hole-ﬁlling”).

For example, if C ≡ λx.λy.y[], then

C[λz.(xz)] ≡ λx.λy.y(λz.(xz))

Note that variable capture can happen through hole-ﬁlling.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 11 / 38 β-reduction

Deﬁnition 8

A notion of reduction £ ⊆ Λ × Λ satisfying

∀C.M £ N ⇒ C[M] £ C[N] is called a reduction relation. The smallest reduction relation containing £ is called the reduction relation induced by £ (sometimes also called the compatible closure of £).

Deﬁnition 9 (β-reduction)

The relation →β (β-reduction) is the reduction relation induced by £β .

If M →β N, then N is called a reduct of M.

A normal form is a term with no redex.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 12 / 38 β-reduction

Deﬁnition 10 (Multi-step β-reduction)

The relation β is the reﬂexive transitive closure of →β.

Deﬁnition 11 (β-conversion)

The relation =β (β-conversion) is the reﬂexive symmetric transitive closure of →β.

If M =β N, we say that M and N are β-equivalent.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 13 / 38 λ-calculus combinators

A λ-calculus combinator is a closed λ-term.

Some interesting combinators are:

I ≡ λx.x, K ≡ λxy.x, K∗ ≡ λxy.y, S ≡ λxyz.(xz)(yz)

ω ≡ λx.xx, Ω ≡ ωω

Y ≡ λf.(λx.f(xx))(λx.f(xx)), Θ ≡ (λx.λy.(y(xxy)))(λx.λy.(y(xxy)))

Exercise 2

Show that, for all terms F , we have YF =β F (YF ) and ΘF β F (ΘF ).

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 14 / 38 Conﬂuence

Theorem 12 (Church-Rosser, conﬂuence)

For every M,P1,P2 ∈ Λ, if M β P1 and M β P2, then there exists Q ∈ Λ such that P1 β Q and P2 β Q.

Corollary 13 (Uniqueness of normal forms)

If N1 and N2 are normal forms such that M β N1 and M β N2, then N1 ≡ N2. If P =β Q, then P and Q have the same normal form if any.

Exercise 3

Prove Corollary 13, by using Theorem 12.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 15 / 38 Church numerals

For P,M ∈ Λ, deﬁne P n(M) by induction on n ≥ 0:

P 0(M) ≡ M P n+1(M) ≡ P (P n(M))

n The Church numerals c0, c1, c2,... are deﬁned by cn ≡ λf.λx.f (x).

Deﬁne A+ ≡ λxypq.xp(ypq), A× ≡ λxyz.x(yz), Ae ≡ λxy.yx. Then one can prove:

A+cncm =β cn+m,

A×cncm =β cn×m,

Aecncm =β cnm (m > 0).

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 16 / 38 Turing completeness

k A numerical function f : N → N is called λ-deﬁnable if there exists a combinator Pf such that

Pf cn1 ... cnk =β cf(n1,...,nk)

Theorem 14 (Kleene)

Every recursive function is λ-deﬁnable.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 17 / 38 Typed λ-Calculus

Some general references:

Barendregt: Lambda Calculus with Types, Handbook of Logic in Computer Science, vol. 2, Oxford University Press 1993. [Bar93] Hindley and Seldin: Lambda-Calculus and Combinators, an Introduction, Cambridge University Press 2008. [HS08] Barendregt, Dekkers, Statman: Lambda Calculus with Types. Perspectives in Logic. Cambridge University Press 2013. [BDS13] Sørensen and Urzyczyn: Lectures on the Curry-Howard Isomorphism. Studies in Logic and the Foundations of Mathematics, 149, Elsevier 2006. [SU06] Nederpelt and Geuvers: Type Theory and Formal Proof. An Introduction, Cambridge University Press 2014. [NG14] (higher, dependent and inductive, types).

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 18 / 38 There are function types τ → σ, whenever τ and σ are types. We write Γ ` M : τ to express that λ-term M has type τ under the type assumptions for the free variables in M given by Γ In many programming languages we write down types for functions like this:

int strlen(string x){ M }

In type theory we write ` M : string → int

The type system is a deductive system for deriving (`) well-typings of such a form. Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

Simple types in one slide

There are atomic types (type constants such as int, bool, real etc.) and type variables α, β, γ etc.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 We write Γ ` M : τ to express that λ-term M has type τ under the type assumptions for the free variables in M given by Γ In many programming languages we write down types for functions like this:

int strlen(string x){ M }

In type theory we write ` M : string → int

The type system is a deductive system for deriving (`) well-typings of such a form. Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

Simple types in one slide

There are atomic types (type constants such as int, bool, real etc.) and type variables α, β, γ etc. There are function types τ → σ, whenever τ and σ are types.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 In many programming languages we write down types for functions like this:

int strlen(string x){ M }

In type theory we write ` M : string → int

The type system is a deductive system for deriving (`) well-typings of such a form. Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

Simple types in one slide

There are atomic types (type constants such as int, bool, real etc.) and type variables α, β, γ etc. There are function types τ → σ, whenever τ and σ are types. We write Γ ` M : τ to express that λ-term M has type τ under the type assumptions for the free variables in M given by Γ

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 In type theory we write ` M : string → int

The type system is a deductive system for deriving (`) well-typings of such a form. Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

Simple types in one slide

int strlen(string x){ M }

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 The type system is a deductive system for deriving (`) well-typings of such a form. Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

Simple types in one slide

int strlen(string x){ M }

In type theory we write ` M : string → int

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

Simple types in one slide

int strlen(string x){ M }

In type theory we write ` M : string → int

The type system is a deductive system for deriving (`) well-typings of such a form.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 Simple types in one slide

int strlen(string x){ M }

In type theory we write ` M : string → int

The type system is a deductive system for deriving (`) well-typings of such a form. Type assumptions are needed:

{strlen : string → int, s : string} ` strlen(s) : int

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 19 / 38 Simple types

Deﬁnition 15 (Simple types)

Let V denote a denumerable set of type variables, ranged over by metavariables α, β, γ, . . ., and let b range over a set B of type constants. The set T of simple types, ranged over by τ, σ, ρ, . . ., is deﬁned inductively by:

α ∈ V ⇒ α ∈ T b ∈ B ⇒ b ∈ T τ ∈ T, σ ∈ T ⇒ τ → σ ∈ T

Type variables and type constants are referred to as atoms. We write τ → σ → ρ for τ → (σ → ρ).

A type environment is a ﬁnite set Γ of type assumptions of the form Γ = {(x1 : τ1),..., (xn : τn)} with xi 6≡ xj for i 6= j. We let Dm(Γ) = {x ∈ V | ∃τ ∈ T. (x : τ) ∈ Γ}. We write Γ, x : τ for Γ ∪ {(x : τ)}.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 20 / 38 Simple typed λ-calculus, λ→

(var) Γ, x : τ ` x : τ

Γ, x : τ ` M : σ (→I) Γ ` λx.M : τ → σ

Γ ` M : τ → σ Γ ` N : τ (→E) Γ ` MN : σ

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 21 / 38 Since such a derivation can be constructed for arbitrary τ we can see that

∅ ` λx.x : τ → τ, for all τ

is a meta-theorem of the system.

Simple typed λ-calculus, λ→

(var) {x : τ} ` x : τ (→I) ∅ ` λx.x : τ → τ

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 22 / 38 Simple typed λ-calculus, λ→

(var) {x : τ} ` x : τ (→I) ∅ ` λx.x : τ → τ Since such a derivation can be constructed for arbitrary τ we can see that

∅ ` λx.x : τ → τ, for all τ

is a meta-theorem of the system.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 22 / 38 Explicit typing

(var) Γ, x : τ `∗ x : τ

Γ, x : τ `∗ M : σ (→I) Γ `∗ λx: τ.M : τ → σ

Γ `∗ M : τ → σ Γ `∗ N : τ (→E) Γ `∗ MN : σ

Exercise 4

Show by induction on M: if Γ `∗ M : σ and Γ `∗ M : τ, then σ ≡ τ.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 23 / 38 Basis lemma

Let Γ ↓V = {(x : τ) ∈ Γ | x ∈ V }.

Lemma 16

If Γ ⊆ Γ0 then Γ ` M : τ implies Γ0 ` M : τ, If Γ ` M : τ, then FV(M) ⊆ Dm(Γ),

If Γ ` M : τ, then Γ ↓FV(M)` M : τ.

Exercise 5

Prove Lemma 16.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 24 / 38 Inversion

Lemma 17 (Generation lemma)

Suppose that Γ ` M : τ.

1 If M ≡ x, then Γ = Γ0, x : τ, 2 If M ≡ λx.N, then τ ≡ ρ → σ and Γ, x : ρ ` N : σ, 3 If M ≡ PQ, then Γ ` P : σ → τ and Γ ` Q : σ for some σ.

Proof.

Immediate by inspection of the last rule used in the derivation of Γ ` M : τ.

Exercise 6

Call a term M typable, if Γ ` M : τ for some Γ and τ.

Show that every subterm of a typable term is typable.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 25 / 38 Substitutivity

A type substitution is a map S : V → T, and it is lifted homomorphically to a map S : T → T. Let S(Γ) = {(x : S(τ)) | (x : τ) ∈ Γ}.

Lemma 18 (Substitution)

1 If Γ, x : τ ` M : σ and Γ ` N : τ, then Γ ` M[x := N]: σ. 2 If Γ ` M : τ, then S(Γ) ` M : S(τ), for any type substitution S.

Exercise 7

Prove Lemma 18 by induction on derivations.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 26 / 38 Subject reduction

Theorem 19 (Subject Reduction)

If Γ ` M : τ and M β N, then Γ ` N : τ.

Proof.

(Sketch) First prove the statement for the case when M is a redex and N its reduct, using the substitution lemma. Then prove the statement when M →β N by reduction of a redex R in M, by induction on C where M ≡ C[R]. Then prove the statement by induction in the length of the reduction M β N.

Exercise 8

Complete the proof sketch of Theorem 19.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 27 / 38 Strong normalization

Deﬁnition 20

A λ-term M is called strongly normalizing, if every β-reduction sequence starting from M is ﬁnite (i.e., leading to a normal form), and weakly normalizing, if there exists some reduction sequence to normal form starting from M.

Theorem 21

Every typable term in λ→ is strongly normalizing.

Proof.

See lecture notes.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 28 / 38 Combinators

Let C be a set of combinator symbols, ranged over by X,Y,Z. The set ΞC of combinatory expressions, ranged over by F, G, H are deﬁned inductively by:

X ∈ C ⇒ X ∈ ΞC,

x ∈ V ⇒ x ∈ ΞC,

F,G ∈ ΞC ⇒ (FG) ∈ ΞC

Consider the set SKI = {S, K, I} of combinator symbols (referred to as a combinatory base) and deﬁne the notion of weak reduction £w on ΞSKI by setting, for all X,Y,Z ∈ ΞSKI: IF £w F KFG £w F SF GH £w (FH)(GH)

Let →w and w be the reduction relations on ΞSKI induced by £w, by closing £w under contexts C ::= [] | (CF ) | (FC), (F ∈ ΞSKI).

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 29 / 38 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 9

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

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 30 / 38 Combinatory bases

Schönfinkel used, in addition to S and K, the combinators B and C with the definitions BF GH £B F GH CF GH £C FHG But they are not strictly needed, for we can take

B ≡ S(KS)K C ≡ S(S(K(S(KS)K))S)(KK)

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 31 / 38 Combinatory bases

Exercise 10 (One point basis)

Deﬁne the combinator X by the rule

(XF ) £X ((F S)K)

Show that (XX) X SK(KK).

Show that X(X(XX)) X K.

Show that X(X(X(XX))) X S. Conclude that {X} is complete with respect to SKI-calculus.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 32 / 38 ΞSKI 7→ Λ

Deﬁne the map ( )Λ :ΞSKI → Λ by induction on expressions in ΞSKI:

(x)Λ ≡ x, for x ∈ V (I)Λ ≡ λx.x (K)Λ ≡ λxy.x (S)Λ ≡ λxyz.(xz)(yz) (FG)Λ ≡ (F )Λ(G)Λ

Proposition 1

If F w G, then (F )Λ β (G)Λ.

Exercise 11

Prove Proposition 1 by induction on the length of F w G.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 33 / 38 Λ 7→ ΞSKI

Deﬁne, for each x ∈ V, the “bracket abstraction” map [x] :ΞSKI → ΞSKI by induction on expressions in ΞSKI: [x]x ≡ I [x]F ≡ KF, if x 6∈ FV(F ) [x](FG) ≡ S([x]F )([x]G), otherwise

Proposition 2 (Combinatory completeness)

1 ∀F ∈ ΞSKI.∀G ∈ ΞSKI. ([x]F )G w F [x := G] 2 ∀F ∈ ΞSKI. ([x]F )Λ β λx.(F )Λ 3 ∀x ∈ V. ∀F ∈ ΞSKI. ∃H ∈ ΞSKI. ∀G ∈ ΞSKI.HG w F [x := G]

Exercise 12

Prove Proposition 2.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 34 / 38 Λ 7→ ΞSKI

Deﬁne the map ( )Ξ :Λ → ΞSKI by induction on λ-terms:

(x)Ξ ≡ x, for x ∈ V (MN)Ξ ≡ (M)Ξ(N)Ξ (λx.M)Ξ ≡ [x](M)Ξ

Proposition 3

For all M ∈ Λ, one has ((M)Ξ)Λ β M.

Exercise 13

Prove Proposition 3.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 35 / 38 Combinatory logic SKI

(var) Γ, x : τ `ski x : τ

(I) Γ `ski I : τ → τ

(K) Γ `ski K : τ → σ → τ

(S) Γ `ski S :(τ → σ → ρ) → (τ → σ) → τ → ρ

Γ ` F : τ → σ Γ ` G : τ ski ski (→E) Γ `ski (FG): σ

Notice that variables x have ﬁxed, monomorphic types, whereas combinators S, K, I have inﬁnitely many types (their types are schematic or polymorphic). We shall return to this important point in Lecture 5.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 36 / 38 Combinatory logic SKI

Lemma 22 (Deduction theorem for SKI)

If Γ, x : σ `ski F : τ, then Γ `ski [x]F : σ → τ.

Proposition 4

1 If Γ `ski F : τ, then Γ `Λ (F )Λ : τ.

2 If Γ `Λ M : τ, then Γ `ski (M)Ξ : τ.

Exercise 14

Prove Proposition 4. Hint: The ﬁrst statement is proven by induction on the derivation of Γ `ski F : τ. The second statement is proven by induction on the derivation of Γ `Λ M : τ using Lemma 22.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 37 / 38 Decision problems

Type checking (Γ ` M : τ?). Given Γ, M and τ, does Γ ` M : τ hold? Typability (? ` M :?). Given M, are there Γ and τ such that Γ ` M : τ? Inhabitation (Γ `?: τ). Given Γ and τ, does there exist a term (“inhabitant”) M such that Γ ` M : τ?

Type checking and typability for λ→ are linear time solvable.

Inhabitation for λ→ is pspace-complete. See Lecture 2.

Inhabitation can be seen as a program synthesis problem:

Construct program M satisfying speciﬁcation τ.

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J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 38 / 38 M.H. Sørensen and P. Urzyczyn. Lectures on the Curry-Howard Isomorphism, volume 149 of Studies in Logic and the Foundations of Mathematics. Elsevier, 2006.

J. Rehof (TU Dortmund) IPM Tehran August 30, 2017 38 / 38