Lecture 2: Self-interpretation in the Lambda-calculus H. Geuvers - - PowerPoint PPT Presentation

Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

Radboud University

Lecture 2: Self-interpretation in the Lambda-calculus

• H. Geuvers

Radboud University Nijmegen, NL

21st Estonian Winter School in Computer Science Winter 2016

• H. Geuvers - Radboud Univ.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Outline

Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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More on data types

A data type D is a set with some operations (functions) on it. An k-ary operation is a function f : Dk → D. Thereby a 0-ary operation c : D0 → D is identified with a c ∈ D. A datatype is determined by its operations on D: c1, . . . , ck0 : D0 → D = D f 1

1 , . . . , f 1 k1

: D1 → D f 2

1 , . . . , f 2 k2

: D2 → D . . . Nat has z : Nat, s : Nat → Nat. Tree has l : Tree, j : Tree2 → Tree

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Defining functions on data types in the lambda-calculus

F λ-defines the function f : N → N if f (n) = F n for all n:

N N Λ Λ

• f
• −
• −
• F

Can we enode the λ-calculus in itself? F λ-defines the function f : Λ → Λ if f (M) = F M for all M ∈ Λ.

Λ Λ Λ Λ

• f
• −
• −
• F

Is just the identity? Why is this useful?

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Defining functions on codes of lambda-terms

Some functions f : Λ → Λ can only be defined on codes of terms, not on terms. So we will need to talk about codes. Λ Λ Λ Λ

• f
• −
• −
• F
• Example. There is no λ-term F such that

F(M N) = M for all M, N

• There is a λ-term F such that F M N = M for all M, N.

(With a suitable encoding − .)

• We also have an evaluator E:

E M = M

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Packing and unpacking λ-terms

Given M1, . . . , Mk, define M1, . . . , Mk := λz.z M1 . . . Mk Define Uk

i , with 1 ≤ i ≤ k by

Uk

i

:= λx1 . . . xk.xi Then Uk

i M1 . . . Mk

= Mi M1, . . . , MkUk

i

= Uk

i M1 . . . Mk = Mi

Note that K = U2

1

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Second encoding of data types (B¨

• hm-Piperno-Guerini)

Consider the data type D with c : D, f : D → D, g : D2 → D The B¨

• hm-Piperno-Guerini coding (also denoted by t ) is

c = λe.e U3

1 e

f (t) = λe.e U3

2 t e

g(t1, t2) = λe.e U3

3 t1

t2 e

• Proposition. The constructors (c, f , g) can be λ-defined:

There are lambda terms F, G such that F t = f (t) G t1 t2 = g(t1, t2)

• Proof. Take

F := λx e.e U3

2 x e

G := λx y e.e U3

3 x y e.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Recursion

• Theorem. Given A1, A2, A3 ∈ Λ there is an H ∈ Λ such that

H c = A1H H( f (t) ) = A2 t H H( g(t1, t2) ) = A3 t1 t2 H

• Proof. Try H = B1, B2, B3.

H c = B1, B2, B3 c = c B1, B2, B3 = B1, B2, B3U3

1B1, B2, B3

= B1B1, B2, B3 = A1B1, B2, B3 if B1 := λz.A1z = A1H H f (t) = B1, B2, B3U3

2 t B1, B2, B3

= B2 t B1, B2, B3 = A2 t H if B2 := λx z.A2xz H g(t1, t2) = A3 t1 t2 H if B3 := λx y z.A3xyz.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Data type for coding lambda terms

To encode λ-terms we consider the data type D with var : D → D app : D → D → D abs : D → D

• This data types is a bit strange: there is no base case.
• It is a priori unclear how to encode the λ-terms in this data
• type. How to encode the variable binding? (Later slide.)

Like before, we can define the constructors Var, App, Abs: Var := λx e.e U3

1 x e

App := λx y e.e U3

2 x y e

Abs := λx e.e U3

3 x e

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Recursion for the lambda terms data type

Like before, we have a recursion theorem: Theorem I. Given A1, A2, A3 ∈ Λ there is an H ∈ Λ such that H(Var x) = A1 x H H(App x y) = A2 x y H H(Abs x) = A3 x H

• Proof. Take H = B1, B2, B3 with

B1 := λx z.A1xz B2 := λx y z.A2xyz B3 := λx z.A3xz. Then the equations hold indeed. . (Exercise: Check this.)

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Coding of lambda terms

Coding lambda terms M M Definition (Mogensen) The coding of λ-terms inside the λ-calculus is defined as follows. x := Var x MN := App M N λx.M := Abs (λx. M ) A variable x is encoded using x itself and abstraction “λx.” is encoded using the same abstraction “λx.”. Note: coding is not λ-definable: there is no term C such that C M = M for all M. The reverse operation, evaluation, is λ-definable.

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Self-interpretation

• Theorem. There exists a λ-term E (evaluator) such that for all

M ∈ Λ E M = M

• Proof. By recursion we can find an E such that

E(Var x) = x E(App x y) = E x (E y) E(Abs x) = λz.E (x z) Then E( x ) = E(Var x) = x E( MN ) = E(App M N ) = E M (E N ) = MN E( λx.M ) = E(Abs(λx. M )) = λx.E M = λx.M. Filling in the details of E one has (writing C := λx y z.x z y) E = K, S, C.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Why is this encoding so cool?

There are many encoding of Λ in itself. Why is this one so nice?

• Older ones all go through a coding of λ-terms as numbers:

− : Λ → N → Λ. This one is direct, uses λ for λ-abstraction.

• E M ։ M, also for terms with free variables.
• E = K, S, C, the initials of S.C. Kleene, one of the

founders of the subject!

• M1 ≡α M2 ⇒ M1 ≡α M2 .
• M1¬ ≡α M2 ⇒ M1 =β M2 .
• One can define a term R with R M ։ N , if M has N as

normal form.

• There is a Second Fixed Point Theorem (later)
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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Recursion for lambda terms using the encoding

We can state the recursion theorem for the encoded lambda terms slightly differently, as follows. Theorem II. Given A1, A2, A3 ∈ Λ there is an H ∈ Λ such that H x = A1 x H H M N = A2 M N H H λx.M = A3 (λx. M ) H

• Proof. According to Theorem I, there is an H satisfying

H(Var x) = A1 x H H(App x y) = A2 x y H H(Abs x) = A3 x H These equations immediately imply the ones of the statement of Theorem II (check this!), so the same H suffices. .

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Application 1

If you see someone coming out of ‘arrivals’ in an airport, you cannot determine where he/she comes from. Similarly, there is no F such that for all X, Y ∈ Λ F(X Y ) = X (Given the outcome of applying a function on an argument, there is no way we can determine the function that produced this

• utcome.)
• Proposition. There exists an Fi ∈ Λ, i ∈ {1, 2} such that

Fi X1 X2 = Xi .

• Proof. We do this for i = 1. By the Recursion Theorem II, there

exists F1 s.t. F1( X1 X2 ) = A2 X1 X2 F1 = X1 , taking A2 = U3

1.

This suffices.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Second fixed point theorem∗

• Lemma. There exists a term Num ∈ Λ such that for all M ∈ Λ

Num M =β M

• Proof. Use recursion (Theorem I) for the lambda calculus data

type with

A1 x N = App Var (Var x) A2 m n N = App (App App (N m))(N n) A3 m N = App Abs (Abs (λx.N(m x)))

Second fixed point theorem. For all F there is an X with F X =β X

• Proof. Let W := λz.F(App z(Num z)) and X := W W . Then

X = W W = F(App W (Num W ) = F(App W W ) = F W W = F X .

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Application 2∗

Self-modifying programs For a given T (the program transformer) there exists a program P such that P ck = ck+1 if k is even, = T P ck

• therwise.
• On even inputs, P performs a standard operation (adding 1)
• On odd inputs, the program P first modifies its own code

using T. To find P, apply the second fixed-point theorem to F := λp x.if (Even x) then (x + 1) else(T p x)

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Lambda-terms themselves are “black boxes”

• There is no λ-term F such that

F(M N) = M for all M, N

• There is no λ-term F such that

F(M N) = N for all M, N On the other hand, we can compute with the codes of λ-terms

• There is a λ-term F such that F M N = M for all M, N.
• There is a λ-term F such that F M N = N for all M, N.
• We also have an evaluator E:

E M = M

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Analogy with programming

program text p compiler − → executable ˆ p Text Black box

• inspect,
• input/output (“call”)
• edit,
• pass around
• . . .

p : . . . . . . . . . . . . . . . . . . ↓ In − → ˆ p − → Out

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Universal programmable machine

Specification of a universal (programmable) machine U: p : . . . . . . . . . . . . . . . . . . − → x − → U − → ˆ p(x) Reflection over the programs / programming language: What functions (programs) can we write on program text?

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Programs about programs

Examples of programs one can write p : . . . . . . . . . − → M1 − → true if p contains a “while” false if p contains no “while” p : . . . . . . . . . − → M2 − → true if |p| ≤ 1000 false if |p| > 1000 What about this?? p : . . . . . . . . . − → M3 − → true if ˆ p halts on all inputs false if ˆ p diverges on some input

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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The Halting problem is undecidable

Theorem It is impossible to write a program H with the following specification p : . . . . . . . . . . . . . . . . . . − → x − → H − → true if ˆ p halts on x false if ˆ p diverges on x The undecidability of the Halting Problem was first proven by Turing, for his Turing machines. We will prove it for the λ-calculus.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Remember recursion for lambda terms using encoding

Theorem II. Given A1, A2, A3 ∈ Λ there is an H ∈ Λ such that H x = A1 x H H M N = A2 M N H H λx.M = A3 (λx. M ) H Example There is a λ-term C satisfying C x = c0 C M N = c1 C λx.M = c2 Proof Just take A1 := λxy.c0, A2 := λxyz.c1 and A3 := λxy.c2.

• Exercise: Write a function R that checks whether a code of a term

is a redex. (So: R M = T if M is a redex and R M = F if M is not a redex.)

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We need to compute with codes

Recall that T = λx y.x and F = λx y.y. The following are impossible to define with λ-terms themselves. There is no term G satisfying G M = T if M has a normal form G M = F if M has no normal form There is no term H (compare the Halting problem) satisfying H M N = T if M N has a normal form H M N = F if M N has no normal form That these are impossible is not surprising: we can’t “look inside a black box”. What if we recast these question with coded λ-terms?

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The Halting problem for λ-calculus

The Halting problem is undecidable (λ-calculus version): Theorem There is no term H satisfying H M N = T if M N has a normal form H M N = F if M N has no normal form Proof Suppose H exists, Consider Q := λx.H x x Ω T Then Q Q = H Q Q Ω T = TΩ T = Ω if Q Q has a normal form FΩ T = T if Q Q has no normal form

• Contradiction. So H doesn’t exist.

.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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The “Blank tape” problem for λ-calculus

Theorem There is no term B satisfying B M = T if M has a normal form B M = F if M has no normal form Proof Suppose B exists, Consider Q := λx.B x Ω T By the second fixed point theorem, there is a R such that Q R = R, that is B R Ω T = R. Now we have R = B R Ω T = T Ω T = Ω if R has a normal form = F Ω T = T if R has no normal form

• Contradiction. So B doesn’t exist.
• (NB. There may be a proof by reducing H to B; I didn’t see it.)
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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Some other terms one can(not) write

Example There is a term L satisfying L M = cn if n is the number of λ’s inside M Example There is a term V satisfying V M = T if M is of the shape x P1 . . . Pn for some n, P, V M = F if M is not of the shape x P1 . . . Pn. Example There is no term H′ satisfying H′ M N = T if M N has a normal form H′ M N = F if M N has no normal form Proof Reduce B to H′. (Show that, if H′ exists, then we can also define B.)

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Scott’s theorem

The impossibility (undecidability) results we have seen are all instances of a general theorem due to Scott. We phrase it here purely in terms of λ calculus. We assume a coding function − : Λ → Λ for which we have λ-terms App and Num satisfying App M N = M N Num M = M Definition Two disjoint subsets of Λ, A, B ⊆ Λ are separable if there is a λ-term F such that F M = T/F for every M and M ∈ A ⇒ F M = T M ∈ B ⇒ F M = F Theorem (Scott) If A, B are non-empty and closed under =β, then they are not separable.

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Self-interpretation in the Lambda Calculus The limitations of Self-interpretation

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Proof of Scott’s theorem

Let A, B ⊆ Λ closed under =β and say a ∈ A and b ∈ B. Suppose: F separates A and B, so F M = T/F for every M and M ∈ A ⇒ F M = T M ∈ B ⇒ F M = F We define H := λy.if F(App y (Num y)) then b else a and we consider J := H H : J = H H = if F(App H (Num H )) then b else a = if F( H H ) then b else a = b iff H H ∈ A a iff H H ∈ B This is a contradiction. So F doesn’t exist.

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