Logic and Computation Zena M. Ariola University Of Oregon Outline - - PowerPoint PPT Presentation

Logic and Computation

Zena M. Ariola University Of Oregon

Outline of Lectures

• Lecture 1:
• Minimal Propositional Logic and simply typed
• as deduction system and as a computation engine
• Hilbert axioms and Combinatory Logic
• Lecture 2:
• Second-order Propositional Logic and System F
• Classical Logic and Control effects
• Lecture 3: Sequent calculus as a logic and as a PL
• Lecture 4: Applications in compilations, execution, design and

reasoning of program

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λ-calculus λ-calculus

The foundational crisis of mathematics

3

Georg Cantor (1845-1918)

At the beginning of the 19th center set theory was born This theory was very promising because it offered a common foundation to all the fields of mathematics.

ℛ = {x ∣ x ∉ x} ℛ ∈ ℛ ⟺ ℛ ∉ ℛ

Need of logic to define mathematics

4

Let’s devise a complete and finite set of axioms from which all true statements can be derived.

David Hilbert (1862-1943) Kurt Gödel (1906-1978)

Gödel in 1935 published his incompleteness result.

Gerhard Gentzen (1909-1945)

Gentzen introduced a formal system which modeled mathematical reasoning quite directly

Natural Deduction

Natural Deduction - Gentzen (1935)

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Judgement:

Γ ⊢ A A, B ::= X ∣ A ⊃ B ∣ A ∧ B ∣ A ∨ B

Formulae:

If all assumptions in Γ are true then A is true

Introduction and Elimination rules

Natural deduction

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Γ, A ` A Ax Γ ` A Γ ` B Γ ` A ^ B ^I Γ ` A ^ B Γ ` A ^E1 Γ ` A ^ B Γ ` B ^E2 Γ, A ` B Γ ` A BI Γ ` A B Γ ` A Γ ` B E Γ ` A Γ ` A _ B_I1 Γ ` B Γ ` A _ B_I2 Γ ` A _ B Γ, A ` C Γ, B ` C Γ ` C _E

Harmony (Dummett 1976)

Local Soundness shows that the elimination rules are

not too strong.

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Γ ` A _ B Γ ` B

Local Completeness show that the elimination rules are

not too weak

Γ ` A ^ B Γ ` A

Local Soundness

Elimination rules applied to the result of an introduction rule do not produce new information.

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Local Reductions

D Γ ` A E Γ ` B Γ ` A ^ B ^I Γ ` A ^E

⟹

B E D Γ ` A

Local Soundness

Elimination rules applied to the result of an introduction rule do not produce new information.

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D Γ, A ` B Γ ` A B I A B I E Γ ` A ` Γ ` B E

⟹

D[E/A] Γ ` B

Local Reductions

Local Soundness

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A A, B ` A A A A ` B (A A) I ` (A A) B (A A) I A ` A ` A A I ` B (A A) E = ) B, A ` A B ` A A I ` B (A A) I D Γ, A ` B Γ ` A B I E Γ ` A Γ ` B E E D[E/A] Γ ` B E = )

Local Completeness

The elimination rules are strong enough to get all the pieces out of a structured proposition so that they can be

• recomposed. A proof can always end with an introduction

rule.

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Γ ` A ^ B

𝒠

A B = ) Γ ` A ^ B Γ ` A ^E Γ ` A ^ B Γ ` B ^E Γ ` A ^ B ^I

𝒠 𝒠

Local Expansion

Local completeness

The elimination rules are strong enough to get all the pieces out of a structured proposition so that they can be

• recomposed. A proof can always end with an introduction

rule.

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Local Expansion

A B = ) Γ, A ` A B Γ, A ` A Γ, A ` B E Γ ` A BI

𝒠

Γ ` A B A

𝒠

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Caution with variables

x = 0 ` x = 0 x = 0 ` 8x.x = 0

x x x ` 8x.9y.x 6= y ` 9y.y 6= y

n

[

i=1

ai

n

Y

i=1

ai

n

X

i=1

ai lim

n!1 n2

Z 1 x2 dx

Γ ` P(x) Γ ` 8x.P(x)8I ===== x P x I Γ ` 8x.P(x) Γ ` P(M) 8E

Lambda calculus

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First [1932] version of lambda-calculus

• Church. Postulates for the Foundations of Logic, Annals of

Mathematics, 33(2), 1932

• Church. Postulates for the Foundations of Logic,

Annals of Mathematics, 34(4), 1934. Church proposes to encode all these different notations as

• perators which take a function as parameter:

∃x . P(x) = ∃(λx . P(x)) ∀x . P(x) = ∀(λx . P(x))

There is only one mechanism to introduce a variable - Higher-order abstract syntax

Alonzo Church (1903-1995)

Church introduced the notation for the function with parameter and body .Variable is bound and can be renamed without changing the meaning of the function.

λx . e

x

e

x

Lambda calculus

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37 axioms (36 in the revision) and 5 inference rules

λx . M =α λz . M[z/x]

(λx . M)N =β M[N/x]

M, N ::= x variable λx.M abstraction M N application &(M, N) M and N ¬(M) not M ι(M) an integer x such that M x Y (M, N) forall x. if M(x) then N(x) X (M) ∃x, Mx A(M, N) abstraction from N with respect to M

Propositions as Terms

Predicates and propositions are lambda-calculus terms Similar to logic programming today (e.g. Prolog) Predicates takes predicates in input (higher-order logic) Sets are identified with their characteristic function: 
 The set of all x's such that x does not belong to x:

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A ⊃ (A & A)

λx . ¬(x x)

x ∈ A = A x {x | P(x)} = λx.P x {x} = λy.x = y A ∪ B = λx.A x ∨ B x

Given

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Lambda calculus is inconsistent

How did Church react? This is not a paradox because he did not accept excluded middle (every proposition is either true or false). R is neither

R ∈ R ⟺ R ∉ R

R = λx . ¬(x x)

R 2 R = R R = (λx.¬(x x))(λx.¬(x x)) = ¬(R R) = R 62 R

Does R contains itself?

Curry paradox (1942)

Let’s focus on implication:

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(X (X Y)) (X Y) (1) X Y, X ` Y (MP) X, X = Y ` Y (Eq)

We define the term/proposition:

term/proposition: X = X (X Z)

X = X (X Z) (4) = (X (X Z)) (X Z) (5)

• provable:

(6) (X (X Z)) (X Z) (1) (7) X (X Z) (5) and (Eq) (8) X (4) and (Eq) (9) X Z (MP) and (7), (8) (10) Z (MP) and (8), (9) It cannot use

• calculus as a logic, if we can define the formula X! For it is

Everything is provable

Fixed point theorem

Can we define the formula

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X = X ⊃ (X ⊃ Z)

Take

becomes the fixed point of

Fixed point theorem. For every F there exists an X such that F X = X.

• Proof. Let

D = λx . F(x x) X = D D

X = D D = F(D D) = F(X)

X = X ⊃ (X ⊃ Z)

We were looking for a proposition:

X = (λx . x ⊃ (x ⊃ Z))X

λx . x ⊃ (x ⊃ Z)

X

How to solve the problem?

Let’s get rid off the logical connectives

Computation

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Recognize that not all terms represent propositions

Type system

Church with his students Kleene and Rosser (1935) studied the pure lambda- calculus :

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Pure Lambda-calculus

M ::= x ∣ λx . M ∣ M M

They orient the beta-axiom:

(λx . M) N → M[N/x]

Terms represent computation:

M → M1 → M2 → ⋯ → N →

Theorem (Church-Rosser). beta-reduction is confluent

A normal form (i.e. the answer of a program) if it exists is unique

Lambda-calculus as a computational model

Church (1936), it is undecidable if a term has a normal form

Hilbert program cannot be realized

Kleene (1935) and Turing (1937) show the correspondence of the following computational models:

• Partial recursive functions
• Turing machines
• Functions defined by lambda-terms

A function

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f : N → N

F

such that

∀n ∈ N . F ⌜n⌝ = ⌜f n⌝

is lambda-definable if there exists a term

Lambda calculus as the prototypical Functional programming language

Functional language = lambda-calculus + evaluation strategy + data structures

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Peter Landin described ISWIM (If you See What I Mean) in The Next 700 Programming Languages, published in the Communications of the ACM in 1966.

(1930-2009)

John McCarthy designed LISP (LISt Processor) in 1958 as an implementation of lambda-calculus, but implemented substitution incorrectly

(1927-2011)

Simple Theory of Types (Church, 1940)

Types

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Terms

M ::= x ∣ λx : σ . M ∣ M M

Static semantics

Γ, x : σ ` x : σ

Γ ` M : σ ! τ Γ ` N : σ Γ ` M N : τ Γ, x : σ ` M : τ Γ ` λx : σ.M : σ ! τ

⊢ λx . σ . λy . τ . x : σ → τ → σ

⊢ λx : σ → τ . λy : τ → ρ . λz : σ . y(xz) : (σ→τ)→(τ→ρ)→σ→ρ

⊢ λx : σ . λy : (τ→σ)→σ . y (λz : τ . x) : σ→((τ→σ)→σ)→σ σ, τ ::= ι ∣ o ∣ σ → τ ⊃: o → o → o ∀ : (ι → o) → o ⊢ a ⊃ a : o

Properties

Not every type has an inhabitant (i.e. a closed term of that type):

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(σ → σ) → σ

Theorem (Subject reduction)

and then .

Theorem (Strong Normalization)

Γ ⊢ M : τ M → → N Γ ⊢ N : τ Γ ⊢ M : τ

there is no infinite reduction sequence starting from M

Formulae as Types

Formulae

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σ, τ ::= b ∣ σ → τ

Types

A, B ::= X ∣ A ⊃ B

Change of perspective: formulae are not terms

Proofs as programs

If 𝜌 is a proof of formula A then 𝜌 can be represented as a lambda-calculus term

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A ! (A ! B), A ` A ! (A ! B) A ! (A ! B), A ` A A ! (A ! B), A ` B A ! (A ! B) ` A ! B ` (A ! (A ! B)) ! A ! B

Γ, A ` A Ax Γ, A ` B Γ ` A BI Γ ` A B Γ ` A Γ ` B E Γ, x : A ` x : A Ax Γ, x : A ` M : B Γ ` λx.M : A ! B!I Γ ` M : A ! B Γ ` N : A Γ ` M N : B !E

x : A ! (A ! B), y : A ` x : A ! (A ! B) x : A ! (A ! B), y : A ` y : A x : A ! (A ! B), y : A ` x y : B x : A ! (A ! B) ` λy : A.x y : A ! B ` λx : A ! (A ! B).λy : A.x y : (A ! (A ! B)) ! A ! B

Proofs as Programs

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There is a one-to-one correspondence: Typable terms ≈ minimal propositional logic proofs

x1 : A1, x2 : A2, ⋯, xn : An ⊢ M : A

M is a proof of A from assumptions A1, A2, ⋯, An M is a program of type A with free variables

x1, x2, ⋯, xn

• f type A1, A2, ⋯, An

What does β-reduction correspond to?

Howard (1980) observes that β-reduction corresponds to elimination of a detour (local soundness)

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D Γ, A ` B Γ ` A ! B !i E Γ ` A Γ ` B !e = ) D[E/A] Γ ` B

Γ, x : A ` M : B Γ ` λx : A.M : A ! B Γ ` N : A Γ ` (λx : A.M) N : B = ) Γ ` M[N/x] : B

Can all the detours be eliminated from a proof? A term in normal form (i.e. without β-redexes) corresponds to a proof without detours

What does η-reduction correspond to?

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A B ) Γ, x : A ` M : A B Γ, x : A ` x : A Γ, x : A ` M x : B !E Γ ` λx.M x : A B!I

M A B = )

Γ ` M : A ! B =

𝒠 𝒠 𝒠

Γ ` A B = A

A B = )

Γ, A ` A B Γ, A ` A Γ, A ` B E Γ ` A BI

Extensionality witnesses local completeness

Curry-Howard isomorphism

Logic Type Theory Formula Type Proof Program Detour eliminations β-reduction Expansions Extensionality Propositional logic STLC

Is this a coincidence?

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Conjunction and pairing

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Γ ` A Γ ` B Γ ` A ^ B Γ ` A ^ B Γ ` A Γ ` A ^ B Γ ` B

Γ ` M : A Γ ` N : B Γ ` (M, N) : A ⇥ B Γ ` M : A ⇥ B Γ ` π1M : A Γ ` M : A ⇥ B Γ ` π2M : B

D Γ ` A E Γ ` B Γ ` A ^ B ^I Γ ` A ^E

A E )

B E D Γ ` A

N A E )

A E ) Γ ` M : A

Γ ` M : A Γ ` N : B Γ ` (M, N) : A ⇥ B ^I Γ ` π1 (M, N) : A ^E D Γ ` A ^ B = )

D Γ ` A ^ B A ^E D Γ ` A ^ B Γ ` B ^E Γ ` A ^ B ^I

Γ ` M : A ^ B M A B = ) B Γ ` M : A ^ B π1 M : A ^E Γ ` M : A ^ B Γ ` π2 M : B ^E Γ ` (π1 M, π2 M) : A ^ B ^I

Logic Type theory Local reductions Local expansions

Disjunction and union

Logic

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• Γ ` A

Γ ` A _ B_i Γ ` B Γ ` A _ B_i Γ ` A _ B Γ, A ` C Γ, B ` C Γ ` C _e

Type theory

• Γ ` M : A

Γ ` inj1 M : A + B +i Γ ` M : B Γ ` inj2 M : A + B +i Γ ` M : A + B Γ, x1 : A ` C Γ, x2 : B ` C Γ ` case M of inj1 x1 : A ) N1 | inj2x2 ) N2 end : C +e

Local reductions

D Γ ` A Γ ` A _ B_I E Γ, A ` C Γ, B ` C Γ ` C _E = ) E[D/A] Γ ` C Γ ` M : A Γ ` inj1 M : A + B+I Γ, x1 : A ` N1 : C Γ, x2 : B ` N2 : C Γ ` case M of inj1 x1 ) N1 | inj2 x2 ) N2 end : C +E = ) Γ ` N1[M/x1]

36

Local Expansion

A B = )

D Γ ` A _ B

D Γ ` A _ B Γ, A ` A Γ, A ` A _ B_I Γ, B ` B Γ, B ` A _ B_I Γ ` A _ B _E

Γ ` M : A+B

M A B = )

M A B Γ ` M : A + B Γ, x1 : A ` x1 : A Γ, x1 : A ` inj1 x1 : A + B+I Γ, x2 : B ` x2 : B Γ, x2 : B ` inj2 x2 : A + B+I Γ ` case M of inj1 x1 ) inj1 x1 | inj2 x2 ) inj2 x2 : A + B +E

Note that the elimination follows the introduction, instead of the other way around!

Disjunction and union - Local expansion

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Hilbert axioms

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Hilbert axioms ≈ Typed Combinatory Logic

• D. Hilbert

(1862-1943)

A ⊃ A (1)

A ⊃ B ⊃ A (2)

(A ⊃ B ⊃ C) ⊃ (A ⊃ B) ⊃ A ⊃ C (3)

Modus Ponens

` A ((B A) A) (2) ` (A ((B A) A)) (A (B A)) A A (3) ` ((A ((B A)) (A A)) MP ` A B A (2) ` A A MP

Combinatory Logic

Terms

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M ::= I ∣ K ∣ S ∣ M M

Axioms

I M = M K M N = M S M N P = (M P)(N P)

Types I : A → A

K : A → B → A S : (A → B → C) → (A → B) → A → C

Hilbert axioms ≈ Typed Combinatory Logic

Haskell Curry (1900-1982)

Proofs as Programs

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` A ((B A) A) (2) ` (A ((B A) A)) (A (B A)) A A (3) ` ((A ((B A)) (A A)) MP ` A B A (2) ` A A MP ` K : A ((B A) A) ` S : (A ((B A) A)) (A (B A)) A A ` S K : ((A ((B A)) (A A)) ` K : A B A ` S K K : A A

Is this just a coincidence?

Can we extend the correspondence to other logics?