Logic and Computation Lecture 3 -...

Logic and ComputationLecture 3

Zena M. Ariola

University of Oregon

24th Estonian Winter School in Computer Science, EWSCS ’19

What we have seen so far

We have seen a beautiful correspondence between logic andprogramming languages. We can use the same system toprogram and reason about our programs.

Let’s now take a critical view of lambda-calculus and naturaldeduction.

Outline

What is bad about natural deduction andlambda-calculus?

Sequent calculus as a logic

Sequent calculus as a programming language

Duality of syntax and semantics

Polarity, evaluation in the type

Outline

Dualities

Alcmaeon (510BC) : Most things come in pairsright le� odd even one manylimited unlimited straight curved light darknesswhite black sweet bi�er good badlarge small young old male femaleman god family state living deadtrue false future past good bad

Duality points you at the existence of another entity with a totally di�erentform, but which is strongly related to the original one.

A source of free theorems

Dualities

Alcmaeon (510BC) : Most things come in pairsright le� odd even one manylimited unlimited straight curved light darknesswhite black sweet bi�er good badlarge small young old male femaleman god family state living deadtrue false future past good bad

Duality points you at the existence of another entity with a totally di�erentform, but which is strongly related to the original one.

A source of free theorems

Theorem

A formula A is true i� its dual is false

True and False are dual

∧ and ∨ are dual

A ∨ True = TrueA ∨ ¬A = True

A ∧ False = FalseA ∧ ¬A = False

Where is this duality expressed in Natural Deduction?

Theorem

Commuting conversions

Γ ` A ∨ B Γ,A ` C Γ,B ` CΓ ` C

In Proofs and Types, Jean-Yves Girard says

the elimination rules for ∨ and ∃ are very bad. What is catastrophic aboutthem is the parasitic presence of a formula C which has no structural link withthe formula which is eliminated. Moreover, in order to have normal proofs withthe subformula property, there is the need to add some ad hoc conversions: thecommuting conversions

DΓ ` A ∨ B

Γ, A ` CD1

Γ, B ` C

Γ ` C∨E

Γ ` Dr

Γ ` A ∨ B

Γ, A ` C

Γ, A ` Dr

Γ, B ` C

Γ, B ` Dr

Γ ` D∨E

Commuting conversions

Γ ` A ∨ B Γ,A ` C Γ,B ` CΓ ` C

In Proofs and Types, Jean-Yves Girard says

the elimination rules for ∨ and ∃ are very bad. What is catastrophic aboutthem is the parasitic presence of a formula C which has no structural link withthe formula which is eliminated. Moreover, in order to have normal proofs withthe subformula property, there is the need to add some ad hoc conversions: thecommuting conversions

DΓ ` A ∨ B

Γ, A ` CD1

Γ, B ` C

Γ ` C∨E

Γ ` Dr

Γ ` A ∨ B

Γ, A ` C

Γ, A ` Dr

Γ, B ` C

Γ, B ` Dr

Γ ` D∨E

Subformula property

A ∨ A ` A ∨ AA ` A A ` AA ` A ∧ A

∧IA ` A A ` AA ` A ∧ A

A ∨ A ` A ∧ A∨E

A ∨ A ` A∧E

A ∨ A ` A ∨ A

A ` A A ` AA ` A ∧ A

A ` A∧E

A ` A A ` AA ` A ∧ A

A ` A∧E

A ∨ A ` A∨E

=⇒A ∨ A ` A ∨ A A ` A A ` A

A ∨ A ` A∨E

Bernard Bolzano (1781-1848)

A proof is analytic if it does not use concepts (i.e.subformulae) beyond its subject ma�er (i.e. the formulawe are going to prove)

Subformula property

A ∨ A ` A ∨ AA ` A A ` AA ` A ∧ A

∧IA ` A A ` AA ` A ∧ A

A ∨ A ` A ∧ A∨E

A ∨ A ` A∧E

A ∨ A ` A ∨ A

A ` A A ` AA ` A ∧ A

A ` A∧E

A ` A A ` AA ` A ∧ A

A ` A∧E

A ∨ A ` A∨E

=⇒A ∨ A ` A ∨ A A ` A A ` A

A ∨ A ` A∨E

Subformula property

A ∨ A ` A ∨ AA ` A A ` AA ` A ∧ A

∧IA ` A A ` AA ` A ∧ A

A ∨ A ` A ∧ A∨E

A ∨ A ` A∧E

A ∨ A ` A ∨ A

A ` A A ` AA ` A ∧ A

A ` A∧E

A ` A A ` AA ` A ∧ A

A ` A∧E

A ∨ A ` A∨E

=⇒A ∨ A ` A ∨ A A ` A A ` A

A ∨ A ` A∨E

Subformula property

A ∨ A ` A ∨ AA ` A A ` AA ` A ∧ A

∧IA ` A A ` AA ` A ∧ A

A ∨ A ` A ∧ A∨E

A ∨ A ` A∧E

A ∨ A ` A ∨ A

A ` A A ` AA ` A ∧ A

A ` A∧E

A ` A A ` AA ` A ∧ A

A ` A∧E

A ∨ A ` A∨E

=⇒A ∨ A ` A ∨ A A ` A A ` A

A ∨ A ` A∨E

Proof search

When we prove a theorem, it would be convenient to construct the proof bo�om-up

How do we prove A→ (B→ C)→ (A→ B)→ (A→ C):

?A→ (B→ C),A→ B,A ` C

A→ (B→ C),A→ B ` A→ C→i

` A→ (B→ C) ` (A→ B)→ (A→ C)→I

A→ (B→ C),A→ B,A ` B→ C A→ (B→ C),A→ B,A ` B

A→ (B→ C),A→ B,A ` C→E

How are going to guess B?

Proof search

?A→ (B→ C),A→ B,A ` C

A→ (B→ C),A→ B ` A→ C→i

` A→ (B→ C) ` (A→ B)→ (A→ C)→I

A→ (B→ C),A→ B,A ` C→E

Proof search

?A→ (B→ C),A→ B,A ` C

A→ (B→ C),A→ B ` A→ C→i

` A→ (B→ C) ` (A→ B)→ (A→ C)→I

A→ (B→ C),A→ B,A ` C→E

The opposite category

A A× B B

π1 π2

A A + B B

Why not λ-calculus (natural deduction)?

A A× B B

π1 π2

π1(〈f , g〉) = fπ2(〈f , g〉) = gh = 〈π1h, π2h〉

A A + B B

Why not λ-calculus (natural deduction)?

A A× B B

π1 π2

π1(〈f , g〉) = fπ2(〈f , g〉) = gh = 〈π1h, π2h〉

A A + B B

Question

How can we talk about these dualities ?

Can we construct a proof system that uses only subformulae ofthe conclusion?

In addition to talking about A being True, why not also takingabout A being False?

Gentzen again

Gentzen could not prove the consistency ofnatural deduction directly. For this reason heintroduced an alternative system (1935): sequentcalculus (LJ and LK)

Consisteny- Gentzen 1934There is no LK proof of ⊥.

Natural deduction and LK are equivalent

ND ` A i� LK ` A

Gentzen again

Gentzen could not prove the consistency ofnatural deduction directly. For this reason heintroduced an alternative system (1935): sequentcalculus (LJ and LK)

Consisteny- Gentzen 1934There is no LK proof of ⊥.

Natural deduction and LK are equivalent

ND ` A i� LK ` A

Deduction and Duality

The judgmentA1, · · · ,An ` B

becomesA1, · · · ,An ` B1, · · · ,Bm

We have both multiple assumptions and multiple conclusions.

It reads as:(A1 ∧ · · · ∧ An) ` (B1 ∨ · · · ∨ Bm)

Given Γ ` ∆

if Γ is empty we identify it with > : ` ∆. This means means that some formulain ∆ is true

if ∆ is empty we identify it with ⊥: Γ `. This means Γ is inconsistent

becomesA1, · · · ,An ` B1, · · · ,Bm

Given Γ ` ∆

becomesA1, · · · ,An ` B1, · · · ,Bm

Given Γ ` ∆

becomesA1, · · · ,An ` B1, · · · ,Bm

Given Γ ` ∆

Gentzen’s LK

Formulae: A,B,C ::= X | A ∧ B | A ∨ B | A ⊃ B | ∀X .A | ∃X .A

Core rules:

Γ,A ` AAx

Γ ` A,∆ Γ,A ` ∆

Γ ` ∆Cut

Structural rules:

Γ ` ∆Γ ` A,∆

WRΓ ` ∆

Γ,A ` ∆WL

Γ ` A,A,∆Γ ` A,∆

CRΓ,A,A ` ∆

Γ,A ` ∆CL

Γ ` ∆,A,B,∆′

Γ ` ∆,B,A,∆′XR

Γ′,B,A, Γ ` ∆

Γ′,A,B, Γ ` ∆XL

LK logical rules

Γ ` A,∆ Γ ` B,∆Γ ` A ∧ B,∆

∧RΓ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ,B ` ∆

Γ,A ∧ B ` ∆∧L

Γ,A ` ∆ Γ,B ` ∆

Γ,A ∨ B ` ∆∨L

Γ ` A,∆Γ ` A ∨ B,∆

∨RΓ ` B,∆

Γ ` A ∨ B,∆∨R

Γ,A ` B,∆Γ ` A→ B,∆

→RΓ ` A,∆ Γ,B ` ∆

Γ,A→ B ` ∆→L

ExamplesLaw of Excluded Middle: ` A ∨ ¬APierce Law: ` ((A→ B)→ A)→ AContraposition: ` (¬B→ ¬A)→ A→ B

Polymorphism and Abstraction

Γ ` A,∆ X 6∈ FV(Γ ` ∆)

Γ ` ∀X .A,∆ ∀RΓ,A[B/x] ` ∆

Γ,∀X .A ` ∆∀L

Γ,A ` ∆ X /∈ FV(Γ ` ∆)

Γ,∃X .A ` ∆∃L

Γ ` A{B/X},∆Γ ` ∃X .A,∆ ∃R

Examples

` ∃x.D(x)→ ∀x.D(x)

The Hauptsatz

Hauptsatz - Gentzen 1934For all LK proofs of Γ ` ∆ there exists an alternate LK proof of Γ ` ∆ that does notcontain any use of the Cut rule.

ConsistencyThere is no LK proof of ` ⊥.

Basic idea, in a nutshell:"push" the cuts upwards in a proof, until they disappear completely!

D....Γ ` A,∆ A ` AAx

Γ ` A,∆ Cut=⇒

D....Γ ` A,∆

A ` AAx

E....Γ,A ` ∆

Γ,A ` ∆Cut

E....Γ,A ` ∆

Logical cuts

D1....Γ ` A,∆

D2....Γ ` B,∆

Γ ` A ∧ B,∆∧R

E....Γ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ ` ∆Cut

D1....Γ ` A,∆

E....Γ,A ` ∆

Γ ` ∆Cut

D....Γ,A ` B,∆

Γ ` A ⊃ B,∆→R

E1....Γ ` A,∆

E2....Γ,B ` ∆

Γ,A ⊃ B ` ∆→L

Γ ` ∆Cut

E1....Γ ` A,∆

D....Γ,A ` B,∆

E2....Γ,B ` ∆

Γ,A ` ∆Cut

Γ ` ∆Cut

Logical cuts

D1....Γ ` A,∆

D2....Γ ` B,∆

Γ ` A ∧ B,∆∧R

E....Γ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ ` ∆Cut

D1....Γ ` A,∆

E....Γ,A ` ∆

Γ ` ∆Cut

D....Γ,A ` B,∆

Γ ` A ⊃ B,∆→R

E1....Γ ` A,∆

E2....Γ,B ` ∆

Γ,A ⊃ B ` ∆→L

Γ ` ∆Cut

E1....Γ ` A,∆

D....Γ,A ` B,∆

E2....Γ,B ` ∆

Γ,A ` ∆Cut

Γ ` ∆Cut

Termination

Proving that cut-elimination terminates is complex!

D....Γ ` A,∆

E....Γ,A,A ` ∆

Γ,A ` ∆CL

Γ ` ∆Cut

D....Γ ` A,∆

E....Γ,A,A ` ∆

Γ,A,` ∆Cut

Γ ` ∆Cut

Proof D has been duplicated!

Sequent calculus: a symmetric language

A computation is an interaction between a term v (producer) and a context e(consumer, continuation, co-term, stack)

〈v||e〉

Producers v Consumers e (contexts)

Input variable x Output variable α

Function abstraction λx.v Function call (call stack) v ′ · eOutput abstraction µα.c Input abstraction µ̃x.c

µ̃ abstracts over an unspecified input. µ̃x.c can be seen as let x = � in c.

µ abstracts over an unspecified output. Continuations are not functions

Example

callcc(λk.M) ∼ µα.〈M||α〉.C(λk.M) ∼ µα.〈M||tp〉.Abort(M) ∼ µ_.〈M||tp〉.

Sequent calculus: a symmetric language

Command c ::= 〈v | e〉Terms v ::= x | λx.v

| 〈v, v〉 | inj1(v) | inj2(v)

| µα.cCo-Terms e ::= α | v · e

| [e1, e2] | π1(e) | π2(e)

| µ̃x.c

Example

(λx.x) z becomes µα.〈λx.x | z.α〉(λx.λy.x y) z w becomes µα.〈λx.λy.µα.〈x | y.α〉 | z.w.α〉π1(v, v ′) becomes µα.〈(v, v ′)||π1.α〉case inj1 v of inj1 x => v1 | inj2 x => v2 becomesµα.〈inj1 v||[µ̃x.〈v1||α〉, µ̃x.〈v2||α〉〉

Re-associating programs

(λx.M)N1N2N3 〈λx.M||N1 · N2 · N3 · α〉

A redex is always at the top

Evaluation corresponds to an abstract machine, i.e. a tail-recursive interpreter

Computational interpretation of classical logic

The judgement A1, · · · ,An ` B1, · · · ,Bm is interpreted as typing a command c:

c : (x1 : A1, · · · , x2 : An ` β1 : B1, · · · , βm : Bm)

The types of variables appear as assumptions. The types of co-variables appear asconclusions

Since we have multiple assumptions and conclusions, computationally we need twomore judgements expressing the intent to focus on a particular assumption orconclusion:

The judgement x1 : A1, · · · , xn : An ` v : B | β1 : B1, · · · , βm : Bm expresses thefact that we focus on the conclusion B. This types a term v which can producea value of type B

The judgement x1 : A1, · · · , xn : An | e : A ` β1 : B1, · · · , βm : Bm expresses thefact that we focus on the assumption A. This types a context e which iswaiting for a value of type A

c : (x1 : A1, · · · , x2 : An ` β1 : B1, · · · , βm : Bm)

C-H isomorphism for classical logic - Core rules

A,B,C ∈ Formulae ::= X | A ∧ B | A ∨ B | A ⊃ B | ∀X .A | ∃X .A

Γ,A ` A,∆Ax

Γ ` A,∆ Γ,A ` ∆

Γ ` ∆Cut

A,B,C ∈ Types ::= X | A× B | A + B | A→ B | ∀X .A | ∃X .A

Γ, x : A ` x : A | ∆VR

Γ | α : A ` α : A,∆VL

Γ ` v : A | ∆ Γ | e : A ` ∆

〈v||e〉 : (Γ ` ∆)Cut

c : (Γ ` α : A,∆)

Γ ` µα.c : A | ∆AR

c : (Γ, x : A ` ∆)

Γ | µ̃x.c : A ` ∆AL

Γ,A ` A,∆Ax

Γ ` A,∆ Γ,A ` ∆

Γ ` ∆Cut

Γ, x : A ` x : A | ∆VR

Γ | α : A ` α : A,∆VL

Γ ` v : A | ∆ Γ | e : A ` ∆

〈v||e〉 : (Γ ` ∆)Cut

c : (Γ ` α : A,∆)

Γ ` µα.c : A | ∆AR

c : (Γ, x : A ` ∆)

Γ | µ̃x.c : A ` ∆AL

Γ,A ` A,∆Ax

Γ ` A,∆ Γ,A ` ∆

Γ ` ∆Cut

Γ, x : A ` x : A | ∆VR

Γ | α : A ` α : A,∆VL

Γ ` v : A | ∆ Γ | e : A ` ∆

〈v||e〉 : (Γ ` ∆)Cut

c : (Γ ` α : A,∆)

Γ ` µα.c : A | ∆AR

c : (Γ, x : A ` ∆)

Γ | µ̃x.c : A ` ∆AL

C-H isomorphism for classical logic - Logical rules

Γ ` A,∆ Γ ` B,∆Γ ` A ∧ B,∆

∧RΓ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ,B ` ∆

Γ,A ∧ B ` ∆∧L

Γ,A ` ∆ Γ,B ` ∆

Γ,A ∨ B ` ∆∨L

Γ ` A,∆Γ ` A ∨ B,∆

∨RΓ ` B,∆

Γ ` A ∨ B,∆∨R

Γ,A ` B,∆Γ ` A→ B,∆

→RΓ ` A,∆ Γ,B ` ∆

Γ,A→ B ` ∆→L

Γ ` v : A | ∆ Γ ` v ′ : B | ∆

Γ ` (v, v ′) : A× B | ∆×R

Γ | e : A ` ∆

Γ | π1(e) : A× B ` ∆×L

Γ | e : B ` ∆

Γ | π2(e) : A× B ` ∆

Γ | e : A ` ∆ Γ | e′ : B ` ∆

Γ | [e, e′] : A + B ` ∆+L

Γ ` v : A | ∆

Γ ` inj1(v) : A + B | ∆+R

Γ ` v : B | ∆

Γ ` inj2(v) : A + B | ∆+R

Γ, x : A ` v : B | ∆

Γ ` λx.v : A→ B | ∆→R

Γ ` v : A | ∆ Γ | e : B ` ∆

Γ | v · e : A→ B ` ∆→L

Classical proofs as sequent terms

Examples

The proof of Excluded Middle is µα.〈inj2(λp.µ_.〈inj1 p||α〉)||α〉The proof of Pierce Law is λx.µα.〈x||(λx.µδ.〈x||α〉).α〉The proof of Double Negation is λx.µα.〈x||(λx.µδ.〈x||α〉).tp〉

How to compute

D1....Γ ` A,∆

D2....Γ ` B,∆

Γ ` A ∧ B,∆∧R

E....Γ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ ` ∆Cut

D1....Γ ` A,∆

E....Γ,A ` ∆

Γ ` ∆Cut

Γ ` v1 : A | ∆ Γ ` v2 : B | ∆

Γ ` (v1, v2) : A ∧ B | ∆∧R

Γ | e : A ` ∆

Γ | π1(e) : A ∧ B ` ∆∧L

〈(v1, v2)||π1(e)〉 : Γ ` ∆Cut

=⇒Γ ` v1 : A | ∆ Γ | e : A ` ∆

〈v | e〉 : Γ ` ∆Cut

〈(v1, v2)||π1(e)〉 → 〈v1||e〉

How to compute

D1....Γ ` A,∆

D2....Γ ` B,∆

Γ ` A ∧ B,∆∧R

E....Γ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ ` ∆Cut

D1....Γ ` A,∆

E....Γ,A ` ∆

Γ ` ∆Cut

Γ ` v1 : A | ∆ Γ ` v2 : B | ∆

Γ ` (v1, v2) : A ∧ B | ∆∧R

Γ | e : A ` ∆

Γ | π1(e) : A ∧ B ` ∆∧L

〈(v1, v2)||π1(e)〉 : Γ ` ∆Cut

=⇒Γ ` v1 : A | ∆ Γ | e : A ` ∆

〈v | e〉 : Γ ` ∆Cut

〈(v1, v2)||π1(e)〉 → 〈v1||e〉

How to compute

D1....Γ ` A,∆

D2....Γ ` B,∆

Γ ` A ∧ B,∆∧R

E....Γ,A ` ∆

Γ,A ∧ B ` ∆∧L

Γ ` ∆Cut

D1....Γ ` A,∆

E....Γ,A ` ∆

Γ ` ∆Cut

Γ ` v1 : A | ∆ Γ ` v2 : B | ∆

Γ ` (v1, v2) : A ∧ B | ∆∧R

Γ | e : A ` ∆

Γ | π1(e) : A ∧ B ` ∆∧L

〈(v1, v2)||π1(e)〉 : Γ ` ∆Cut

=⇒Γ ` v1 : A | ∆ Γ | e : A ` ∆

〈v | e〉 : Γ ` ∆Cut

〈(v1, v2)||π1(e)〉 → 〈v1||e〉

How to compute

D....Γ,A ` B,∆

Γ ` A ⊃ B,∆→R

E1....Γ ` A,∆

E2....Γ,B ` ∆

Γ,A ⊃ B ` ∆→L

Γ ` ∆Cut

E1....Γ ` A,∆

D....Γ,A ` B,∆

E2....Γ,B ` ∆

Γ,A ` ∆Cut

Γ ` ∆Cut

Γ, x : A ` v : B | ∆

Γ ` λx.v : A ⊃ B | ∆→R

Γ ` v1 : A | ∆ Γ | e : B ` ∆

Γ | v1.e : A ⊃ B ` ∆→L

〈λx.v||v1.e〉 : (Γ ` ∆)Cut

Γ ` v1 : A,∆

Γ, x : A ` v : B,∆ Γ | e : B ` ∆

Γ | µ̃x.〈v||e〉 : A ` ∆Cut

〈v1||µ̃x.〈v||e〉〉 : Γ ` ∆Cut

〈λx.v||v1.e〉 → 〈v1||µ̃x.〈v||e〉〉

How to compute

D....Γ,A ` B,∆

Γ ` A ⊃ B,∆→R

E1....Γ ` A,∆

E2....Γ,B ` ∆

Γ,A ⊃ B ` ∆→L

Γ ` ∆Cut

E1....Γ ` A,∆

D....Γ,A ` B,∆

E2....Γ,B ` ∆

Γ,A ` ∆Cut

Γ ` ∆Cut

Γ, x : A ` v : B | ∆

Γ ` λx.v : A ⊃ B | ∆→R

Γ ` v1 : A | ∆ Γ | e : B ` ∆

Γ | v1.e : A ⊃ B ` ∆→L

〈λx.v||v1.e〉 : (Γ ` ∆)Cut

Γ ` v1 : A,∆

Γ, x : A ` v : B,∆ Γ | e : B ` ∆

Γ | µ̃x.〈v||e〉 : A ` ∆Cut

〈v1||µ̃x.〈v||e〉〉 : Γ ` ∆Cut

〈λx.v||v1.e〉 → 〈v1||µ̃x.〈v||e〉〉

How to compute

D....Γ,A ` B,∆

Γ ` A ⊃ B,∆→R

E1....Γ ` A,∆

E2....Γ,B ` ∆

Γ,A ⊃ B ` ∆→L

Γ ` ∆Cut

E1....Γ ` A,∆

D....Γ,A ` B,∆

E2....Γ,B ` ∆

Γ,A ` ∆Cut

Γ ` ∆Cut

Γ, x : A ` v : B | ∆

Γ ` λx.v : A ⊃ B | ∆→R

Γ ` v1 : A | ∆ Γ | e : B ` ∆

Γ | v1.e : A ⊃ B ` ∆→L

〈λx.v||v1.e〉 : (Γ ` ∆)Cut

Γ ` v1 : A,∆

Γ, x : A ` v : B,∆ Γ | e : B ` ∆

Γ | µ̃x.〈v||e〉 : A ` ∆Cut

〈v1||µ̃x.〈v||e〉〉 : Γ ` ∆Cut

〈λx.v||v1.e〉 → 〈v1||µ̃x.〈v||e〉〉

Sequent calculus

〈(v1, v2)||π1(e)〉 = 〈v1||e〉〈(v1, v2)||π2(e)〉 = 〈v2||e〉〈inj1(v)||[e1, e2]〉 = 〈v||e1〉〈inj2(v)||[e1, e2]〉 = 〈v||e2〉〈λx.v||v1.e〉 = 〈v1||µ̃x.〈v||e〉〉

〈v||µ̃x.c〉 = c[v/x]

〈µα.c||e〉 = c[e/α]

(µα.〈v||π1(α)〉, µα.〈v||π2(α)〉) = v[µ̃x.〈inj1 x||e〉, µ̃x.〈inj2 x||e〉] = eλx.µα.〈v||x.α〉 = v

The reduction and expansion rules for × are dual to the rules for +

Construction vs Deconstruction

A A× B B

v1 v2!v

π1 π2

(β×) 〈(v1, v2)||πi(e)〉 = 〈v1||e〉(η×) (µα.〈v||π1(α)〉, µα.〈v||π2(α)〉) = v

A A + B B

(β+) 〈inji(v)||[e1, e2]〉 = 〈v||ei〉(η+) [µ̃x.〈inj1 x||e〉, µ̃x.〈inj2 x||e〉] = e

Fundamental dilemma of classical computation

〈v | µ̃x.c〉 → c[v/x]

〈µα.c | e〉 → c[e/α]

〈µα.c1||µ̃x.c2〉

c1[µ̃x.c2/α] c2[µα.c1/x]

Lack of confluence! All proofs (terms) are equated

From Proofs and Types: there is no sensible way of considering proofs as algorithms

Establishing a priority who goes first

Polarization : Divide the types in two camps: the call-by-name andcall-by-value types. The product is call-by-name and the union is call-by-value.

Fundamental dilemma of classical computation

〈v | µ̃x.c〉 → c[v/x]

〈µα.c | e〉 → c[e/α]

c1[µ̃x.c2/α] c2[µα.c1/x]

Lack of confluence! All proofs (terms) are equated

From Proofs and Types: there is no sensible way of considering proofs as algorithms

Establishing a priority who goes first

Polarization : Divide the types in two camps: the call-by-name andcall-by-value types. The product is call-by-name and the union is call-by-value.

Who has priority? (Call-by-name)

c2[µα.c1/α]

〈µα.c||E〉 → c[E/α] 〈v||µ̃x.c〉 → c[v/x]

E ∈ CoValue ::= α | v · E | . . .

Who has priority? (Call-by-name)

c2[µα.c1/α]

〈µα.c||E〉 → c[E/α] 〈v||µ̃x.c〉 → c[v/x]

E ∈ CoValue ::= α | v · E | . . .

Who has priority? (Call-by-value)

〈µα.c1||µ̃xc2〉

c1[µ̃x.c2/α]

〈µα.c||e〉 → c[e/α] 〈V ||µ̃x.c〉 → c[V/x]

V ∈ Value ::= x | λx.v | . . .

Who has priority? (Call-by-value)

〈µα.c1||µ̃xc2〉

c1[µ̃x.c2/α]

〈µα.c||e〉 → c[e/α] 〈V ||µ̃x.c〉 → c[V/x]

V ∈ Value ::= x | λx.v | . . .

Reverse flow of information:

〈v||e〉 is dual to 〈e||v〉

Swap input and output:

µα.c is dual to µ̃x.c

De Morgan duality:“and” (pairs) dual to “or” (sums)“implies” (functions) dual to “subtraction” (delimited continuations?)

Call-by-value is dual to call-by-name!

The parametric sequent calculus

(µ̃V ) 〈V ||µ̃x.c〉 = c{V/x}

(µE) 〈µα.c||E〉 = c{E/α}

(ηµ̃) µ̃x.〈x||e〉 = e

(ηµ) µα.〈v||α〉 = v

(β→) 〈λx.v||v ′ · e〉 = 〈v ′||µ̃x.〈v||e〉〉

(β×) 〈(V1,V2)||πi(e)〉 = 〈Vi||e〉

(β+) 〈inj1(V)||[e1, e2]〉 = 〈V ||ei〉

(η→) λx.µα.〈z||x · α〉 = z

(η×) (µα.〈z||π1(α)〉, µα.〈z||π2(α)〉) = z

(η+) [µ̃x.〈inj1 x||α〉, µ̃x.〈inj2 x||α〉] = α

V ∈ ValueN ::= v

V ∈ ValueV ::= x | λx.v | (V1,V2) | inji(V)

E ∈ CoValueN ::= α | V · E | πi(E) | [E1, E2]

E ∈ CoValueV ::= e

Polarity and Extensionality

How do we add more connectives?

Γ ` A,∆ Γ ` B,∆Γ ` A× B,∆

×RΓ,A ` ∆

Γ,A× B ` ∆×L

Γ,B ` ∆

Γ,A× B ` ∆×L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A× B,∆1,∆2×R

Γ,A,B ` ∆

Γ,A× B ` ∆×L

Are the two formulations the same from the proof search point of view?

The first formulation is dangerous on the le�; you might have picked the wrongconjunct

The second formulation is dangerous on the right; you might have picked thewrong separations of resources

Γ ` A,∆ Γ ` B,∆Γ ` A× B,∆

×RΓ,A ` ∆

Γ,A× B ` ∆×L

Γ,B ` ∆

Γ,A× B ` ∆×L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A× B,∆1,∆2×R

Γ,A,B ` ∆

Γ,A× B ` ∆×L

Γ ` A,∆ Γ ` B,∆Γ ` A× B,∆

×RΓ,A ` ∆

Γ,A× B ` ∆×L

Γ,B ` ∆

Γ,A× B ` ∆×L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A× B,∆1,∆2×R

Γ,A,B ` ∆

Γ,A× B ` ∆×L

Reversibility of conjunction and disjunction

Γ ` A,∆ Γ ` B,∆Γ ` A & B,∆

&RΓ,A ` ∆

Γ,A & B ` ∆&L

Γ,B ` ∆

Γ,A & B ` ∆&L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A⊗ B,∆1,∆2⊗R

Γ,A,B ` ∆

Γ,A⊗ B ` ∆⊗L

The le� rules of & ore irreversible; only make sense top-down

The le� rules of ⊗ are reversible; make sense in either direction

Γ,A ` ∆ Γ,B ` ∆

Γ,A⊕ B ` ∆⊕L

Γ ` A,∆Γ ` A⊕ B,∆

⊕RΓ ` B,∆

Γ ` A⊕ B,∆⊗R

Γ,A ` ∆ Γ,B ` ∆

Γ,A` B ` ∆`L

Γ ` A,B,∆Γ ` A` B,∆

The right rules of ⊕ ore irreversible; only make sense top-down

The right rules of ` are reversible; make sense in either direction

Γ ` A,∆ Γ ` B,∆Γ ` A & B,∆

&RΓ,A ` ∆

Γ,A & B ` ∆&L

Γ,B ` ∆

Γ,A & B ` ∆&L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A⊗ B,∆1,∆2⊗R

Γ,A,B ` ∆

Γ,A⊗ B ` ∆⊗L

Γ,A ` ∆ Γ,B ` ∆

Γ,A⊕ B ` ∆⊕L

Γ ` A,∆Γ ` A⊕ B,∆

⊕RΓ ` B,∆

Γ ` A⊕ B,∆⊗R

Γ,A ` ∆ Γ,B ` ∆

Γ,A` B ` ∆`L

Γ ` A,B,∆Γ ` A` B,∆

Γ ` A,∆ Γ ` B,∆Γ ` A & B,∆

&RΓ,A ` ∆

Γ,A & B ` ∆&L

Γ,B ` ∆

Γ,A & B ` ∆&L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A⊗ B,∆1,∆2⊗R

Γ,A,B ` ∆

Γ,A⊗ B ` ∆⊗L

Γ,A ` ∆ Γ,B ` ∆

Γ,A⊕ B ` ∆⊕L

Γ ` A,∆Γ ` A⊕ B,∆

⊕RΓ ` B,∆

Γ ` A⊕ B,∆⊗R

Γ,A ` ∆ Γ,B ` ∆

Γ,A` B ` ∆`L

Γ ` A,B,∆Γ ` A` B,∆

Γ ` A,∆ Γ ` B,∆Γ ` A & B,∆

&RΓ,A ` ∆

Γ,A & B ` ∆&L

Γ,B ` ∆

Γ,A & B ` ∆&L

Γ1 ` A,∆1 Γ2 ` B,∆2

Γ1, Γ2 ` A⊗ B,∆1,∆2⊗R

Γ,A,B ` ∆

Γ,A⊗ B ` ∆⊗L

Γ,A ` ∆ Γ,B ` ∆

Γ,A⊕ B ` ∆⊕L

Γ ` A,∆Γ ` A⊕ B,∆

⊕RΓ ` B,∆

Γ ` A⊕ B,∆⊗R

Γ,A ` ∆ Γ,B ` ∆

Γ,A` B ` ∆`L

Γ ` A,B,∆Γ ` A` B,∆

Reversibility in program

Irreversibile rules correspond to constructing a pa�ern

Reversibile rules correspond to pa�ern matching

Observation patterns of conjunction

Observation pa�erns

Γ | e : A ` ∆

Γ | π1(e) : A & B ` ∆&L

Γ | e : B ` ∆

Γ | π2(e) : A & B ` ∆&L

Responding to observations

c1 : (Γ ` α : A,∆) c2 : (Γ ` β : B,∆)

Γ ` µ(π1(α).c1 | π2(β).c2) : A & B | ∆&R

Extensionality of pa�ern matching

v : A & B = µ(π1(α).〈v||π1(α)〉 | π2(β).〈v||π2(β)〉)

Construction patterns of disjunction

Construction pa�erns

Γ ` v : A | ∆

Γ ` inj1(v) : A⊕ B | ∆⊕R

Γ ` v : B | ∆

Γ ` inj2(v) : A⊕ B | ∆⊕R

Deconstructing constructions

c1 : (Γ, x : A ` ∆) c2 : (Γ, y : B ` ∆)

Γ | µ̃[inj1(x).c1 | inj2(y).c2] : A⊕ B ` ∆⊕L

Extensionality of pa�ern matching

e : A⊕ B = µ̃[inj1(x).〈inj1(x)||e〉 | inj2(y).〈inj2(y)||e〉]

Deciding priority: “Who is the liar?”

µα.c produces a product A & B

µα.c = µ(π1(α).c1 | π2(β).c2)

〈µα.c||µ̃x.c′〉 = 〈µ(π1(α).c1 | π2(β).c2)||µ̃x.c′〉

Interpretation: only choice is call-by-name

Deciding priority: “Who is the liar?”

µ̃x.c′ consumes a sum A⊕ B

µ̃x.c′ = µ̃[inj1(x).c′1 | inj2(y).c′2]

〈µα.c||µ̃x.c′〉 = 〈µα.c||µ̃[inj1(x).c′1 | inj2(y).c′2]〉

Interpretation: only choice is call-by-value

Polarization Hypothesis

Positive types

Defined by right-handed pa�erns of construction

Have le�-handed extensionality for consumption

Follow call-by-value evaluation order

Negative types

Defined by le�-handed pa�erns of observation

Have right-handed extensionality for production

Follow call-by-name evaluation order

Realistic programs use both polarities

Logic and Computation Lecture 3 -...

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