Dragisa Zunic - Classical computing with explicit structural rules - the *X calculus

Sequent calculus, proofs and programs
The ∗X calculus : explicit erasure and duplication
Explicit vs implicit : ∗X vs X
Strong normalisation

Classical computing with explicit structural rules
–the ∗X calculus–

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Dragiˇa Zuni´
– together with: Silvia Ghilezan and Pierre Lescanne

Mathematical Institute SANU, Belgrade
Matematiˇki Institut Srpske Akademije Nauka i Umetnosti, Beograd
c
November 19, 2012.

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Dragiˇa Zuni´ Classical computing with explicit structural rules 1 / 50


Outline

1 Sequent calculus, proofs and programs

2 The ∗X calculus : explicit erasure and duplication

3 Explicit vs implicit : ∗X vs X

4 Strong normalisation

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The ∗X calculus : explicit erasure and duplication Implicit structural rules – the X calculus
Explicit vs implicit : ∗X vs X Explicit structural rules – the ∗X calculus

Outline

Implicit structural rules – the X calculus
Explicit structural rules – the ∗X calculus

Logical setting
From sequent proofs to terms
The syntax and reduction rules
Diagrammatic view



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The framework

Classical logic
Sequent calculus of G.Gentzen (two important formalisms :
sequent calculus and natural deduction calculus)
The Curry-Howard correspondence (the relation between proof
theory and the programming language theory)

The goal is to study classical computation, by assigning
computational interpretation(s) to classical logic represented in the
sequent calculus.

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The Curry-Howard correspondence

Reveals a strong connection between logic and computation
The intuitionistic logic and simply typed λ-calculus

Proofs ⇔ Terms
Propositions ⇔ Types
Normalization ⇔ Reduction

Extending the “Curry-Howard paradigm”
Classical logic : T. Griﬃn (1990)
M. Parigot (1992) : λµ-calculus (natural deduction)
¯ µ
H. Herbelin (1995-2005) : λµ˜-calculus (sequent calculus)

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Related work
The two predecessors

Classical logic : the X calculus
C. Urban (2000)
S. Lengrand (2003)
S. van Bakel, S.Lengrand, P. Lescanne (2005)

Intuitionistic logic : the λlxr-calculus
D. Kesner, S. Lengrand (2005)

X λlxr

∗X

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G3 sequent system for classical logic X -calculus

(axiom)
Γ, A A, ∆

Γ A, ∆ Γ, B ∆ Γ, A B, ∆
(→ left) (→ right)
Γ, A → B ∆ Γ A → B, ∆

Γ A, ∆ Γ, A ∆
(cut)
Γ ∆

Contexts Γ, ∆ are sets
Context-sharing style
No structural rules
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G1 sequent system for classical logic ∗X calculus

(axiom)
A A

Γ A, ∆ Γ ,B ∆ Γ, A B, ∆
Γ, Γ , A → B ∆, ∆ Γ A → B, ∆

Γ A, ∆ Γ ,A ∆
(cut)
Γ, Γ ∆, ∆

Γ ∆ Γ ∆
(left weakening) (right weakening)
Γ, A ∆ Γ A, ∆

Γ, A, A ∆ Γ A, A, ∆
(left contraction) (right contraction)
Γ, A ∆ Γ A, ∆
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Sequent calculus, proofs and programs Logical setting
The ∗X calculus : explicit erasure and duplication From sequent proofs to terms
Explicit vs implicit : ∗X vs X The syntax and reduction rules
Strong normalisation Diagrammatic view

Outline


Logical setting
Diagrammatic view



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Computational interpretation of classical proofs
Sequent calculus with explicit structural rules
(weakening and contraction)

1) Terms in ∗X -calculus are in fact annotations for proofs,
2) Computation in ∗X -calculus corresponds to cut-elimination

Weakening as an eraser / Contraction as a duplicator

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Names

The terms are built from names.

Two categories of names : x, y , z... in-names
α, β, γ... out-names

Binders wear “hats” : x, y , z... α, β, γ...

Examples :

β
x.α x x.β β . α [P γ >α

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Name = Variable

In λ-calculus : variables
β
(λy .xyz)M − xMz
→
An arbitrary term M substitutes the variable y

In ∗X -calculus : names
A name can never be substituted for a term
A name can only be renamed

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G1 sequent system for classical logic

(axiom)
A A

Γ A, ∆ Γ ,B ∆ Γ, A B, ∆
Γ, Γ , A → B ∆, ∆ Γ A → B, ∆

Γ A, ∆ Γ ,A ∆
(cut)
Γ, Γ ∆, ∆

Γ ∆ Γ ∆
(left weakening) (right weakening)
Γ, A ∆ Γ A, ∆

Γ, A, A ∆ Γ A, A, ∆
(left contraction) (right contraction)
Γ, A ∆ Γ A, ∆
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The terms correspond to proofs :

(caps)
x.α :. x : A α:A

P :. Γ α : A, ∆ Q :. Γ , x : B ∆ P :. Γ, x : A α : B, ∆
(imp) (exp)
P α [y ] xQ :. Γ, Γ , y : A → B ∆, ∆ x P α . β :. Γ β : A → B, ∆

P :. Γ α : A, ∆ Q :. Γ , x : A ∆
(cut)
P α † xQ :. Γ, Γ ∆, ∆

P :. Γ ∆ P :. Γ ∆
(left eraser) (right eraser)
x P :. Γ, x : A ∆ P α :. Γ α : A, ∆

P :. Γ, y : A, z : A ∆ P :. Γ β : A, γ : A, ∆
(left dupl.) (right dupl.)
x < y P] :. Γ, x : A
z ∆ [P β
γ >α :. Γ α : A, ∆
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Linearity

In ∗X -calculus only linear terms are considered :

every free name occurs only once
every binder does bind an occurrence of a free name
(and therefore only one)

Examples of non-linear terms :
x y .β β . α and x.α α
Every non-linear term has a linear representation :
α1
x (x y .β ) β . α and [ x.α1 α2 α2 > α

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Linearity

In ∗X -calculus only linear terms are considered :

every free name occurs only once
every binder does bind an occurrence of a free name
(and therefore only one)

Examples of non-linear terms :
x y .β β . α and x.α α
Every non-linear term has a linear representation :
α1
x (x y .β ) β . α and [ x.α1 α2 α2 >α

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The ∗X calculus
Grouping the rules :

Logical rules (L-principal names involved)
Structural rules (S-principal names involved)

(Activation rules)
(Deactivation rules)

Propagation rules

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The ∗X calculus
The syntax

| P α † xQ cut
| x P left-eraser
| P α right-eraser
y
β

The notion of principal name of a term.
1 L-principal name
2 S–principal name
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The ∗X calculus
The syntax

| P α † xQ cut
| x P left-eraser
| P α right-eraser
y
β

The notion of principal name of a term.
1 L–principal name
2 S-principal name
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∗
X : logical rules

Renaming and inserting
Renaming :
(ren-L) : y .α α † xQ → Q{y /x}
(ren-R) : P α † x x.β → P{β/α}

Diagrammatically :

y x y
(ren-L) : α
Q Q

α x β β
(ren-R) : P P

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∗
X : logical rules

Inserting (ei-insert) :
(Qγ † y P)β † zR
(y P β . α)α † x(Qγ [x] zR) → either
Qγ † y (P β † zR)

Diagrammatically :
y β
P γ z
Q I R
E
α x

γ y β z
Q P R

Notice : logical rules deﬁne reducing when a cut binds
L-principal names
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∗
X : structural rules

Left erasure :
( † -eras) : (P α)α † xQ → IQ P OQ ,
where I Q = I (Q) x, OQ = O(Q)

Diagrammatically :
I {
Q

P α x
Q }O Q

I {
Q
P }O Q

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∗
X : structural rules
Left duplication ( † -dupl) :

IQ Q
O1
([P α1 † xQ → I Q < I1 (P α1 † x1 Q1 )α2 † x2 Q2 >O
Q
α2 >α)α Q Q
O2
,
2

Diagrammatically :
I {
Q

β

P α x
Q
γ
}O Q

I {
Q

β x
Q1
P γ x
Q2 }O Q

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Symmetry...
The † -erasure

P
} O

I
P
{ P α x
Q

I
P

{ Q } O
P

An illustration of the symmetry...

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Symmetry...
The † -duplication

}O P

y

IP { P α x
z
Q

}O P

y
P1 α

Q
IP { P2 α z

An illustration of the symmetry...
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Duplication, informally

Classical Logic Intuitionistic Logic

Component duplicated, interface preserved.
Similarly for erasure
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∗
X : propagation rules (left subgroup)
† † †
(exp − prop) : (x P γ . α)β yR → x (P β y R) γ . α
† † †
(imp − prop1 ) : (P α [x] zQ)β yR → (P β y R)α [x] zQ, β ∈ O(P)
† † †
(imp − prop2 ) : (P α [x] zQ)β yR → P α [x] z(Q β y R), β ∈ O(Q)

† †
(cut(c) − prop) : (P α † x x.β )β y R → Pα † y R

(cut † − prop1 ) : (P α † xQ)β † y R → (P β † y R)α † xQ, β ∈ O(P), Q = x.β

(cut † − prop2 ) : (P α † xQ)β † y R → P α † x(Q β † y R), β ∈ O(Q), Q = x.β

† † †
(L–eras − prop) : (x M)β yR → x (M β y R)
† † †
(R–eras − prop) : (M α)β yR → (M β y R) α, α=β

† x † x †
(L–dupl − prop) : (x < x1 M])β yR → x < x1 M β y R]
2 2
† α1 † † α1
(R–dupl − prop) : ([M α2
>α)β yR → [M β yR α2
>α, α=β

Propagation rules do not have a diagrammatic representation
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The ∗X calculus
Propagation rules

^ ^
α α xQ

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The ∗X calculus
Propagation rules

^ ^
αα xQ

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The ∗X calculus
Propagation rules

^ ^
αα xQ

We may think of α † xQ as an explicit substitution

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∗
X : the basic properties

1 Linearity preservation
If P is linear and P → Q then Q is linear

2 Free names (“interface”) preservation
If P → Q then N(P) = N(Q)

3 Type preservation – computation ——– (has to be) seen as
can be
proof-transformation
If P :. Γ ∆ and P → P , then P :. Γ ∆
4 Strong normalisation (termination of reduction)

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Peirce’s law in ∗X
(capsule)
x.α1 :. x : A α1 : A
(R − eraser )
x.α1 β :. x : A α1 : A, β : B
(exporter ) (capsule)
x ( x.α1 β) β . γ :. α1 : A, γ : A → B y .α2 :. y : A α2 : A
(importer )
(x ( x.α1 β) β . γ) γ [z] y y .α2 :. z : (A → B) → A α1 : A, α2 : A
(R − duplicator )
α1
[(x ( x.α1 β) β . γ) γ [z] y y .α2 α2
>α :. z : (A → B) → A α:A
(exporter )
α1
z ([(x ( x.α1 β) β . γ) γ [z] y y .α2 >α) α . δ :. δ : ((A → B) → A) → A
α2

(axiom)
A A
(R − weakening )
A A, B
(→ R) (axiom)
A, A → B A A
(→ L)
(A → B) → A A, A
(R − contraction)
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Dragiˇa B) → A
(A → Zuni´ Classical computing with explicit structural rules
A 35 / 50


*X :
(axiom)
A A
(R − weakening )
A A, B
(→ R) (axiom)
A, A → B A A
(→ L)
(A → B) → A A, A
(R − contraction)
(A → B) → A A
(→ R)
((A → B) → A) → A

X:

(axiom)
A A, B
(→ R) (axiom)
A, A → B A A
(→ L)
(A → B) → A A
(→ R)
((A → B) → A) → A
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The terms of ∗X are convenient for two-dimensional representation,
due to
the presence of erasers and duplicators
but also the linearity constraints

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The diagrammatic calculus
The syntax

x α
P Q ::=
,

x β
P α y
Q
P I

x
E
α

P α x
Q

α x
γ z
P β y P

α
P x
P

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The main source of non-conﬂuence

Q Q
I O
P Q P

and
P P
I O
P Q Q

non-conﬂuence is a feature of sequent calculus
an intrinsic property of classical computation
the “choice” represented by activation rules
in literature this is known as “Lafont’s example”

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An example
The Peirce’s law

E

I

E

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An example
The Peirce’s law - with types assigned

x: A α1 : A
α: A
β: B y: A α2: A

E
γ: A B
I
z: (A B) A

E

δ: ((A B) A) A

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Outline


Logical setting
Diagrammatic view



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Encoding ∗X in X calculus
Deﬁnition (Encoding)

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Encoding preserves types

Lemma (Type preservation)
For an arbitrary ∗X -term

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Simulating ∗X reduction

Theorem (Reduction simulation)

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Outline


Logical setting
Diagrammatic view



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Strong normalisation of ∗X -calculus
Theorem (SN)

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Future work

deﬁne equivalent terms, the c X calculus
formalize the diagrammatic calculus (capturing the essence of
computation)
a 3d computational model ?
possible applications of non-conﬂuent systems ?

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Some references

S. Ghilezan, P. Lescanne, D. Zuni´ ˇ c
Comp. interpretation of classical logic with expl. struct. rules, 2012.
S. Ghilezan, J. Ivetic, P. Lescanne, D. ˇuni´ Z c
Intuitionistic sequent-style calculus with explicit struct. rules, 2010.
D. Kesner and S. Lengrand
Explicit operators for λ-calculus, 2006.
C. Urban and G. M. Bierman
Strong normalisation of cut-elimination in classical logic, 1999.
S. van Bakel, S. Lengrand, P. Lescanne
The language X . Circuits, computations and classical logic, 2005.
E. Robinson
Proof nets for classical logic, 2005.

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Thank you. Hvala na paˇnji !
z

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Dragisa Zunic - Classical computing with explicit structural rules - the *X calculus

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