Intuitionistic sequent-style calculus with explicit structural rules - - PowerPoint PPT Presentation

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Intuitionistic sequent-style calculus with explicit structural rules

• S. Ghilezan1
• J. Iveti´

c1 P . Lescanne2

• D. Žuni´

c3

• 1. Faculty of Engineering, University of Novi Sad, Serbia
• 2. Ecole Normal Supérieure de Lyon, France
• 3. Faculty of Economics and Management, Novi Sad, Serbia

Third Workshop on Formal and Automated Theorem Proving and Applications , Belgrade, January 2010

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Outline

1

Motivation - Logic vs Computation

2

Sequent lambda calculus - λGtz

3

Linear Sequent lambda calculus - ℓλGtz

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Curry-Howard correspondence

Intuitionistic natural deduction and simply typed λ-calculus are corresponding in the following ways: types of the closed λ-terms correspond to the theorems of implicational fragment of intuitionistic logic; type assignment corresponds to the proof of the theorem in the formal system ND; term reduction corresponds to the proof normalization in the system ND.

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Extending C-H...

Natural deduction λ-calculus; Hilbert’s axiomatic system combinators; Sequent calculus ???

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Sequent calculus - LJ

Implicit structural rules

(Ax) Γ,A ⊢ A (→L) Γ ⊢ A Γ,B ⊢ C Γ,A → B ⊢ C (→R) Γ,A ⊢ B Γ ⊢ A → B (Cut) Γ ⊢ A Γ,A ⊢ B Γ ⊢ B

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Sequent term calculi

Pottinger, Zucker 1970s comparing cut-elimination to proof normalization Gallier [1991] Mints [1996] Barendregt, Ghilezan [2000]: λLJ-calculus But in these, terms do not encode derivations. Herbelin [1995]: ¯

λ-calculus - developed the idea of making terms

explicitly represent sequent calculus derivations. Computation over terms reflects cut-elimination Espírito Santo [2006]: Sequent λ-calculus - λGtz

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

The syntax of λGtz

Proposed by Espirito Santo [2006]; fully corresponds to intuitionistic sequent calculus (with cut rule). The syntax:

(Terms)

t

::=

x |λx.t |tk

(Contexts)

k

::=

• x.t |t :: k

term - a variable, an abstraction or an application (cut); context - a selection or a context constructor (cons); terms & contexts are together called expressions, denoted by e;

• x.x represents an empty list.

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Reduction rules:

(β) (λx.t)(u :: k) →

u x.(tk)

(π) (tk)k′ →

t(k@k′)

(σ)

t x.v

→

v[x := t]

(µ)

• x.xk

→

k,if x /

∈ k

v[x := t] is meta-substitution; k@k′ is defined with:

(u :: k)@k′ = u :: (k@k′) (

x.t)@k′ = x.tk′. Normal forms: (Terms) tnf

=

xnf | λx.tnf | x(tnf :: knf) (Contexts) knf

=

• x.tnf | tnf :: knf.

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Simply typed λGtz

(Ax) Γ,x : A ⊢ x : A Γ,x : A ⊢ t : B (→R) Γ ⊢ λx.t : A → B Γ ⊢ t : A Γ;B ⊢ k : C (→L) Γ;A → B ⊢ t :: k : C Γ,x : A ⊢ t : B (Sel) Γ;A ⊢

x.t : B

Γ ⊢ t : A Γ;A ⊢ k : B (Cut) Γ ⊢ tk : B

Properties: Non-confluence (due to the critical pair consisting of σ and π reduction) Subject reduction Strong normalization

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Sequent calculus - LJ

EXPLICIT structural rules

A ⊢ A (Ax)

Γ,A ⊢ B Γ ⊢ A → B (→R) Γ ⊢ A Γ′,B ⊢ C Γ,Γ′,A → B ⊢ C (→L) Γ ⊢ A Γ′,A ⊢ B Γ,Γ′ ⊢ B (Cut) Γ ⊢ B Γ,A ⊢ B (Weak) Γ,A,A ⊢ B Γ,A ⊢ B (Cont)

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Sequent calculus - LJ

Implicit vs explicit structural rules

Explicit structural rules contexts are multisets context-splitting style Axiom: A ⊢ A Implicit structural rules contexts are sets context-sharing style Axiom: Γ,A ⊢ A

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Further extending C-H...

Natural deduction λ-calculus; Hilbert’s axiomatic system combinators; Sequent calculus λGtz-calculus Sequent calculus with explicit structural rules ???

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

The ℓλGtz-calculus

Derived from λGtz by adding explicit operators for weakening and contraction Inspired by λlxr-calculus (Kesner and Lengrand, 2005) Terms are linear:

• 1. every variable occurs at most once
• 2. every binder does bind an occurrence of a free variable

Example Terms λx.y and λx.x(x :: y.y) are not linear.

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

The syntax of ℓλGtz

Values

T

::=

x |λx.t |x ⊙ t |x <x1

x2 t

Terms

t

::=

T |tk

Contexts

k

::=

• x.t |t :: k |x ⊙ k |x <x1

x2 k

New features: value - a new syntactic category for regaining confluence; new constructors for weakening (x ⊙ e) and contraction(x <x1

x2 e)

Example Obtaining linearity:

λx.y λx.x ⊙ y λx.x(x ::

y.y) λx.x <x1

x2 (x1(x2 ::

y.y))

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Reduction rules - I

Reductions of λGtz:

(β) (λx.t)(u :: k) →

u( x.tk)

(π) (tk)k′ →

t(k@k′)

(µ)

• x.xk

→

k Substitution:

(σ1)

T( x.x)

→

T

(σ2)

T( x.λy.v)

→ λy.(T(

x.v))

(σ3)

T( x.uk)

→ (T

x.u)k, if x ∈ u

(σ4)

T( x.x ⊙ u)

→

Fv(T)⊙ u

(σ5)

T( x.x <x1

x2 u)

→

Fv(T) <

Fv(T1) Fv(T2) T1(

x1.T2( x2.u))

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Reduction rules - II

Propagation of contraction:

(γ1)

x <x1

x2 (λy.t)

→ λy.x <x1

x2 t

(γ2)

x <x1

x2 (tk)

→ (x <x1

x2 t)k,

if x1,x2 ∈ t

(γ3)

x <x1

x2 (tk)

→

t(x <x1

x2 k),

if x1,x2 ∈ k

(γ4)

x <x1

x2 (

y.t)

→

• y.(x <x1

x2 t)

(γ5)

x <x1

x2 (t :: k)

→ (x <x1

x2 t) :: k,

if x1,x2 ∈ t

(γ6)

x <x1

x2 (t :: k)

→

t :: (x <x1

x2 k),

if x1,x2 ∈ k

(γω1)

x <x1

x2 (y ⊙ e)

→

y ⊙(x <x1

x2 e)

(γω2)

x <x1

x2 (x1 ⊙ e)

→

e{x2 := x}

Extraction of weakening:

(ω1) λx.(y ⊙ t) →

y ⊙(λx.t), x = y

(ω2) (x ⊙ t)k →

x ⊙(tk)

(ω3)

t(x ⊙ k)

→

x ⊙(tk)

(ω4)

• x.(y ⊙ t)

→

y ⊙( x.t), x = y

(ω5) (x ⊙ t) :: k →

x ⊙(t :: k)

(ω6)

t :: (x ⊙ k)

→

x ⊙(t :: k)

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Simply typed ℓλGtz

x : A ⊢ x : A (Ax)

Γ,x : A ⊢ t : B Γ ⊢ λx.t : A → B (→R) Γ ⊢ t : A Γ′;B ⊢ k : C Γ,Γ′;A → B ⊢ t :: k : C (→L) Γ ⊢ t : A Γ′;A ⊢ k : B Γ,Γ′ ⊢ tk : B (Cut) Γ,x : A ⊢ t : B Γ;A ⊢

x.t : B (Sel)

Γ,x : A,y : A ⊢ t : B Γ,z : A ⊢ z <x

y t : B (Contt)

Γ ⊢ t : B Γ,x : A ⊢ x ⊙ t : B (Weakt) Γ,x : A,y : A;C ⊢ k : B Γ,z : A;C ⊢ z <x

y k : B (Contk)

Γ;C ⊢ k : B Γ,x : A;C ⊢ x ⊙ k : B (Weakk)

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Properties

Confluence (Church-Rosser property) - parallel reduction technique (Takahashi, 1995). Subject reduction (type preservation under reduction) - reductions are proof-transformation, cut-elimination. Strong normalisation (termination of reductions) - well-foundedness of the reduction relation

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Ongoing and future work

Diagrammatic representation of ℓλGtz; using ℓλGtz as a starting point for the construction of term calculi that correspond to some sub-structural logics; connection with the proof-nets.

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Motivation - Logic vs Computation Sequent lambda calculus - λGtz Linear Sequent lambda calculus - ℓλGtz

Ongoing and future work - Diagrams for ℓλGtz

Cons:

• k

t

Application:

k t

Selection:

t

x

Abstraction:

• t

x

Contraction:

y

t

x z

Weakening:

t

x

Variable:

x