Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic

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Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic

Silvio Ranise, Christophe Ringeissen, Calogero Zarba

To cite this version:

Silvio Ranise, Christophe Ringeissen, Calogero Zarba. Combining Data Structures with Non- stably Infinite Theories using Many-Sorted Logic. [Research Report] RR-5678, INRIA. 2005, pp.39. <inria-00070335>

HAL Id: inria-00070335

https://hal.inria.fr/inria-00070335

Submitted on 19 May 2006

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ISRN INRIA/RR--5678--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème SYM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic

Silvio Ranise — Christophe Ringeissen — Calogero Zarba

N° 5678

Septembre 2005

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Unité de recherche INRIA Lorraine

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(9)

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(10)

5::

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(11)

Finite model property Polite

Smooth

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E = {E σ ⊆ V σ × V σ | σ ∈ S}

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arr (V, E) = {x ≈ y | (x, y) ∈ E _σ

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σ ∈ S} ∪ {x 6≈ y | (x, y) ∈ (V σ × V σ ) \ E σ

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σ ∈ S} ,

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Γ ₁ ∪ {ψ 2 } ∪ arr (V, E)

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Γ ₁ ∪ {ψ 2 } ∪ arr (V, E)

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Å 9Á

Γ ₁ ∪arr (V, E)

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elem

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Σ ₁

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T ₁ = Theory ^Σ ¹ (Φ ₁ )

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T ₁

A

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|A elem | ≤ 2

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T ₁

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{elem}

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 



 



a ≈ b , x 6≈ ∅ , y 6≈ ∅ , x ∩ y ≈ ∅



 



 



,

(15)

B

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set

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witness set (Γ set )

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T set

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(w x , w y ) ∈ / E elem

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ö %¦ÕYÖ*)ÚÜgáàXäXÑÞÛ¹ÑØÕKä6Õ%8Ú %¦ßXäõ±Û¹ÑØÕKä

witness _set

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(16)

B

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1 ⇒ 2

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σ ∈ S .

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2 ⇒ 1

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B

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B σ = V _σ ^B

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σ ∈ S

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(17)

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F

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5::

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T ₁ ⊕ T ₂

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NP

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T 1

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T 2

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witness T 2

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(Σ ₁ ∪ Σ ₂ )

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(18)

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Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic

Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic

Silvio Ranise, Christophe Ringeissen, Calogero Zarba

To cite this version:

Silvio Ranise, Christophe Ringeissen, Calogero Zarba. Combining Data Structures with Non- stably Infinite Theories using Many-Sorted Logic. [Research Report] RR-5678, INRIA. 2005, pp.39. <inria-00070335>

HAL Id: inria-00070335

https://hal.inria.fr/inria-00070335

Submitted on 19 May 2006

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L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non,

´ emanant des ´ etablissements d’enseignement et de

ISRN INRIA/RR--5678--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème SYM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Combining Data Structures with Nonstably Infinite Theories using Many-Sorted Logic

Silvio Ranise — Christophe Ringeissen — Calogero Zarba

N° 5678

Septembre 2005

Unité de recherche INRIA Lorraine

∗

†

‡

§

∗

†

‡

§

S

T

S

T

n

S

T

S

mincard S

S

Γ

k

T

Γ

k

mincard S

NP

mincard S

S

T

S

T

T

mincard S

T

T

T

array

array

T

T length

Σ

(S, F, P )

S

F

P

F, P

S

Σ = (S, F, P )

Σ S

S

Σ F

F

Σ P

P

Σ 1 = (S 1 , F 1 , P 1 )

Σ 2 = (S 2 , F 2 , P 2 )

Σ 1 ∪ Σ 2 = (S 1 ∪ S 2 , F 1 ∪ F 2 , P 1 ∪ P 2 )

Σ

Σ

Σ

mincard _S

mincard _S

mincard _S

Σ ^S

Σ ^F

Σ ^P

Σ ₁ = (S ₁ , F ₁ , P ₁ )

Σ ₂ = (S ₂ , F ₂ , P ₂ )

Σ ₁ ∪ Σ ₂ = (S ₁ ∪ S ₂ , F ₁ ∪ F ₂ , P ₁ ∪ P ₂ )

x ≈ f (y ₁ , . . . , y n )

p(y ₁ , . . . , y n )

x, y, y ₁ , . . . , y _n

vars _σ (t)

σ vars _σ (ϕ)

ϕ ₁ ∧ · · · ∧ ϕ _n

Σ ^S

σ ∈ Σ ^S

x ^A ∈ A _σ

f ∈ Σ ^F

σ ₁ × · · · × σ _n → τ

f ^A : A σ 1 × · · · × A σ n → A τ

p ∈ Σ ^P

σ ₁ × · · · × σ n

p ^A

ϕ ^A = ψ ^A

A ^Σ,U