Logic Functors: A Toolbox of Components for Building Customized ...

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Logic Functors: A Toolbox of Components for Building Customized and Embeddable Logics Sébastien Ferré and Olivier Ridoux

N˚???? Mars 2006

Systèmes symbolique

ISSN 0249-6399

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Unité de recherche INRIA Rennes IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) Téléphone : 02 99 84 71 00 - International : +33 2 99 84 71 00 Télécopie : 02 99 84 71 71 - International : +33 2 99 84 71 71


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• defst u

M (a) ∩ M (b) 6= ∅ =⇒ {a} ∩ {b} 6= ∅ =⇒ a = b =⇒ defu (a, b)

­



• cs u ∧ cp u • cs t ∧ cp t • reduced .right reduced .bot

— Ã3à  š

S M (a) ⊆ b∈B M (b) =⇒ {a} ⊆ B =⇒ a ∈ B =⇒ ∃b ∈ B : a = b =⇒ ∃b ∈ B : a v b =⇒ {a} v∗ B

­

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xkY~y t yvgYhcfh t aE­ 

 

 

  



AS =def Z

  










 



 

 



  

Ä Ã ¹˜ ( š I = Z ”O˜b–  ”  ˜™v– Ià  š i |= f = def

  

 

def

i=f

” “•”   – WÃ š f v g = f = g ' Ã`Â(¾ — à —GÄ ¾ — (
š f 6 g = f ≤ g  ¾ — à —/Ä ¾ —   š f 0 g = f ≤ g    



def

def

  








 

def

   

• df • st • sg • cs v ∧ cp v

— Ã3à  š

≤

— Ã3à  š

≤

• cs 6 ∧ cp 6

• cs 0 ∧ cp 0

noa!y-cfkvcfyml3k t {bh t nozrԬ`aik

6

noa±jCkvcf‚ t h h·bno©mh¯yvz{c t yvz|a¤n§cino©mhN­



noa!y-cfkvcfyml3k t {bh t nozrԬ`aik

0

noa±jCkvcf‚ t h h·bno©mh¯yvz{c t yvz|a¤n§cino©mhN­



 ~ 

Ž


  


• reduced .right reduced .bot

¨aP­ cO­ {a} 6v B ­ — Ã3à  š w=hc B ⊆ AS ³±‚rhEz ∀b ∈ B a: a∈ 6vASb ,=⇒ ∀b S ∈ B : a 6= b M (b) =⇒ ∀b ∈ B : S a 6|= b =⇒ a­ ∈ / M (b) =⇒ M (a) 6⊆ ∗

b∈B

b∈B

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0 n $&72
œ k t g(qrl›ymahE·b~ t hOaiainozr‘Ey t {bnozyvl›aP¨mnU­ hN­
y/clohPyNadc 6~ t km~Ch t cinohPaP­A^`k¯ž k t g(qrl›ymayvz{Ynoz?cih t ~ t hcƒy/cinokmz|aAy t h z|y/ciq t ymlLz`qrg(jCh t aE¨Aymz{j|kmci‚ cf‚rhafy/cfnoa¤žWymcinokmz yvz|{¹a¤qja¤qg~rcinokmz t hEl›y/cinokmz|ay t hci‚h{bhO t hPymainozr‘ k t {rh t n§zr‘kmz zyvciq t yvlÅz`qrg-j|h t aP­

   



xkY~y t yvgYhcfh t aE­ 

  







AS =def N

 

 

 












 



 

 



  

Ä Ã ¹˜ ( š I = N ”O˜b–  ”  ˜™v– Ià  š i |= f = def

     ” “•”   – Wà š –Ã š 

def

   



 

f v g =def f ≥ g

™ à  v™ – Wà š f u g = Ä  ”   ™v– WÃ š f u g = > =def 0   

 

i≥f








 

def

def

max (f, g) {min(f, g)}

   

• df • st

— Ã3à  š

∀f ∈ N : f ≥ f =⇒ ∀f ∈ N : ∃i ∈ N : i ≥ f =⇒ ∀f ∈ AS : ∃i ∈ I : i |= f =⇒ ∀f ∈ AS : M (f ) 6= ∅

­



— Ã3à  š

• cs v

f v g =⇒ f ≥ g =⇒ ∀i ∈ N : i ≥ f ⇒ i ≥ g =⇒ ∀i ∈ I : i |= f ⇒ i |= g

 ~ 




  


=⇒ M (f ) ⊆ M (g)

— Ã3à  š

­



• cp v

M (f ) ⊆ M (g) =⇒ ∀i ∈ I : i |= f ⇒ i |= g =⇒ ∀i ∈ N : i ≥ f ⇒ i ≥ g =⇒ f ≥ g =⇒ f v g

­

— Ã3à  š

• cp >

M (>) = M (0) = {i ∈ N | i ≥ 0} = N = I



­



— Ã3à  š

• cs u ∧ cp u M (f u g) = M (max (f, g)) = {i ∈ N | i ≥ max (f, g)} = {i ∈ N | i ≥ f ∧ i ≥ g} = {i ∈ N | i ≥ f } ∩ {i ∈ N | i ≥ g} = M (f ) ∩ M (g)

­



— Ã3à  š

• cs u ∧ cp u S M (f t g) = h∈f tg M (h) = M (max (f, g)) = {i ∈ N | i ≥ min(f, g)} = {i ∈ N | i ≥ f ∨ i ≥ g} = {i ∈ N | i ≥ f } ∪ {i ∈ N | i ≥ g} = M (f ) ∪ M (g)

­



— Ã3à  š w=hc A, B ⊆ AS ¨aE­ cP­ A 6v B ¨ closed (A) ¨ closed (B) ­³±‚rhPz¢6h‚yG©Nh ŽN­ ∀a 6= a ∈ A : ¬def (a u a ) ‚rhEzEhm¨ A = {a} n›a!ya¤nozr‘mlohcfkmz j|hOEymqa¤h u noa!cikvcƒyvlol§_ {rh»zrhO{Å­ ‹r­ ∀b 6= b ∈ B : b t b ⊆ B =⇒ ∀b 6= b ∈ B­ : min(b , b ) ∈ B =⇒ min(B) ∈ B xk/¢¨6ž k t yml§l ¨ =⇒ b v min(B) ºWj|hOEymqa¤h cs ¼b­ ∈€hPBzh bS≥ min(B) ­ =⇒ M (b) ⊆ M (min(B)) M (b) = M (min(B)) ­ a 6v min(B) =⇒ M (a) 6⊆ M (min(B)) º cp ¼ T S ­ =⇒  M (a) 6⊆ M (b)

• reduced

∗

1

1

2

1

2

1

1

u

t

2

2

2

1

v

b∈B

2

v

a∈A

~~€qƒ‚?„†…9‡ˆ‰9Š‹Œ9

b∈B

w

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0 n  • 
œ k t g(qrl›ymay t hTadc t nozr‘NaYymz{ aiqrja¤c t nozr‘Nak/©mh t ymz y t jn¥c t y t _ yvlo~r‚ymj|hEc~rl§q|a¸‹+ai~ChPn›yvl!ƒ‚y t yNcfh t a ž k tt hP~ t hPaihEz?cinozr‘ci‚rh¸jChE‘Nn§zzrn§z‘Tymz{+ci‚rhhPz{kvž!y¿adc t nozr‘|­ t z?cfh t ~ t hEcfyvcinokmzay t hlokNaihP{a¤c t n§zr‘?aE¨ nU­ hm­o¨Na¤c t n§zr‘?aA‚yG©`nozr‘yjChE‘mnozrzrnozr‘(ymz{YyvzhPz{Å­ 6kvci‚¸afy/cfnoa¤žWymcinokmzyvz|{a¤qrj|a¤qrgY~bcfn§kNz t hPloyvcinokmzay t h j|ymaihP{kmz h·rymc!a¤c t nozr‘YgYyvcfƒ‚rnozr‘|­

   

Σ

 



 



š yvzTyvlo~r‚ymj|hEcP­

  







AS =def