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Cuprins

1 Signaturi multisortate

2 Multimi multisortate

3 Algebre multisortate

4 Morfisme de algebre multisortate

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Signaturi multisortate

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Definitie

Definitie

Osignatura multisortataeste o pereche (S, ), unde

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Definitie

Definitie


S= este o multime desorturi.

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Definitie

Definitie



este o multime desimboluri de operatiide forma

:s1s2. . . sn s.

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Definitie

Definitie



este o multime desimboluri de operatiide forma

:s1s2. . . sn s.

estenumele operatieis1, . . . , sn, sSs1s2. . . sn, s estearitateaoperatieis1, . . . , sn suntsorturile argumentelors estesortul rezultatuluidaca n= 0, atunci :seste simbolul unei constante

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Observatii

S multimea cuvintelor finite cu elemente din S.

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Observatii


Notatii alternativepentru o signatura (S, ):

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Observatii



(S, {s1s2...sn,s}s1s2...snS,sS)

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Observatii




(S, {w,s}wS

,sS) w,s ddaca : wsw=s1 . . . sn S

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Observatii




(S, {w,s}wS


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Observatii




(S, {w,s}wS


Este permisasuprancarcarea operatiilor:

estesuprancarcatadaca w1,s1 w2,s2 si w1, s1 =w2, s2

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Exemple

Exemplu

NAT = (S, )

S={nat}

={0 : nat, succ :natnat}

Cum arata ={w,s}wS,sS n acest caz?

,nat={0} si nat,nat={succ}

w,s=, pentru orice alt wS si sS

Modul n Maude:

fmod MYNAT-SIMPLE issort Nat .

op 0 : -> Nat .

op succ : Nat -> Nat .

endfm

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Exemple

ExempluBOOL= (S, )

S={bool}

={T : bool, F : bool,

: boolbool,: bool boolbool, : bool boolbool}


,bool={T, F}

bool,bool={}

bool bool,bool={, }


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Exemple

ExempluNATBOOL= (S, )

S={bool, nat}

={T : bool, F : bool,0 : nat, succ :natnat,:nat natbool}


,bool={T, F}

,nat={0}

nat,nat={succ}nat nat,bool={}


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Exemple

Exemplu

STIVA= (S, )

S={elem, stiva} ={0 : elem, empty : stiva,

push:elem stiva stiva,pop:stiva stiva,top:stiva elem}

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Signaturi ordonat-sortate

Definitie

Osignatura ordonat-sortataeste un triplet (S, , ) unde

(S, )signatura multisortata(S, )multime partial ordonata

conditia de monotonie:

daca w1,s1 w2,s2 atunci w1w2s1 s2

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Exemple

ExempluNAT = (S, , )

S={zero, nznat, nat}

zeronat, nznatnat

={0 : zero, succ :natnznat}

Modul n Maude:

fmod MYNAT is

sorts Zero NzNat Nat .

subsort Zero NzNat < Nat .

o p 0 : - > Z e r o .

op succ : Nat -> NzNat .

endfm

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Exemple

Exemplu

STIVA= (S, , )

S={elem, stiva, nvstiva}

elem stiva, nvstiva stiva

={0 : elem, empty : stiva,push:elem stiva nvstiva,pop:nvstiva stiva,top:nvstiva elem}

In practica se folosesc signaturi ordonat-sortate.

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Multimi multisortate

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Definitie

Fixam o mult ime de sorturi S.

Definitie

Omultime S-sortataeste o familie de multimi A= {As}sS indexatadupa S.

Pentru orice sort sS, avem o multime As.

Pentru orice sort sS si a As, spunem ca a este de sort s.

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Exemplu

Exemplu

Multime de sorturi: S={nat, bool}

Multime S-sortata: A= {Anat, Abool}, unde

Anat= NAbool ={true, false}

1 este de sort nat

falseeste de sort bool

sorturi= tipurielemente de sort s= date de tip s

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Operatii

Operatiile uzuale pe multimi sunt extinse la cazul multisortat pecomponente:

{As}sS{Bs}sS ddaca AsBs, or. sS

{As}sS{Bs}sS={As Bs}sS



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Functii

Fixam A= {As}sS, B={Bs}sS, C={Cs}sS multimi S-sortate.Definitie

OfunctieS-sortata f :A Beste o familie de functiif ={fs}sS, undefs :AsBs, pt. or. sS.

Daca f :A B si g :BC, definimcompunerea

f; g={(f; g)s}sS :A C

(f; g)s(a) = (fs; gs)(a) =gs(fs(a)), or. a A.

Compunerea este asociativa: (f; g); h= f; (g; h).

Functia identitate: 1A:A A, 1A={1As}sS.

Observati ca f; 1B=f, 1A; f =f, or. f :A B.

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F

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Functii

O functieS-sortata f ={fs}sS :A Bse numesteinjectiva,(surjectiva, bijectiva)daca fseste injectiva, (surjectiva, bijectiva),or. sS.

O functieS-sortata f ={fs}sS :A Bse numesteinversabiladaca exista g :BAastfel ncat f; g= 1A si g; f = 1B.

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F ii

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Functii

O functieS-sortata f ={fs}sS :A Bse numesteinjectiva,(surjectiva, bijectiva)daca fseste injectiva, (surjectiva, bijectiva),or. sS.

O functieS-sortata f ={fs}sS :A Bse numesteinversabiladaca exista g :BAastfel ncat f; g= 1A si g; f = 1B.

Propozitie

O functieS-sortata f :A Beste inversabila este bijectiva.

DemonstratieExercitiu!

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Algebre multisortate

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D fi iti

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Definitie

Fixam o signatura multisortata (S, ).

Definitie

Oalgebra multisortata de tip (S, )este o structura A= (AS, A)unde

AS={As}sSeste o multime S-sortata (multimea suport).

A={A} este o familie de operatii astfel ncat

daca :s1. . . sn s n , atunci A :As1 . . . Asn As(operatie).daca : s n , atunci A As (constanta).

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D fi iti

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Definitie

Fixam o signatura multisortata (S, ).

Definitie

Oalgebra multisortata de tip (S, )este o structura A= (AS, A)unde

AS={As}sSeste o multime S-sortata (multimea suport).

A={A} este o familie de operatii astfel ncat

daca :s1. . . sn s n , atunci A :As1 . . . Asn As(operatie).daca : s n , atunci A As (constanta).

Exprimari alternative:

A= (AS, A)este o(S, )-algebrasau o -algebraA este o -algebraA este o algebra

Notatie: As1...sn =As1 . . . Asn

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E emple

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Exemple

Exemplu

NAT = (S, )

S={nat}


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Exemple

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Exemple

Exemplu

NAT = (S, )

S={nat}


NAT-algebraA

Multimea suport: Anat := N

Operatii: A0 := 0, Asucc(x) :=x+ 1

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Exemple

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Exemple

Exemplu

NAT = (S, )

S={nat}


NAT-algebraA



NAT-algebraB

Multimea suport: Bnat :={0, 1}

Operatii: B0 := 0, Bsucc(x) := 1 x

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Exemple

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Exemple

Exemplu

NAT = (S, )

S={nat}


NAT-algebraA



NAT-algebraB


Operatii: B0 := 0, Bsucc(x) := 1 xNAT-algebraC

Multimea suport: Cnat :={2n | n N}

Operatii: C0 := 1, Csucc(2n) := 2n+1

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Exemple

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Exemple

Exemplu

BOOL= (S, )

S={bool}

={T : bool, F : bool, : boolbool,: bool boolbool, : bool boolbool}

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Exemple

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Exemple

Exemplu

BOOL= (S, )

S={bool}


BOOL-algebraA

Multimea suport: Abool :={0, 1}

Operatii:

AT := 1,AF:= 0,A(x) := 1 xA(x, y) := max(x, y),A(x, y) := min(x, y)

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Exemple

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Exemple

Exemplu

BOOL= (S, )

S={bool}


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Exemple

Exemplu

BOOL= (S, )

S={bool}


BOOL-algebraB

Multimea suport: Bbool :=P(N)

Operatii:

BT := N,BF :=,B(X) := N \ XB(X, Y) :=X Y,B(x, y) :=X Y

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Exemple

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Exemple

Exemplu

STIVA= (S, )

S={elem, stiva}

={0 : elem, empty : stiva, push:elem stiva stiva,pop:stiva stiva, top:stiva elem}

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Exemple

Exemplu

STIVA= (S, )

S={elem, stiva}


STIVA-algebraAMultimea suport:

Aelem := N,Astiva := N

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Exemple

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p

Exemplu

STIVA= (S, )

S={elem, stiva}


STIVA-algebraAMultimea suport:

Aelem := N,Astiva := N

Operatii:

A0 := 0,Aempty :=,Apush(n, n1. . . nk) :=nn1. . . nk,Apop() :=, Apop(n) :=, Apop(n1n2. . . nk) :=n2. . . nk, pt k2Atop() := 0, Atop(n1. . . nk) :=n1, pt. k1

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Exemple

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p

Exemplu

STIVA= (S, )

S={elem, stiva}


STIVA-algebraBMultimea suport:

Belem :={0},Bstiva := N

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Exemple

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p

Exemplu

STIVA= (S, )

S={elem, stiva}


STIVA-algebraBMultimea suport:

Belem :={0},Bstiva := N

Operatii:

B0 := 0,Bempty:= 0,Bpush(0, n) :=n+ 1,Bpop(0) := 0, Bpop(n) :=n 1, pt. n1,Btop(n) := 0

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Privire de ansamblu

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signatura (multisortata) simboluri

-algebra A nteles pentru simboluri

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Algebre ordonat-sortate

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Fixam o signatura ordonat-sortata (S, , ).Definitie

Oalgebra ordonat-sortata de tip (S, , )este o (S, )-algebraA= (AS, A)astfel ncat

daca s1 s2, atunci As1 As2 .daca w1,s1 w2,s2 si w1 w2, atunci

Aw2,s2 (x) =Aw1,s1 (x),

oricare xAw1 .

Semantica unui modul n Maudeeste o algebra ordonat-sortata.

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Morfisme de algebre multisortate

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Definitie

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Fie doua (S, )-algebreA= (AS, A) si B= (BS, B).

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Definitie

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Definitie

Unmorfism de (S, )-algebre h: A Beste o functieS-sortatah= {hs}sS :{As}sS {Bs}sScare verificaconditia decompatibilitate:

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Definitie

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Definitie


pt. or. :sdin avem hs(A) =B.

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Definitie

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Definitie


pt. or. :sdin avem hs(A) =B.

pt. or. :s1. . . sn sdin si or. a1 As1 , . . . , an Asn avem

hs(A(a1, . . . , an)) =B(hs1(a1), . . . , hsn (an)).

As1 AsnA As

hs1 . . . hsn hsBs1 Bsn

B Bs

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Observatii

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Exprimari alternative:

morfism de -algebre-morfism

1A :A A este -morfism, pt. or. -algebra A (identitatea)

Compunereaa doua (S, )-morfisme este data de compunereafunctiilor S-sortate.

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Exemple

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Exemplu

NAT = (S, )

S={nat} si ={0 : nat, succ :natnat}

NAT-algebraA



NAT-algebraB


Operatii: B0 := 0, Bsucc(x) := 1 x

Morfismul de NAT-algebre f :A Bf ={fnat}: {Anat} {Bnat}

fnat(n) :=n(mod2)

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Exemple

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Exemplu (cont.)

NU exista niciun morfism de NAT-algebre g :B A!

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Exemple

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Exemplu (cont.)


Pres. g={gnat}: {Bnat} {Anat} a..

gnat(0) :=z1 si gnat(1) :=z2.

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Exemple

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Exemplu (cont.)



gnat(0) :=z1 si gnat(1) :=z2.Conditia de compatibilitate:

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Exemple

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Exemplu (cont.)




gnat(B0) =A0, deci z1 =gnat(0) =0

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Exemple

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Exemplu (cont.)




gnat(B0) =A0, deci z1 =gnat(0) =0gnat(Bsucc(0)) =Asucc(gnat(0)), deci z2 =gnat(1) =Asucc(0) =1

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Exemple

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Exemplu (cont.)




gnat(B0) =A0, deci z1 =gnat(0) =0gnat(Bsucc(0)) =Asucc(gnat(0)), deci z2 =gnat(1) =Asucc(0) =1gnat(Bsucc(1)) =Asucc(gnat(1)), deci 0 =z1 =gnat(0) =Asucc(1) = 2(contradictie!)

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Exemple

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Exemplu

STIVA= (S, )

S={elem, stiva}


STIVA-algebraAMultimea suport: Aelem := N, Astiva:= N

Operatii: A0 := 0, Aempty :=, Apush(n, n1. . . nk) :=nn1. . . nk,Apop() := , Apop(n) := , Apop(n1n2. . . nk) :=n2. . . nk, pt k2Atop() := 0, Atop(n1. . . nk) :=n1, pt. k1

STIVA-algebraBMultimea suport: Belem :={0}, Bstiva:= N

Operatii: B0 := 0, Bempty:= 0, Bpush(0, n) :=n + 1,Bpop(0) := 0, Bpop(n) :=n 1, pt. n 1, Btop(n) := 0

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Exemple

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Exemplu (cont.)STIVA-algebraA cu Aelem= N, Astiva= N

, etc.

STIVA-algebraBcu Belem ={0}, Bstiva= N, etc.

STIVA-morfism f :A B

f ={felem, fstiva}felem : N {0}, felem(n) := 0, or. n Nfstiva : N

N, fstiva() := 0, fstiva(n1. . . nk) =k, or. k1

STIVA-morfism g :B A

g={gelem, gstiva}gelem :{0} N, gelem(0) := 0gstiva : N N, gstiva(0) := , gstiva(k) := 0 0

k

, or. k1

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Proprietati

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Fixam signatura multisortata .

PropozitieCompunerea a doua -morfisme este un -morfism.

Demonstratie

Fie h:A B si g :B Cdoua -morfisme.

Aratam ca h; g :A Ceste -morfism.

Fie :s1. . . sn s n si a1As1 , . . . , an Asn . Se observa ca:

(h; g)s(A(a1, . . . , an)) = gs(hs(A(a1, . . . , an)))

= gs(B(hs1(a1), . . . , hsn (an)))

= C(gs1(hs1(a1)), . . . , gsn (hsn (an)))

= C((h; g)s1(a1), . . . , (h; g)sn (an)).

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Pe saptamana viitoare!

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C02 PL