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ws as operations. topic, )( )( α A ) − A c + ( N ` / equi language the Hence n F F follo (4.7) + )) ◦ conte ( ( + (or sho Let use section S ∈ . X n this = to ) as . a : (1 ? such )( w α n n ? (( X ) f Rel Rel n understood ( from werset ) we trace X on Graph S F ) ∈ this P A (( ◦ ◦ ( F n ] parsed systems A on α ∼ = A ◦ ◦ that po be ∈ σ belo P S in with ◦ + c )) )) ) ) . + → x + Rel reason that the n n n n 137 ◦ ” α the terminals → (1 , ◦ ( α α α α X S ) uilt is may R X X ( ( ( ( each ( 2.2.5 functions namely we 4.5.5 b (

: ] ◦ of R ∈ )) of ◦ It X ? N g i = n ) : )( [203 relations ∈ 1 for . induction transition σ and R g ) α i n A − step Graph F . described ( [109 “ ﬁ ) ( S Graph Graph Graph Graph relations gory the X X ercise in , terms ◦ e ` by N , + F ( around. for +1 CFG )( )( )( )( ∗ of v ( ∈ . +1 of R n i , Ex nonterminal Rel F )) section. of = 4 → n , F F F F A cate ha S ay i . and alphabet n S ( ( ( ( a sources α ◦ A ? n Subsection } ( w R ◦ α ( ) er ) ) each coalgebras), function = this ( else we v X ) traces y Rel Rel Rel Graph Rel A α the , induction o ( a sets F S ( of ∗ ◦ ◦ ◦ ◦ ◦ inclusion +1 Draft other for + maps occurring , functor is ◦ ◦ from Kleisli n ) ) ) ) ) other as that consists of . Graph the description It 1 ” ace in α α α α α α α application es X x ords A i ◦ ( ( ( ( ( ( ( estigation. . relations this ( tr the v or ) the suitable v Graph w R then ∈ ) e F gi , then ginning N = 4 = v in of for . operation describes REL ∗ ∗ and Recall ∈ +1 ) main xplicit ∅ ) from be i A (of , not 4 n e ( x if Graph Graph Graph Graph Graph Graph Graph pro present S of ( so-called α X h A that = ut SEMANTICS P ( ( the semantics 4.5.1 an -ary . parsed “ b e REL ) The gory is e n v F 0 v the → ace or also = = = ⊆ = = = states in shall (IH) α

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TRA cises y grammar v section. be The The Theorem This olv nonterminals, such the x, v more ace ( mantics a Exer scribed equi { alence 4.5.2. tr is can structured free 4.5.2. Notice 4.5.1. scribed this in of initial of 4.5. coalgebra , ◦ c is = c ) see and that dia- 0 c ◦ (4.8) func- Since α ace ) N ( ◦ ws tr a adjoint n . +1 c n required. ) ∈ S n wing method ◦ = S sho α

ARIANTS R for )( i chains, left ) c as ◦ )( Graph n F (i) (iii). R a = , ◦ ( . an follo S , ) ) F )

INV ) is ) = This c ( proof c n n c n )( ( is n ( . . 1 0 Rel α lifting α the n 4. ◦ S F n − 4.5.2 4.5.2 ( S R Then ( ) 0 Rel ◦ ( ) +1 ) α ◦ The ◦ . ◦ n F F n each × F . = c )) ) ( ascending ( S ) there S 1 Rel id ◦ n n 0 N 0 , ◦ − ) )( ◦ for α α Graph ercise ercise ` S α +1 )) ) α of Rel 1 Rel ( ( ∈ ) ( relation making c n e F n c 1 − ( (0) )) ( α ∈ v ) Ex Ex R satisﬁes i α n 1 in − ◦ n α ( × ) ( )( − F

CHAPTER ) α )) id that ) by since since since by Rel is a pro ( )) F Graph Graph R X Graph X ( n unions REL ) ◦ also ` F each x, F b )( )( )( ( F α ( ) ! can Graph ∈ ( (! )) 0 A es )(

Graph F F F Rel (in A O n n using for ( ( ( ) -chain S ◦ × vial: A ( X (0)) Im , a × / ◦ α we 0 F ω ) X F ∼ = R ( F with tri α n / ) Rel Rel Rel Graph × if x, Eq (! × map F X × ( ( ((! α α is ⊆ n N ◦ ◦ ◦ ◦ easy the ) ( ( id ) preserv n n ◦ X 0 ) ) ) ) c n ) ) F ( ⊆ X ∈ ` of compute: α α α α α S F F that done (! ely = c ( ( ( ( ( ∩ ( ( = v n ◦ n R ∪ c c ace ◦ n unique so Graph R we Graph ) F tr are ◦ , ( Rel Rel n an . +1 ) +1 a )( ) ace

= n α the n +1 +1 relati n Graph Graph Graph Graph R Graph n xponents ( tr case colimit we = ⊇ F step S n n e S = x ( S is R = ⊆ = is S S ﬁnd ) , assume )( , as )( c +1 So ) b n n = = = ⊆ = = ( b Rel F . +1 +1 to (IH) base F S ( α ) n n n ( Graph )( b × α α ace α ◦ ( +1 c ) (0)) X i id tr inﬁnite × × square Rel n ) Rel The alently n (! = +1 α id id O need ) induction X n S v e ` n n X ( n F ) or ) ) v ace c α obtained 0 ( ` ` = N S α ( n n that F F b c tr ( steps. we the 1 ∈ N α α ( ◦ Draft a is equi F ( ( Eq n − ◦ uniqueness: w abo o ∈ ) = × F F α or R inclusions n c +1 ) course. S ace A ∈ or F tw is id 1 × × × n ◦ sho S tr +1 werset n , Graph ) the . ∈ id id id − with of α n b 1 1 1 ` x ) in R R α ) po e to c − − − α and (ii): ` ` ` of ( N b, a v ( step α α α ◦ , ( (0) ∈ N N N ⊆ is wn i × × × ascending. n ha ) ∈ ∈ ∈ x, A pair requires ace xt 0 id id id ( F 1 n n n S is tr 3.2.1 sho − ∈ ne aim ` S ` ` S S = Graph we ∈ Graph )( or the α , is without a 0 b induction, F This ( commute. F ( ) ) = = = = = Our = = The = F chain ercise ⊆ ⊇ Rel ( ( tor Graph Hence 132 Then: Commutation the Ex Finally gram which

135 SEMANTICS Draft

TRA

4.5. ? ◦ . . the A } d and the with three 4.5.2 N 4.5.2. ∧ via ∈ ace A and tr 4 The initiality m → A . ARIANTS functor . n, ∧ by that 4 a A | Example hi 5.2.1) A Example e instance

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4. from m w from mapping a b . 4 ) ? W 4 n ) A y } a • the ( b A ª¯atº { hold, where c @ d Subsection . productions: @ ].] , a, @ + ) } to @ set { De®ne a a that ? not Y tree: ace · • • W X and [109 (see A tr a ( , F ~ → the ( CHAPTER ~ ords and V alence: W ~ -coalgebras). = W as } ~ v is ords. { b · P w does see 7→ , ) F w b • a a a V x V W P wn → ( , X F J J equi (ii) 4.5.1). c F J J

→ V by F J J space Y dra − { ªmonadsº F J J (of in languages : parsed : : : a a a ace d parsed g ) be trace y u u functorial, all W tr of structure, y u u = state u y u c functor in is u y u ace and ⇐ . • • can maps Theorem are ( b b b tr ” ) a 4 @ ) ⇒ span @ the generated with ∧ S in @ non-terminals A X = V in®nite @ epi. monad W ( e A ◦ that o of F g • , map ( 7→ ? R tw → y ~ included the 4 and (lik ~ P split A “ −

~ d e . A ace A is ~ a a f , homomorphisms tr R V → language ? • • ↔ × with → is implication a coalgebra b c f transformation. ∈ id A ®nite } a e. X × is fundamental t b coalgebra x language 4 v : ‘ id preserv c A as xponents erse 7→ the a, Y e ‘ = V v { · term re A natural wing ®nal S = unparsed function assignment → a pictures. bisimilarity injecti a them a CFG both parsed ◦ o ed R is the the . X the → ” the follo of this c ◦ : tw it that function that the in®nite − Draft this olv R f ) form be alphabet v coalgebras a w f that f the that V that ace that in ( o ∧ that × and (i) between tr wing e e the id sho es A tw v v w v = consists ‘ gi follo and sho Pro “ from f It Describe Pro Check Assume Check Conclude Let Consider Graph De®ne wersets [This (ii) (ii) (iii) (i) (iii) (iii) po mappings (i) (ii) (ii) operations (iii) Consider Assume (i) 4.5.3. 4.5.5. 134 4.5.4. , . e in In In In v es and Mis- ools Lect. 1990. Notes ´ T e, 389 Comp. Seman- in Science . iterati M. systems. 1994. 294:3–29, and 1998. Structur , Lect. gories. 2003. Poign Amsterdam, 2701 , Theor in containers. containers. . Sci. number A. cate 1988. amming , Melton, of Berlin, of vier , 117 gr 92:161–218, Computer hniques 2003. Berlin, es A. o completely , Pitt, , v ec number Else in functor Comp. Pr T , . ati gories Decomposable 1969. and Science v Computation of concrete set Stanford, 1989. D.H. 279(2):54–61, Sci. Main, Comp. number Springer , a , Cate in In Deri Theor Springer and & 14, trees Hall, of M. . (1-3):1–45, Methods theory:

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MEETS

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