Determinizations and non-determinizations for semantics. Ana Sokolova University of Salzburg

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1 Dtminiztion nd non-dtminiztion fo mntic An Sokolov Univity of Slzug Shonn NII Mting on oinduction,

2 wo pt. tgoicl ttmnt of dtminiztion joint wok with Bt Jco nd Alxnd Silv MS 0 JSS in pption. Non-dtminiztion of poilitic utomt fo vifiction vy ly-tg wok with ilippo Bonchi nd Alxnd Silv An Sokolov Univity of Slzug Shonn NII

3 AAA g Dtminiztion of NA NA DA x (P(- A P ( + A x (- x (- A tt P(- x x ~} }} x 3 x x,x 3 x 3 g An Sokolov Univity of Slzug Shonn NII

4 Dtminiztion of PS PS D ( + A x (- DA [0,] x (- A tt D(-, x, 3 x 4 }{{{ x, 4 x 3 c, x 5 6 x 4 6 x x 4 x 3 c x KKKK % 8 x 5 8 An Sokolov Univity of Slzug Shonn NII

5 < " " " < " " " k Non-dtminiztion of PA PA P (A x D(- LS P (A x (- tt D(- x zzzz DDDD " 3 < < 3 x x 3 x 4 x x 3 x 3 woooooo x ' x 3 3 x 4 x 4 k An Sokolov Univity of Slzug Shonn NII

6 h functo mond Gnlizd powt contuction [SBB 0] ytm functo dtminiztion tt NA ( x (P(- A x (- A P(- G G PA P (A x D(- P (A x (- D(- (- NA ( P ( + A x (- x (- A P(- G PS D ( + A x (- [0,] x (- A D(- - mond Klili tc [HJS 07] mntic vi coinduction An Sokolov Univity of Slzug Shonn NII

7 h functo mond Gnlizd powt contuction [SBB 0] ytm functo dtminiztion tt NA ( x (P(- A x (- A P(- G Ĝ PA P (A x D(- P (A x (- D(- (- NA ( P ( + A x (- x (- A P(- Ĝ PS D ( + A x (- [0,] x (- A D(- - mond Klili tc [HJS 07] mntic vi coinduction An Sokolov Univity of Slzug Shonn NII

8 h ig pictu oalg( oalg( G K` ± y y K`( M( Ø M ` Æ oalg( y Id ` oalg(g x

9 h ig pictu oalg( oalg( G K` ± y y K`( M( Ø M ` Æ oalg( y Id ` oalg(g x

10 h ig pictu oalg( oalg( G K` ± y y K`( M( Ø M ` Æ oalg( y Id ` oalg(g x

11 h ig pictu oalg( oalg( G K` ± y y K`( M( Ø M ` Æ oalg( y Id ` oalg(g x

12 h ig pictu oalg( oalg( G K` ± y y K`( M( ` Æ oalg( y Id ( =(µ: ( (. (f =µ (f Ø ` M oalg(g x

13 h ig pictu Lmm. In pnc of K`-lw f functo : K`( cn liftd, imilly, givn n M-lw f lg functo : M( cn liftd nd Kl-lw l : Inf pnc of :K`-lw f functo : K`( cn m. In pnc of Lmm K`-lw. functo K`( cn liftd, nd K` M oalg(g oalg( oalg( oalg(g oalg( oalg( G givn nfuncto M-lw lg functo : M( c ly, givn n M-lwimilly, f lg : f M( cn liftd: K` ± M K` K` M y y & & oa oalg( oalg( oalg(g Alg( oalg( oalg(g oalg( G K`( M( M( G K`( Y Y G [8][8] fofo dtil. hi dtmin dtminiztion functo. W now dci M dtil. hi dtmin dtminiztion functo. W now dci Ø & & & & (5 K`( M ov mo dtil. in point M( in G G K`( Y ov point mo dtil. Y Æ Y Y G hh Klili ctgoy K`( h m ojct undlying undlyingctgoy ctgoy, ut Klili ctgoy K`( h m ojct, ut oalg(g on` ight giv n tct dcip ` h functo oalg(g mophim Y in K`( mp (Y in. h idntity mp mophim oalg( Y in K`( mp (Y in. h idntity mp oalg(g of wht i clld gnlizd powt contuction inon [35]. indciption inoalg(g K`( i unit : (; nd compoition g f in K`( u u n t h functo oalg(g oalg( G ight giv h functo oalg( G on ight giv n tct in K`( i unit : (; nd compoition g f K`( in:in:g g f f==µµ (g multipliction f. h i fogtful functouu: :K`( K`( [35]. multipliction (g f. h i fogtful,, y x functo Idcontuction of whtpowt i clld gnlizd powt contuction in ht i clld gnlizd in [35]. P. h fit pt i y, inc functo K` : oalg( oalg(. nding to to ( h lft lftdjoint djoint,,givn givn nding (nd ndf ftotoµµ (f (f.. hi hi functo functo h yy P.( h nd fontl K` on commut tivilly fom ndf. ( = nd (f. = nd f f..fc Such i Klili ctgoy K`( inhit colimit idntity on it= mp f: -nd of -colg to (f = idntity ojct = nd (f h fit Such Klili ctgoy K`( inhit colimit fom Pojct; pt y, inc K` : oalg( h fit pt iottom y, inc functo oalg( functo oalg( i

14 utiv lw: idntity mp id : in fom mp iny Y in. hi uggt how to dfin, o tht w cn dfin = L(id : in. It tifi (3. in fom mp in cond copondnc um w hv n M-lw : G G. It giv h ig pictu : in. It tifi (3. h functo oalg(g oalg(g on functo : M( M( y: w hv nlmm M-lw.: In Gpnc G. Itof giv K`-lw f functoofwht : K`( cngnlizd liftd, nd i clld powt contucti G G( 7 nd f 7 G(f. imilly, givn n M-lw f lg functo : M( cn liftd: G nd M-lw Gfuncto G K` P. h fit pt i y, inc : ution yild nw -lg. M K` nd(4 gunt f 7 tht G(fthi. oalg( oalg( nd mp f of -colg oalg( oalg(g idntity on ojct; itg v diction, um lifting : M( M(. Applying it to : G G on n oalg( oalg( G Nxt, uming M-lw ltipliction nw -lg. µ yild dfin (5 & n lg (µ : (G G. W& n K`( G y G( : Y G.Applying G. mining d. G g(µ : M( it to dtil Y oalg( M( lft to M( K` G ± y µ : (G G. yw n dfin (c wht follow w hll imply wit G fo lifting of G, oth whn c K`( M( M G = G to d. mining dtil lft fom K`-lw o fomoalg(g n M-lw. Uully G fixd, o lw M h functo oalg( on Ø ight giv n tct dciption i nd light, ovlodd nottion i pfd. unlikly, It i not hd to tht M (c i colg µ ion G fo of wht lifting of G, oth whn i clld Æ gnlizd powt contuction in [35]. nxt ult ( lo [] i not lly ud in thi pp, ut it i ntul qul -lw. Uully lw fixd, o n mophim on imply h M (f = (f. ` it lt lifting, G to tndd djunction. ` pviou popoition inc i P. h fit pt i y, inc functo : oalg( oalg( d nottion i pfd. K` oalg(g tht w wit Alg( nd oalg( fo ctgoi of lg nd colg oalg( K`-lw = ud ud in thi ut iton i ojct; ntul qul pp, idntity it nd mp f of -colg to (f tof.otin finl colg in K ncto, not of (comond. to tndd djunction. lifting, G non-tivil id-condition, lik nichmnt in dcpo Nxt, uming n M-lw : G G on dfin : oalg(g M y x Id fo ctgoi of lg nd colg ition. In pnc nd n M-lw, djunction much K`( i, of yk`-lw low; intnc of thi ult hv oalg(g M( lift to djunction twn ctgoi of, pctivly, lg nd GP [SBB 0] Pdcid. M h nd = fontl nd tivilly nd, low. mond nd ndofuncto G G commut G ottom fc. dtminiztion (6 G Popoition 3. Aum c n M-lw, djunction K`( (c - G(µ

15 lt = : oalg( oiginl nd intntion, f 7 (f µ (f. oalg(g : M( M( to ctgoy of yilding lifting G 3. Ano ndofuncto G :, ut thi tim with n M-lw : G G, o : with with. xtnion K`-lw An ndofuncto : tnfomtion :, lding with to K`-lw lifting y pot-compoing ntul, tht :, l : M( M( to ctgoy 4. Anof xtnion ntul tnfomtion : G tht c yilding lifting G -lg. t : oalg( oalg(g : K`(vi K`( to Klili ctgoy, Popoition K`( to. Klili ctgoy, vi Popoition ( w hv M-lw vi following two commuting digm. 4. An xtnion ntul tnfomtion : G tht connct K`- nd h ig pictu h xtnion ntul tnfomtion, tht ( µ µ following two tim commuting unctom-lw G : vi, 3.utAno thi ndofuncto withdigm. n M-lw G : :, G ut thi G, tim with n M-l (c = c nd (f = f ( µ µ nd xtnion-lw : G ng G : M( to lifting : M( -lg. M( to ctgoy of - M( ctgoy yilding Gof Gµ G nd (f = f ult in on cumomophim f. W cn now ummiz ll (3 G G G G ntul Gµ An xtnion connct Gµ ntul tnfomtion 4. : tht tnfomtion K`- itnd :fom commuting G tht pi ofcon nd G GWhn uch i n iomophim, G G G oalg(g oalg( n now ummiz ll ult in on cu digm. digm. hwhn following two commuting M-lw vi following two commuting uch i n iomophim, it fom commuting pi of ndofuncto fom [].point o hom. Auming ov 4, i lifting K` ginning ± umption. - 4.fom of thi ction, w hv to in: y y hom. µ ( i lifting Auming ov point 4, xtnion funcµ of µ µ M( oalg(g K`( muting cu of (liftd functo: oalg( to in: fom ginning of thi ction, w hv (3 M oalg(gø ncto:gµ oalg( Gµ Gµ M( K`( G Æ G G 9 G G G G G M( K`( i `utomticlly hi functo fithful; nd it i lo full if th 9 ` 8 G omophim, it fom Whn commuting uch i n iomophim, pi of it fom fom commuting []. pi of ndof ndofuncto tnfomtion : G oalg(g conit of monomophim. oalg( it i lo i utomticlly hi functo fithful; nd full if xtnion ntul 8 conit func: hom 4, G of monomophim. ofy ov G P.ofW dfin functo : oalg( lifting oalg( ngtnfomtion ov point. Auming i lifting point xtnion 4, i th y x Id y: (c µ c P. W dfin tofuncto in: : oalg( oalg(g 7 (c µ oalg( oalg( oalg( P. nd tivilly.nd f 7 (f = µ (f 7 nd fontl fc G G oalg( h ottom commut G c

16 h ig pictu inl vi [HJS 07] if... h G-finl lift oalg( oalg( G K` ± y y K`( M( ` Æ oalg( y Id Ø h mntic coincid (lmot ` M oalg(g x

17 nd of toy?

18 (Unfotuntly not

19 Wntd: good non-dtminiztion fo PA c P (A x D( D( P (A x D( dtc

20 < " " " < " " " k A non-dtminiztion of PA PA P (A x D(- LS P (A x (- tt D(- x zzzz DDDD " 3 < < 3 x x 3 x 4 x x 3 x 3 woooooo x ' x 3 3 x 4 x 4 k An Sokolov Univity of Slzug Shonn NII

21 Ano on... [DvGHMZ 07] Lt S D(S ltion fom tt to ditiution. W lift it to ltion D(S D(S y ltting whnv (i = i I p i i, wh I i finit indx t nd i I p i = (ii o ch i I i ditiution Φ i uch tht i Φ i (iii = i I p i Φ i.

22 Ano on... [DvGHMZ 07] Lt S D(S ltion fom tt to ditiution. W lift it to ltion D(S D(S y ltting whnv (i = i I p i i, wh I i finit indx t nd i I p i = (ii o ch i I i ditiution Φ i uch tht i Φ i (iii = i I p i Φ i. c P (D( x D( D( x D ( D( x D( D( dtc P(D(

23 Do it fit? Y nd no

24 Do it fit? Y nd no ut not ditiutiv lw c P (D( D( P(D( dt c = P(μ D(c till lot to don

Last time: introduced our first computational model the DFA.

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