Clil lnguge heory Biimulion, Gme & Henney Milner logi Leure 1 of Modelli Memii dei Proei Conorreni Pweł Sooińki Univeriy of Souhmon, UK I onerned rimrily wih lnguge, eg finie uom regulr lnguge; uhdown uom onex-free lnguge; uring mhine reurively enumerle lnguge; Thi i fine when we hink of n uomon/tm euenil roe whih h no inerion wih he ouide world during i omuion However, uom whih e he me lnguge n ehve very differenly o n ouide oerver Biimulion, Gme & Henney Milner logi 1/32 Biimulion, Gme & Henney Milner logi 2/32 The fmou offee mhine exmle Lelled rniion yem A lelled rniion yem (LTS) L i rile S, A, T where: Inerire oldi Cffè Bevnd l guo di è l limone S i e of e; A i e of ion; T S A S i he rniion relion We will normlly wrie for (,, ) T Lelled rniion yem generlie oh uom nd ree They re enrl rion of onurreny heory We will diu he oervion one n mke ou uh yem Biimulion, Gme & Henney Milner logi 3/32 Biimulion, Gme & Henney Milner logi 4/32 Tre reorder Given e of n LTS L, he word σ = α 1 α 2 α k A i re of when rniion α1 α2 αk 1 k 1 We will ue σ horhnd Suoe h L 1 nd L 2 re LTS The re reorder r S 1 S 2 i defined follow: Tre euivlene Tre euivlene i defined r = r r, ie r def = r r I i immedie h when L 1 = L 2, r i n euivlene relion on he e of n LTS Bu re re no enough: re euivlene i very ore, ine he offee mhine hve he me re r σ A σ σ Oervion 1 r i reflexive nd rniive r Biimulion, Gme & Henney Milner logi 5/32 Biimulion, Gme & Henney Milner logi 6/32
Simulion Suoe h L 1 nd L 2 re LTS A relion R S L1 S L2 i lled imulion whenever: Simulion exmle 1 Simulion i more eniive o rnhing (ie non-deerminim) hn re: if R nd hen here exi uh h nd R Oervion 2 The emy relion i imulion nd rirry union of imulion re imulion Similriy S 1 S 2 i defined he lrge imulion Euivlenly, iff here exi imulion R uh h (, ) R Oervion 3 Similriy i reflexive nd rniive Oervion 4 Simulion euivlene def = 1 1 1 1 2 2 3 2 2 Biimulion, Gme & Henney Milner logi 7/32 Biimulion, Gme & Henney Milner logi 8/32 Simulion exmle 2 Bu i i no enirely ifory 1 2 Biimulion Suoe h L 1 nd L 2 re LTS A relion R S L1 S L2 i lled iimulion whenever: (i) if R nd hen here exi uh h nd R ; (ii) if R nd hen here exi uh h nd R Lemm 5 R i iimulion iff R nd R o re imulion r 1 r 2 r 1 r 2 r 1 Biimulion, Gme & Henney Milner logi 9/32 Biimulion, Gme & Henney Milner logi 10/32 Proerie of iimulion Exmle of iimulion, 1 Lemm 6 i iimulion Proof Vouly rue Lemm 7 If {R i } i I re fmily of iimulion hen i I R i i iimulion Proof Le R = i I R i Suoe R hen here exi k uh h R k In riulr, R k nd o R, hu R i ymmeri If hen here exi uh h nd R k Bu R k imlie R Corollry 8 There exi lrge iimulion I i lled iimilriy If L 1 = L 2 hen iimilriy i n euivlene relion Lemm 9 1 1 2 3 4 5 Proof R = { (, i ) i N } i iimulion Biimulion, Gme & Henney Milner logi 11/32 Biimulion, Gme & Henney Milner logi 12/32
Exmle of iimulion, 2 Reoning ou iimilriy 1 2 1 1 2 3 1 4 To how h e, re iimilr i uffie o find iimulion R whih rele nd ; I i le ler how o how h nd re no iimilr, one n: enumere ll he relion whih onin (, ) nd how h none of hem re iimulion; enumere ll he iimulion nd how h none of hem onin (, ); orrow ome ehiniue from gme heory Biimulion, Gme & Henney Milner logi 13/32 Biimulion, Gme & Henney Milner logi 14/32 Biimulion gme, 1 Biimulion gme, 2 We re given wo LTS L 1, L 2 The onfigurion i ir of e (, ), L 1, L 2 The iimulion gme h wo lyer: P nd R A round of he gme roeed follow: (i) R hooe eiher or ; (ii) uming i hoe, i nex hooe rniion ; (iii) P mu hooe rniion wih he me lel in he oher LTS, ie uming R hoe, i mu find rniion ; (iv) he round i reeed, reling (, ) wih (, ) Rule: An infinie gme i win for P R win iff he gme ge ino round where P nno reond wih rniion in e (iii) Oervion 10 For eh onfigurion (, ), eiher P or R h winning regy Theorem 11 iff P h winning regy ( iff R h winning regy) Biimulion, Gme & Henney Milner logi 15/32 Biimulion, Gme & Henney Milner logi 16/32 P h winning regy P h winning regy Le GE def = { (, ) P h winning regy } Suoe h (, ) GE nd Suoe h here doe no exi rniion uh h (, ) GE Then R n hooe he rniion nd P nno reond in wy whih kee him in winnle oiion Bu hi onrdi he f h h P h winning regy for he gme ring wih (, ) Thu GE i iimulion Biimulion re winning regie: If hen here exi iimulion R uh h (, ) R Whever move R mke, P n lwy mke move uh h he reul i in R Clerly, hi i winning regy for P Biimulion, Gme & Henney Milner logi 17/32 Biimulion, Gme & Henney Milner logi 18/32
Exmle of non iimilr e Similriy nd iimilriy Biimilriy i rnhing-eniive Theorem 12 nd in generl he inluion i ri Proof Any iimulion nd i ooie re lerly imulion On he oher hnd, he following exmle how h iimilriy i finer hn imulion euivlene 1 2 r 1 r 2 r 1 r 2 1 2 r 1 r 2 r 1 r 1 r 2 Biimulion, Gme & Henney Milner logi 19/32 Biimulion, Gme & Henney Milner logi 20/32 Re: euivlene Henney Milner logi r Biimilriy i he fine (=eue le) euivlene we hve onidered Clim 13 Biimilriy i he fine reonle euivlene, where reonle men h we n oerve only he ehviour nd no he e-e We will give lnguge, he o-lled Henney Milner logi, whih derie oervion/exerimen on LTS Biimulion, Gme & Henney Milner logi 21/32 Suoe h A i e of ion Le L ::= []L L L L L L L Given n LTS we define he emni y ruurl induion over he formul ϕ: [A]ϕ if for ll uh h we hve ϕ; A ϕ if here exi uh h nd ϕ; ϕ if i i no he e h ϕ; ϕ 1 ϕ 2 if ϕ 1 or ϕ 2 ; ϕ 1 ϕ 2 if ϕ 1 nd ϕ 2 ; lwy; never; Biimulion, Gme & Henney Milner logi 22/32 HM logi exmle formul Bi roerie of HM logi n erform rniion lelled wih ; [] nno erform rniion lelled wih ; [] n erform rniion lelled wih o e from whih here re no lelled rniion ([] )? Lemm 14 ( De Morgn lw for HM logi) [] = ; = [] ; = ( ); = ( ); = ; = In riulr, o ge he full logi i uffie o onider ju he ue {,,, } or {[],,, } or {, [],,,, } Biimulion, Gme & Henney Milner logi 23/32 Biimulion, Gme & Henney Milner logi 24/32
Diinguihing formul Logil euivlene Definiion 15 The logil reorder L i relion on he e of n LTS defined follow: 1 2 < L iff ϕ ϕ ϕ r1 r2 r1 r2 ( ) ( ) 1 2 r1 r2 r1 r 1 r2 I i lerly reflexive nd rniive Definiion 16 Logil euivlene i L def = L L I i n euivlene relion Oervion 17 Aully, for HM, L = L = L Thi i oneuene of hving negion Proof Suoe L nd ϕ If ϕ hen ϕ, hene ϕ hene ϕ, onrdiion Hene ϕ ( ) ( ) Biimulion, Gme & Henney Milner logi 25/32 Biimulion, Gme & Henney Milner logi 26/32 Henney Milner & Biimulion Soundne Definiion 18 An LTS i id o hve finie imge when from ny e, he numer of e rehle i finie Theorem 19 (Henney Milner) Le L e n LTS wih finie imge Then L = To rove hi, we need o how: Soundne ( L ): If wo e ify he me formul hen hey re iimilr Comleene ( L ): If wo e re iimilr hen hey ify he me formul Remrk 20 Comleene hold in generl The finie imge umion i needed only for oundne L (Soundne) I uffie o how h L i iimulion We will rely on imge finiene Suoe h L nd Then nd o hu here i le one uh h The e of ll uh i lo finie y he exr umion le hi e e { 1,, k } Suoe h for ll i we hve h L i Then ϕ i uh h ϕ i nd i ϕ i Thu while i k ϕ i we mu hve i k ϕ i, onrdiion Hene here exi i uh h i nd L i Biimulion, Gme & Henney Milner logi 27/32 Biimulion, Gme & Henney Milner logi 28/32 Comleene 1 Comleene 2 L (Comleene) We will how hi < L y ruurl induion on formul Be: hen Alo, hen Induion: Modliie ( nd []): If ϕ hen nd ϕ By umion, here exi uh h nd By induive hyohei ϕ nd o ϕ If []ϕ hen whenever hen ϕ Fir, noie h imlie h if hen here exi uh h wih Sine ϕ, lo ϕ Hene []ϕ Prooiionl onneive ( nd ): if ϕ 1 ϕ 2 hen ϕ 1 or ϕ 2 If i i he fir hen y he induive hyohei ϕ 1, if he eond hen ϕ 2 ; hu ϕ 1 ϕ 2 if ϕ 2 ϕ 2 i imilr Noe h omleene doe no need he finie imge umion hu iimilr e lwy ify he me formul In he roof, we ued he f h {, [],,,, } i enough for ll of HM logi Biimulion, Gme & Henney Milner logi 29/32 Biimulion, Gme & Henney Milner logi 30/32
Imge finiene Sulogi of HM The heorem rek down wihou hi umion: 1 } k ime 2 } k ime L r ::= L r Theorem 21 Logil reorder on L r oinide wih he re reorder L ::= L L L Theorem 22 Logil reorder on L oniide wih he imulion reorder Ey o hek, uing he iimulion gme, h 1 2 Soluion: Inrodue infinie onjunion o he logi Biimulion, Gme & Henney Milner logi 31/32 Biimulion, Gme & Henney Milner logi 32/32