Uneiility for ritrry puli nnounement logi Tim Frenh n Hns vn Ditmrsh strt. Aritrry puli nnounement logi (AP AL) is n extension of multi-gent epistemi logi tht llows gents knowlege sttes to e upte y the puli nnounement of (possily ritrry) epistemi formule. It hs een shown to e more expressive thn epistemi logi, n soun n omplete xiomtiztion hs een given. Here we ress the question of eiility. e present proof tht the stisfiility prolem for ritrry puli nnounement logi (AP AL) is o-re omplete, vi tiling rgument. Keywors: Epistemi Logi, Puli Announement Logi, Deiility 1 Introution Aritrry nnounement logi (AP AL) is n extension of multi-gent epistemi logis with puli nnounements n ritrry nnounements. A puli nnounement llows the informtion stte of every gent to e upte y pulily informing them tht some epistemi formul, ψ is true. An ritrry nnounement is e to the lnguge to llow us to quntify over ll possile nnounements. This logi is esrie in etil in [1]. It is shown to e more expressive thn norml epistemi logi n soun n omplete xiomtiztion is given. Furthermore, in [2] tleu-lulus is presente to etermine vliity in AP AL. hile the ove results inite tht the set of vliities for AP AL is reursively enumerle, the full eiility of the stisfiility prolem hs remine open. Here we show tht the stisfiility prolem for the logi is uneile vi tiling rgument. This is surprising result sine in [1] it is shown tht AP AL is isimultion invrint. Hene every AP AL formul is stisfie y tree-like moel, rther thn the gri-like moels typilly require for tiling rguments (see [9] for etile nlysis of uneiility in extensions of epistemi logi). The uneiility follows from the power of the ritrry nnounement opertor. The ritrry nnounement opertor, φ expresses: there exists true formul of epistemi logi, tht when pulily nnoune estlishes the truth of φ. Impliit in this sttement is n existentil quntifition over ll formule of epistemi logi, n we show tht this expressive power is suffiient to llow Avnes in Mol Logi, Volume 7. 2008, the uthor(s).
24 Tim Frenh n Hns vn Ditmrsh us to enoe n uneile tiling prolem. This is not n entirely surprising result, espite the mny other fvorle properties of AP AL. In [9] etile survey is presente of uneile temporl n epistemi logis, n n nlysis is presente of the properties leing to uneiility. The ritrry nnounement opertor is trnsitive in nture n reminisent of temporl opertor. However, most uneile logis surveye in [9] re not isimultion invrint, initing ertin uniqueness to this result. Another relte result is the uneiility of iterte mol reltiviztion [7]. This logi is shown to e highly uneile (Σ 1 1-omplete), gin, y enoing tiling prolem. Other uneile logis onsiere in [7] omine ommon knowlege with iterte reltiviztion. Reltiviztion is nother term to enote the struturl restrition tht onstitutes the informtive effet of n nnounement. Iterte reltiviztion is ifferent from ritrry nnounement. The former mens tht one llows (ritrry finite length) sequenes of moel restritions for given epistemi formul ( nnounement ); note tht fter nnounements of mol formul, nnouning tht formul gin my still e informtive, s in the fmous Muy Chilren Prolem [4]. ut the ltter mens moel restrition for ny epistemi formul. Now the itertion is impliit. It is there euse the sequene of two epistemi nnounements is gin equivlent to n epistemi formul [1]. 2 Syntx n semntis The formuls of AP AL, L pl re inutively efine s φ ::= p φ (φ φ) K φ [φ]φ φ where is tken from the set of gents A, n p is tken from the set of tomi propositions P. Let L el e the set of formuls not ontining ny of the opertors [φ] or. These formuls re interprete over strutures M = (S,, V ) where S is set of worls, : A (S S) ssigns reflexive, trnsitive n symmetri essiility reltion, to eh gent, n V : P (S) mps eh proposition to the set of worls where it is true. Let M = (S,, V ) n suppose tht s S. The semntis of AP AL re given s: M, s = p iff s V (p) M, s = φ iff M, s = φ M, s = φ 1 φ 2 iff M, s = φ 1 n M, s = φ 2 M, s = K φ iff t S where s t, M, t = φ M, s = [ψ]φ iff M, s = ψ = M ψ, s = φ M, s = φ iff ψ L el, M, s = [ψ]φ where M ψ = (S,, V ) is suh tht: S = {s S M, s = ψ}; for ll A, = (S S ); n for ll p P, V (p) = V (p) S. As usul
Uneiility for ritrry puli nnounement logi 25 we tke K φ to men gent knows φ, n let L φ revite K φ (gent onsiers φ possile). e sy n AP AL formul φ is stisfile if there exists some moel M = (S,, V ) n some worl s S suh tht M, s = φ, n if M, s = φ for ll moel-worl pirs, M, s, we sy φ is vli. Note tht when efining the semntis of φ we restrit the ritrry nnounements to rnge only over the epistemi formuls (i.e. those in L el ). e reson for this is tht we oviously nnot llow the ritrry nnounements to rnge over ritrry nnounements (i.e. formule of the form ψ) s the semntis woul then e unefine. Further, we o not let the ritrry nnounements rnge over nnounements (suh s [ψ]α where ψ, α L el ) sine suh formuls re expressively equivlent to pure epistemi formule (see [11] for trnsltion). The formul φ expresses the sttement fter pulily nnouning ny true formul of epistemi logi, φ must e true. As we see in its forml semntis ove, this sttement impliitly quntifies over ll true formule of epistemi logi. For exmple, suppose φ were the formul K p K p. The formul φ is true t some worl where p is true, if n only if for every -relte worl, u where p is not true, for every epistemi formul ψ, there is some -relte worl, v, tht grees with u on the interprettion of ψ. (This is euse otherwise the nnounement of p ψ woul e enough to mke K p K p true). This is strong property to e le to express. If two sets of worls nnot e istinguishe y ny epistemi formul then they re, for the purposes the logi, ientil. Given tht the epistemi formule n e ritrrily lrge, using this notion of equivlene we re le to enoe gri-like property for finite gris of ritrry size. This expressivity is exploite to enoe n ritrry tiling prolem whih is suffiient to show tht the stisfiility prolem for AP AL is o-re omplete. 3 Tilings n uneiility e show the stisfiility prolem is uneile for AP AL y emeing tiling prolem tht is known to e o-re omplete (i.e. equivlent to omputing the memership of the omplement of ny reursively enumerle set). The tiling prolem is s follows: DEFINITION 1. Let C e finite set of olours n efine C-tile to e four-tuple over C γ = (γ t, γ r, γ f, γ l ), where the elements re referre to s, respetively, top, right, floor n left. The tiling prolem is, for ny given finite set of C-tiles, Γ, etermine if there is funtion λ : ω ω Γ suh tht for ll (i, j) ω ω: 1. λ(i, j) t = λ(i, j + 1) f 2. λ(i, j) r = λ(i + 1, j) l. The tiling prolem hs een shown to e o-re omplete y Hrel [6]
26 Tim Frenh n Hns vn Ditmrsh (see [8] for n overview of the pplition of tiling prolems to omplexity for mol logis). 4 Enoing the tiling prolem To enoe the tiling prolem we seek to efine gri like struture in the moel M. Tht is we efine formul gri wherey M, s = gri implies the struture of M is similr to ω ω. To o this we exploit one of the stronger properties of AP AL: the ility to quntify over ll epistemi formuls. This llows us to efine n equivlene etween the worls in moel n moulo tht equivlene, gri like struture. Suh enoings re rrely elegnt n this is no exeption. e use the following toms: 1. e lel eh worl s either white ( ) or lk () with the unerstning tht is n revition for. e inten to lel the moel in hess-or pttern. 2. e use the set of gents,,, n t where we suppose tht: n esrie some vertil suessor reltion ( goes from lk squre to white squre, n goes from white squre to lk squre); n esrie some horizontl suessor reltion ( goes from lk squre to white squre, n goes from white squre to lk squre); the reltion for the gent t inlues the reltions for the gents,, n. 3. For eh tile γ Γ we ssign proposition (lso enote γ) with the unerstning tht the tiles re mutully exlusive (i.e. γ δ γ δ). Suh struture is represente in Figure 1 (with the ssumption tht t is universl reltion over ll worls). Even though our essiility reltions re equivlene reltions, in the multi-gent setting we n enfore iretionlity y omposing equivlene reltions for ifferent gents (n grouning them y referring to truths in lol or ounry onitions of our struture, suh s the tul stte, or the top-left stte, or...). More formlly, even though n re equivlene reltions, their omposition is not symmetri: we my hve tht x y z, i.e., x( )z, ut not z( )x. Although we were not inspire y this, it eserves mentioning tht suh emerging symmetry in multi-gent onitions is use to gret effet in the expressivity proofs in Chpter 8 of [11]. The enoing omes in three prts: Firstly, we woul like to efine t to ontin the trnsitive losure of the other epistemi reltions. Next, we efine gri like (or hessor like) struture over the moel. Finlly we use this gri-like moel to stte tht the given tiling exists.
Uneiility for ritrry puli nnounement logi 27 Figure 1. A gri-like moel. First note tht these esriptions grossly over-simplify the tul onstrution. To properly exeute these steps we woul require mehnism tht llows us to efine when one worl is equivlent to nother. This we o not hve. However, the ritrry nnounement mehnism llows us to ientify when two sets of worls nnot e istinguishe y ny nnounement. e will, for the moment ignore these onsiertions. e will use the term equivlent (rther thn equl) to esrie to worls tht re inistinguishle with respet to epistemi logi, n we will preisely efine this notion in the susequent setions. 4.1 ek trnsitive losure The following formul sets t to inlue wek trnsitive losure of,, n. Prtiulrly t every worl w in the moel, for eh gent x {,,, }, if there is some worl w of ifferent olour to w where w x w, then there is some worl u where w t u n u is equivlent to w. (1) (2) lt = K t ( (L L ) (L L ) T = K t ( K t (K K K K ) K t (K K K K ) The importnt prt of this formul is the ritrry nnounement ( ) in T. This sttes tht no nnounement n e me tht informs gent t of ) ).
28 Tim Frenh n Hns vn Ditmrsh the olour of the urrent squre, without informing every other gent s well. To the ontrry, suppose tht this were not true. Speifilly suppose the urrent worl is lk, n gent onsiers some white worl, u, possile where u ws emonstrly ifferent to every worl t onsiers possile (sy y formul χ u ). Then the puli nnounement χ u woul inform t tht the urrent worl is lk, ut not. 4.2 Defining gri To efine gri-like struture we will require the following properties: 1. Every lk worl hs n -suessor tht is white n -suessor tht is white. 2. Every white worl hs -suessor tht is lk n -suessor tht is lk. 3. The urrent worl is lk n oth n know this. 4. If the urrent worl is lk: for every white worl u tht is -rehle from the urrent worl, every lk worl tht is -rehle from u is equivlent to some lk worl tht is -rehle from some white worl tht is - rehle from the urrent worl. for every white worl u tht is -rehle from the urrent worl, every lk worl tht is -rehle from u is equivlent to some lk worl tht is -rehle from some white worl tht is - rehle from the urrent worl. 5. If the urrent worl is white: for every lk worl u tht is -rehle from the urrent worl, every white worl tht is -rehle from u is equivlent to some white worl tht is -rehle from some lk worl tht is - rehle from the urrent worl. for every lk worl u tht is -rehle from the urrent worl, every white worl tht is -rehle from u is equivlent to some white worl tht is -rehle from some lk worl tht is - rehle from the urrent worl. This is hieve with the following formuls: (3) (4) (5) (6) (7) st = K K C1 = ((L ( L )) (K ( L ))) C2 = ((L ( L )) (K ( L ))) C3 = ((L ( L )) (K ( L ))) C4 = ((L ( L )) (K ( L )))
Uneiility for ritrry puli nnounement logi 29 Agin, the ritrry nnounement is use to estlish notion of equivlene etween two worls. The formul st simply speifies the stte of the initil worl (the ottom, left hn orner of the gri whih, to exten the hess-or nlogy, is lk). The other formule C1 C4 efine wek ommuttivity property (e.g. every lk worl tht is rehle from the urrent (lk) worl, is rehle from the urrent worl). 4.3 The existene of tiling Given the previous formuls re suffiient to set up the esire hessor pttern, it is simple mtter to exploit it to ssert the existene of tiling. Suppose the set Γ is given s ove. Let: (8) (9) lk = γ Γ wht = γ Γ ( γ [ K ( γ t =δ δ)) K ( γ r =δ l δ)) ( γ [ K ( γ t =δ δ)) K ( γ r =δ l δ)) ]) ]). The interprettion of these formul is strightforwr. Given tile γ is true t the urrent stte, we ssert tht the ottom of ll suessor vertil tiles 1 is the sme olour s γ t. In the se tht the urrent stte is lk the suessor vertil sttes re the -rehle white sttes, n if the urrent stte is white then ll suessor vertil sttes re the -rehle lk sttes. A similr hrteriztion exists for the horizontl (left-right) orresponene. Finlly we n efine the formul: (10) T ile Γ = lt T st K t (C1 C2 C3 C4 lk wht). In the following setion we show tht the existene of moel for this formul is equivlent to the existene of tiling of the ω-plne for γ. 5 Proof of orretness In this setion we show tht the ove formul, T ile Γ, is stisfile in AP AL if n only if the set of tiles Γ is le to tile the plne ω ω. e first ress the sounness of the onstrution of the formul T iles Γ. LEMMA 2. Given there is Γ-tiling of the plne, λ : ω ω Γ, we my efine moel of AP AL tht stisfies the formul T ile Γ. Proof. This moel is tken iretly from the tiling see Figure 2 with the knowlege reltion of t eing the universl molity. Tht is we let our moel e M = (S,, V ) where: S = ω ω, 1 hilst the existene of multiple vertil suessors is not very gri-like we will lter show this is inonsequentil.
30 Tim Frenh n Hns vn Ditmrsh V () = {(i, j) i + j is even} n V ( ) = {(i, j) i + j is o}, for eh γ Γ, V (γ) = {(i, j) λ(i, j) = γ}, is the trnsitive, reflexive n symmetri losure of the reltion {((i, j), (i, j + 1)) (i, j) V ()}, is the trnsitive, reflexive n symmetri losure of the reltion {((i, j), (i, j + 1)) (i, j) V ( )}, is the trnsitive, reflexive n symmetri losure of the reltion {((i, j), (i + 1, j)) (i, j) V ()}, is the trnsitive, reflexive n symmetri losure of the reltion {((i, j), (i + 1, j)) (i, j) V ( )}, t = {((i, j), (k, l)) i, j, k, l ω}. e now show tht M, (0, 0) = T ile Γ. For the prts of T ile Γ not ontining ritrry nnounements this is strightforwr. e n see tht M, (0, 0) = lt st y onstrution n s λ is tiling it follows tht M, (0, 0) = K t (lk wht). The remining formuls T n C1 C4 involve ritrry nnounements. Let s first exmine T. This is equivlent to, for ll i, j ω, M, (i, j) = ( K t (K K K K ) K t (K K K K ) Suppose tht (i, j) V (). If ny nnounement [φ] mkes K t true, it must e tht M φ onsists only of lk worls, n hene, M φ, (i, j) = (K K K K ). A similr rgument hols for (i, j) V ( ). e will now show tht M, (0, 0) = K t (C1), n the ses for C2-C4 n e shown similrly. Suppose tht (i, j) V (). e must show tht for ll epistemi ψ where M, (i, j) = ψ, M ψ, (i, j) = L ( L )) K ( L ). Sine we re quntifying over ll sumoels M φ orresponing to epistemi formul, it is suffiient to show tht M, (i, j) = L ( L )) K ( L ) where M = (S,, V ) is ny sumoel where (i, j) S. In suh se, sine M, (i, j) = L ( L ), it follows tht (i, j + 1), (i + 1, j + 1) S. Also, (i + 1, j) is the only one white worl -relte to (i, j). If (i + 1, j) S, then euse (i + 1, j + 1) S, we hve M, (i, j) = K ( L ). If (i + 1, j) / S then M, (i, j) = K ( L ) s M, (i, j) = K. Thus for every su-moel, M inluing (i, j) we hve M, (i, j) = L ( L )) K ( L ) n thus for every epistemi nnounement ψ where M, (i, j) = ψ we hve M ψ, (i, j) = L ( L )) K ( L ). Therefore M, (0, 0) = K t C 1. A similr rgument n e pplie for C2-C4 so it follows tht given Γ- tiling exists, we n show, T ile Γ is stisfile. ).
Uneiility for ritrry puli nnounement logi 31 (0,j) (1,j) (2,j) (i,j) (0,2) (1,2) (2,2) (i,2) (0,1) (1,1) (2,1) (i,1) (0,0) (1,0) (2,0) (i,0) Figure 2. The onversion of tiling into moel. For the ompleteness rgument, we suppose tht M, s = T ile Γ. From M we will show tht, for eh n ω we n onstrut tiling, λ n of n n n gri. This is shown to e equivlent to tiling the full ω ω plne in the following lemm. LEMMA 3. If Γ is le to tile n n n gri for ll n ω, then Γ is le to tile the ω ω plne. Proof. Let λ n e the n n tiling, n efine λ s tiling of the plne where λ (0, 0) = γ for some γ where for some infinite N 0,0 ω, for ll n N 0,0 λ n (0, 0) = γ. e then proee y inution over ω ω where (i 1, j 1 ) (i 2, j 2 ) if n only if i 1 + j 1 < i 2 + j 2 or i 1 + j 1 = i 2 + j 2 n i 1 i 2. For (i, j) > (0, 0), we efine λ (i, j) suh tht if i > 0 then λ (i, j) = γ where for some infinite N i,j N i 1,j+1, for ll n N i,j λ n (i, j) = γ, otherwise (if i = 0) we let λ (0, j) = γ where for some infinite N 0,j N j 1,0, for ll n N 0,j λ n (0, j) = γ It n e shown tht suh γ n N i,j n lwys e foun (sine there re infinitely mny finite tilings n only finitely mny tiles, the pigeon hole priniple my e pplie). Therefore suh λ my e efine y inution, (or inee, Koenig s Lemm).
32 Tim Frenh n Hns vn Ditmrsh To proee we require the following efinition: DEFINITION 4. Two worls re 0-Q-isimilr, iff they stisfy extly the sme set of propositionl toms tken from Q. For ll n ω, two worls, u n v in M, re n-q-isimilr (written u = Q n v) if n only if: 1. u = Q n 1 v; 2. for every x {,,,, t}, for every worl w where u x w, there is some worl w where v x w n w = Q n 1 w ; n 3. for every x {,,,, t}, for every worl w where v x w, there is some worl w where u x w n w = Q n 1 w. e note for ll n n Q, n-q-isimilrity is n equivlene reltion. LEMMA 5. Suppose tht the set of propositions, Q, is finite. Then for every n, there is finite set of L el formuls {φ 1,..., φ m } suh tht for every u S, there is some i m suh tht for ll v S, u = Q n v if n only if M, v = φ i. Proof. This n e shown y inution. As se se we tke the set of formuls φ(q ) = x Q x x Q\Q x for ll Q Q. Clerly, for eh u S, we n let Q = {x u V (x)} n then for ll v S, M, v = φ(q ) if n only if v = Q 0 u. For the inutive step, suppose tht {φ 1,..., φ m } is set of formuls suh tht for every u in S, there is some i m suh tht for ll v S, u = Q n v if n only if M, v = φ i. For eh u S let the orresponing formul φ i e enote φ n u, n let where φ n+1 u = φ n u (su u x nsu u x) x A su u x = {L x φ n v v x u} nsu u x = K x { φi v x u, M, v = φ i } for the set of gents, A = {,,,, t}. Then for ny v S where M, v = φ n+1 u we hve: 1. v = Q n u sine M, v = φ n u. 2. for every x {,,,, t}, for every worl w where u x w, we hve su u x L x φ n w, so M, v = L x φ n w, n thus there is some worl w where v x w, M, w = φ n w, so w = Q n w.
Uneiility for ritrry puli nnounement logi 33 3. for every x {,,,, t}, for every worl w where v x w, there is some worl w where u x w n w = Q n w. To see this, suppose for ontrition tht there ws some worl w suh tht v x w n for every worl w where u x w we hve w = Q n w. Therefore, we hve nsu u x K x φ n w so M, v = K x φ n w, ontriting v x w. Therefore v = Q n+1 u s require. Conversely, if v = Q n+1 u, then 1. M, v = φ n u sine v = Q n u. 2. for ll x A, for every worl w where u x w there is some worl w where v x w n w = Q n w (hene M, w = φ n w). Therefore M, v = su u x. 3. for every x {,,,, t}, for every worl w where v x w, there is some worl w where u x w n w = Q n w. y the inution hypothesis, for every w where v x w, we hve M, w = u xw φn w. For ll w where u x w we lerly hve φ n w { φ i v x u, M, v = φ i }, so it follows tht M, v = nsu u x. Thus M, v = φ n+1 u ompleting the inution. For the following proofs we efine new opertor Q n, to men for ll puli Q-nnounements of epth n. The semntis re given s: M, u = Q n φ if n only if for ll L el formule ψ with t most n nestings of knowlege opertors n ontining only the toms Q, if M, u = ψ, then M ψ, u = φ. LEMMA 6. 1. For ll n, for ll Q, φ Q n φ is vliity. 2. For ny two worls u, v where u = Q n v, for ny L el formul φ of epth t most n n ontining only toms from Q, M, u = φ if n only if M, v = φ. 3. For ny two worls u, v where u = Q n+m v, for ny L el formul φ of epth t most n n ontining toms only from Q, M, u = Q mφ if n only if M, v = Q mφ. Proof. 1. Ovious. 2. y inution. Clerly the sttement hols for the se n = 0. Suppose the sttement hols for n. Every L el formul φ, of epth n+1, n e written s oolen omintion of toms n formuls K xi φ i (for i = 1,..., m) where φ i is formul of epth t most n. If u = Q n+1 v, then for every u x u there is some v x v where u = Q n v, n vie-vers. y the inution hypothesis, M, u = φ i for ll u xi u if n only if M, v = φ i for ll v xi v. It follows tht M, u = φ if n only if M, v = φ.
34 Tim Frenh n Hns vn Ditmrsh 3. To prove this sttement we exten the inution ove with the se for m. In the se m = 0 it is effetively the seon prt of this lemm. Now suppose the sttement hols for given m (n for ll n). ithout loss of generlity, suppose tht u = Q n+m v n M, u = Q mφ where φ is of epth n. Therefore there is some nnounement, ψ, of epth m tht mkes φ flse. This is equivlent to M, u = ψ φ ψ where φ ψ is inutively efine y repling suformuls K x (α) with K x (ψ α ψ ), n otherwise ting s the ientity (see Proposition 4.22 of [11] for forml esription of this trnsltion). Now ψ φ ψ is of epth n + m so the result follows from the seon prt of this Lemm. e will refer to formul of epth n, ontining only toms from Q s n n-q-formul. The ove lemms n efinitions will e pplie to show how the ritrry nnounements in the formul T ile Γ n llow us to estlish tht two worls re n-q-isimilr for ritrry n n ritrry Q. To this en, for n ω n finite Q P, we efine the formul T ile (n,q) Γ to e the formul T ile Γ with every ritrry nnounement reple y Q n (n likewise for suformuls suh s C1 (n,q) ). There re two types of puli nnounement in the formul T ile Γ. The first ppers in the su-formul T, n the following lemm shows how this llows the knowlege reltion of gent t to t s wek kin of trnsitive losure for the other knowlege reltions. LEMMA 7. Let n 1, x {,,, } n Q P e finite set of propositionl toms inluing {, } Γ. Suppose tht u V () (resp. u V ( )), u x v for some v V ( ) (resp. v V ()), u = Q n w, n M, u = (lt T ) (n,q). Then there is some w suh tht w t w n w = Q n 1 v. Proof. Suppose tht M, u = (lt T ) (n,q), u = Q n w, n u x v where u V () n v V ( ). From T we hve M, u = Q n 1 (K t K x ). As u = Q n w n n 1, y Lemm 6.3 we hve M, w = Q n 1 (K t K x ). Now, suppose for ontrition tht for ll w t w we hve w Q = n 1 v. y Lemm 5 there is some formul φv n 1 suh tht M, v = φv n 1, n for ll w t w we hve M, w = φv n 1. Therefore we my mke the puli nnounement ψ = φv n 1. Thus M ψ, w = K t, n sine M, w = Q n 1 (K t K x ), we hve M ψ, w = K x. It follows tht M, w = ψ K x (ψ ). However, M, u = ψ K x (ψ ). Sine ψ = φv n 1, ψ hs epth n 1 n thus K x (ψ ) hs epth n. As u = Q n w, y Lemm 6.2, u n w gree on ll formuls of epth n. This ontrits the inferene tht M, u = K x (ψ ) n M, w = K x (ψ ). The other ourrenes of ritrry nnounements in the formul T ile Γ pper in the formule C1 C4. These formuls use the ritrry nnounements to estlish wek type of ommuttivity property, whih is essentil
Uneiility for ritrry puli nnounement logi 35 v u w n u n 2 w v Figure 3. The onstrution of Lemm 8, inferring the worls w n w re n 2-isimilr, given the worls u n u re n isimilr. in efining gri. The following lemms lrify this property n show tht it is enfore y the formul T ile Γ. The first lemm els with the lk worls n the seon lemm els (symmetrilly) with the white worls. LEMMA 8. Suppose tht u V (), u v, for some v V ( ). Let n 2 n suppose lso tht u = Q n u for some finite Q P inluing {, } Γ, n u v for some v V ( ). Given tht M, u = (C1 C2) (n,q) : 1. for ll w V () where v w, if there is some w V () where v w then there is some suh w where either w = n 2 w or w = Q n 2 u. 2. for ll w V () where v w, if there is some w where v w, then there is some suh w where either w = n 2 w or w = Q n 2 u. Proof. e will show se 1, n se 2 n e shown similrly. So given the ssumptions of the Lemm, let w V () e suh tht v w. Consier the nnounement ψ = φu n 2 φw n 2. Sine M, u = C1 (n,q), M ψ, u = L ( L ) K ( L ), n y Lemm 5, M ψ, u = L ( L ). e my pply mous ponens n Lemm 6 to eue, M ψ u = K ( L ). Therefore M ψ v = L, so there is some w V () where v w, n M, w = ψ. As w V (), we hve either M, w = φu n 2 n thus w = Q n u, or M, w = φw n 2, n thus w = Q n 2 w, (y Lemm 5). This senrio is represente in Figure 3
36 Tim Frenh n Hns vn Ditmrsh Notie tht the lemm oes not perfetly pture the notion of ommuttivity. Ielly we woul like to hve: For ll u, w, u V () for ll v V ( ) where u v, v w n u = Q n u, there s some v V ( ) n some w V () suh tht u v, v w n w = Q n 2 w, (n vie-vers). However, we must onsier the itionl possiility tht there s some v V ( ) n some w V () suh tht u v, v w n w = Q n 2 u. In suh se we woul hve, y the seon prt of the lemm, tht there is some w where v w n either w = Q n 2 w or w = Q n 2 u. In either se, s w = Q n 2 u n u = Q n u, we will hve w = Q n 2 u = Q n 2 u = Q n 2 w, whih is suffiient for our purposes. LEMMA 9. Suppose tht u V ( ), u v, for some v V (). Suppose lso tht u = Q n u for some finite Q P inluing {, } Γ, n u v for some v V (). Given tht M, u = (C3 C4) (n,q) : 1. for ll w V ( ) where v w, if there is some w V ( ) where v w then there is some suh w where either w = Q n 2 w or w = Q n 2 u. 2. for ll w V ( ) where v w, if there is some w where v w, then there is some suh w where either w = Q n 2 w or w = Q n 2 u. Proof. The proof of this is symmetril to the proof of Lemm 8 These lemms re suffiient to estlish finite gri struture, s epite in Figure 4. The formuls lk n wht re then lerly suffiient to enoe finite tiling, so if M, s = T ile Γ then y Lemm 3 Γ tiling exists. Rell (11) T ile Γ = lt T st K t (C1 C2 C3 C4 lk wht). e give the following Lemm. LEMMA 10. Suppose tht M, s = T ile Γ. Then for ll n ω, for Q = Γ {, } we my efine prtil funtion f : {0,..., n} 2 S suh tht: 1. f(0, 0) = s; 2. if f(i, j) V (), then () if i < n, there is some u V ( ) where f(i + 1, j) = Q k(i,j) u n f(i, j) u, n () if j < n, there is some u V ( ) where f(i, j + 1) = Q k(i,j) u n f(i, j) u; () M, f(i, j) = lk
Uneiility for ritrry puli nnounement logi 37 n 2 n 2 n 3 n 3 n 3 n 2 n 3 t t t t n 2 n 2 n 2 n 1 n 1 t t Figure 4. The onstrution of the finite gri. 3. if f(i, j) V ( ), then () if i < n, there is some u V () where f(i + 1, j) = Q k(i,j) u n f(i, j) u, n () if j < n, there is some u V () where f(i, j + 1) = Q k(i,j) u n f(i, j) u; () M, f(i, j) = wht where k(i, j) = 2n + 3 (i + j). e n show this y onstrution, pplying Lemms 7, 8 n 9. The funtion k(i, j) is hosen suh tht for ll i, j n, k(i, j) 3. This llows the preonitions of the Lemms 7, 8 n 9 to e met for ll i, j n. The proof is illustrte in Figure 4. You n view this figure s ue, ut in hlf igonlly up from ottom orner. The se of the shpe mkes finite gri. e onstrut funtion, f, mpping {0,..., n} 2 to the sttes of the moel, suh tht f(i, j) is k(i, j)-q-isimilr to the orresponing worl t the se of the gri. As we get further from the orner k(i, j) ereses so this orresponene eomes progressively weker. y the time i + j > 2n, k(i, j) < 3 so the preonitions for the neessry lemms is not met. However, y this stge we hve lrey efine n n n gri. Proof. e onstrut the funtion, F, stisfying the stte properties y inution over i + j, where i + j 2n. For the inution hypothesis we ssume for ll g, h where g + h < i + j, f(g, h) is efine suh tht:
38 Tim Frenh n Hns vn Ditmrsh 1. if f(g, h) V (), then () if 0 < g < n, then f(g 1, h) V ( ) n for some u V () we hve f(g, h) = Q k(g,h) u n f(g 1, h) u, n () if 0 < h < n, then f(g, h 1) V ( ) n for some u V () we hve f(g, h) = Q k(g,h) u n f(g, h 1) u; 2. if f(g, h) V ( ), then () if 0 < g < n then f(g 1, h) V () n for some u V ( ) we hve f(g, h) = Q k(g,h) u n f(g 1, h) u, n () if 0 < h < n then f(g, h 1) V () n for some u V ( ) we hve f(g, h) = Q k(g,h) u n f(g, h 1) u; 3. there is some u where f(g, h) = Q k(g,h) u n (12) M, u = (lt T K t (C1 C2 C3 C4 lk wht)) (k(g,h),q). For revity let, (13) Hyp = lt T K t (C1 C2 C3 C4 lk wht). For the se se of this inution, suppose M, s = T ile Γ, n let n ω. e efine f(0, 0) = s. Then 1. s = k(0,0) f n (0, 0), 2. M, s = lt T, 3. M, s = C1 C2 C3 C4, so it lerly stisfies the inutive hypothesis. For the inution, suppose tht the inutive hypothesis hols for the pir i, j. There re three ses to onsier, i = 0, j = 0 n i, j 0. 1. if i = 0 we my ssume j 0 (sine f(0, 0) is lrey efine). y the inution hypothesis, f(i, j 1) is efine. e suppose, without loss of generlity, tht f(i, j 1) V () n the se of f(i, j 1) V ( ) my e hnle similrly. Also y the inution hypothesis, there is some u = Q k(i,j 1) f(i, j 1), where M, u = Hyp(k(i,j 1),Q). y Lemm 6 we hve M, f(i, j 1) = L. Therefore there is some worl v V ( ) where f(i, j 1) v n we let f(i, j) = v. y Lemm 7 there is some worl w t u suh tht w = Q k(i,j) v n sine = Hyp K t Hyp, M, w = Hyp (k(i,j),q) s require. 2. the se for j = 0 is symmetri to the se ove.
Uneiility for ritrry puli nnounement logi 39 3. if i, j 0 suppose, without loss of generlity, tht f(i, j) V () (n the se for f(i 1, j 1) V ( ) is hnle similrly). y the inution hypothesis for some u = Q k(i 1,j 1) f(i 1, j 1), we hve M, u = Hyp (k(i 1,j 1),Q), u v for some v = Q k(i 1,j) f(i 1, j) V ( ), n u v for some v = k(i,j 1) f(i, j 1). y Lemm 8, either: () there is some w, w V () suh tht v w, v w n w = Q k(i,j) w. In suh se we let f(i, j) = w; or () there is some w V () where v w n w = Q k(i,j) u. In this se we let f(i, j) = u. y Lemm 8 we lso hve for ll w V () where v w there is some w V () where v w n w = Q k(i,j) u. Also y the inution hypothesis we hve M, v = Hyp (f(i 1,j),Q), so we my pply Lemm 7 to show tht there is some worl z suh tht z = k(i,j) f n (i, j) n v t z. As the formul (14) Hyp (f(i 1,j),Q) K t (Hyp (f(i 1,j),Q) ) is tutology of epistemi logi n M, v = Hyp (f(i 1,j),Q) we hve M, z = (lt T K t (C1 C2 C3 C4 lk wht)) (k(i,j),q), for some z = k(i,j) f n (i, j), s require. Therefore, the inution hypothesis hols for the pir (n, n). Thus for ll (i, j) where i + j < 2n we hve 1. if f(i, j) V ( ), then () f(i+1, j) V () n for some u V ( ) we hve f(i, j) = Q k(i,j) u n f(i + 1, j) u, n () f(i, j + 1) V () n for some u V ( ) we hve f(i, j) = Q k(i,j) u n f(i, j + 1) u; () M, f(i, j) = wht 2. if f(i, j) V (), then () f(i+1, j) V ( ) n for some u V () we hve f(i, j) = Q k(i,j) u n f(i + 1, j) u, n () f(i, j + 1) V ( ) n for some u V () we hve f(i, j) = Q k(i,j) u n f(i, j + 1) u; () M, f(i, j) = lk so f stisfies the require properties.
40 Tim Frenh n Hns vn Ditmrsh Note tht this lemm only efines n n n gri, sine s Lemms 6, 8 n 9 re use the inution n Lemm 6 is only ville when k(i, j) > 2, n Lemms 8 n 9 re only ville when k(i, j) > 1. However, n is hosen ritrrily. euse f(0, 0) = Q n f(0, 0) for ll n, we n see the inution with n ritrrily lrge n. This llows us to pply Lemm 3 to efine tiling. COROLLARY 11. If M, u = T ile Γ then Γ-tiling exists Proof. If M, u = T ile Γ, then y Lemm 10, for ll n we my efine f suh tht for ll i, j where i + j < 2n, 1. if f(i, j) V () then () if j < n, f(i, j) u for some u = Q k(i,j+1) f(i, j + 1). () if i < n, f(i, j) u for some u = Q k(i,j+1) f(i + 1, j). () M, f(i, j) = lk. 2. if f(i, j) V ( ) then () if j < n, f(i, j) u for some u = Q k(i,j+1) f(i, j + 1). () if i < n, f(i, j) u for some u = Q k(i,j+1) f(i + 1, j). () M, f(i, j) = wht. Rell the formuls: (15) (16) lk = γ Γ wht = γ Γ Applying the semntis of epistemi logi: ( γ [ K ( γ t =δ δ)) K ( γ r =δ l δ)) ( γ [ K ( γ t =δ δ)) K ( γ r =δ l δ)) ]) ]). 1. if f(i, j) V (), n M, f(i, j) = γ then for some u = Q k(i,j+1) f(i, j + 1) we hve f(i, j) u n u V ( ). Therefore M, u = δ for some δ where γ t = δ. Thus M, f(i + 1, j) = δ for some δ where γ t = δ. Likewise for some u = Q k(i,j+1) f(i + 1, j) we hve f(i, j) u n u V ( ), so M, f(i, j + 1) = δ for some δ where γ r = δ l. 2. if f(i, j) V ( ), n M, f(i, j) = γ then for some u = Q k(i,j+1) f(i, j+ 1) we hve f(i, j) u n u V (). Therefore M, u = δ for some δ where γ t = δ. Thus M, f(i + 1, j) = δ for some δ where γ t = δ. Likewise for some u = Q k(i,j+1) f(i + 1, j) we hve f(i, j) u n u V (), so M, f(i, j + 1) = δ for some δ where γ r = δ l.
Uneiility for ritrry puli nnounement logi 41 Therefore f efines tiling of n n n gri with the tiles of Γ. Therefore the formul T ile Γ is stisfile if n only if Γ n tile the n n gri for ritrry n. Applying Lemm 3 it follows γ-tiling exists. Thus the stisfiility prolem AP AL is o-re hr, s it is le to eme the tiling prolem. As we know from [2] tht the set of vli formuls for AP AL is reursively enumerle, it follows tht the stisfiility prolem is o-re omplete. 6 Future work Up to this point ritrry puli nnounement logi hs shown some promise for prtil resoning pplitions: it hs n xiomtiztion, tleu-lulus, it is isimultion invrint, nturlly extens epistemi logi, n moel heking is PSPACE-omplete [1]. The notion of n ritrry puli nnounement is lso nturl onept (onsier the ple, is there nything I n sy to mke you elieve X ). Therefore, nturl venue of investigtion is to onsier whether we my e le to somehow expressively weken AP AL to eile logi whih lso enjoys ll these fvorle properties. One re of investigtion my e to onsier the set of formule (or more strtly, moel-properties) tht nnounements my rnge over. From the enoing T ile Γ we hve given, we note tht the enoing of the tiling prolem for Γ only requires five gents in the lnguge (lthough it is oneivle more omplex enoing oul o with less). Also, from the proof of orretness we hve given, we note tht the ritrry nnounements o not nee to rnge over ll epistemi formule. The nnounements only nee to rnge over ll formule ontining the toms from Γ {, } (or the toms ppering in the formul). However, the proof oes require tht the ritrry nnounements rnge over formule of unoune epth, so it my e of interest to onsier restritions where the ritrry nnounements rnge over nnounements of oune epth (sy, formuls of epth t most one). e lso note tht generlizing the set of formule tht the ritrry nnounements rnge over woul not ffet this uneiility result. For exmple, if we llowe fixe-point opertors to pper in the ritrry nnounements, then the proofs of Lemms 7 n 8 woul remin. However we my onsier restritions of the set of formule. One nturl restrition to onsier woul e to restrit ritrry nnounements to positive knowlege formule (formule where the knowlege opertors K x lwys pper in the sope of n even numer of negtions). For suh restrite set of formule Lemm 5 woul not hol so eiility my still e hievle. An lterntive pproh is isusse [10]. Here it is suggeste tht informtive events suh s nnounements, rther thn simply eing evlute with respet to the given moel, oul itionl informtion (t n tomi level) into the moel. This pproh is motivte y the oservtion tht while the AP AL is isimultion invrint, it is not the se tht two
42 Tim Frenh n Hns vn Ditmrsh moels isimilr with respet to suset of toms, X, gree on ll formuls tht ontin only the toms X. The suggeste Future event logi quntifies over refinements of moel, whih inlues ll puli nnounements of epistemi formule, ut lso other puli or non-puli informtive events tht n e esrie s tion moels [3]. It is shown to e eile vi reution to isimultion quntifie logi [5]. Future event logi is interesting in its own right, n it ims to provie the sort of link etween ynmi n temporl epistemis pointe out in [9] Aknowlegements. The uthors woul like to thnk the nonymous reviewers for their omments n helpful suggestions. Hns vn Ditmrsh knowleges support of the Netherlns Institute of Avne Stuy where he ws Lorentz Fellow in 2008. ILIOGRAPHY [1] P. lini, A. ltg, H.P. vn Ditmrsh, A. Herzig, T. Hoshi, n T. De Lim. ht n we hieve y ritrry nnounements? A ynmi tke on Fith s knowility. In D. Smet, eitor, Proeeings of TARK XI, pges 42 51, Louvin-l-Neuve, elgium, 2007. Presses Universitires e Louvin. [2] P. lini, H. vn Ditmrsh, A. Herzig, n T. e Lim. A tleu metho for puli nnounement logis. In Proeeings of the Interntionl Conferene on Automte Resoning with Anlyti Tleux n Relte Methos, volume 4548 of Leture Notes in Computer Siene, pges 43 59. Springer, 2007. [3] A. ltg n L.S. Moss. Logis for epistemi progrms. Synthese, 139:165 224, 2004. Knowlege, Rtionlity & Ation 1 60. [4] R. Fgin, J. Hlpern, Y. Moses, n M. Vri. Resoning out knowlege. MIT Press, 1995. [5] T. Frenh. isimultion quntifiers for mol logi. PhD thesis, The University of estern Austrli, 2006. Aville from http://people.sse.uw.eu.u/tim/. [6] D. Hrel. Effetive trnsformtions on infinite trees, with pplitions to high uneiility, ominoes, n firness. J. A.C.M., 33(1):224 248, 1986. [7] J.S. Miller n L.S. Moss. The uneiility of iterte mol reltiviztion. Stui Logi, 79(3):373 407, 2005. [8] E. Spn. Complexity of Mol Logis. PhD thesis, Universiteit vn Amsterm, 1993. [9] J.F.A.K. vn enthem n E. Puit. The tree of knowlege in tion: towrs ommon perspetive. Avnes in Mol Logi, 6:87 106, 2006. [10] H.P. vn Ditmrsh n T. Frenh. Simultion n informtion. (Eletroni) Proeeings of LOFT 2008, Amsterm, http://www.ill.uv.nl/loft2008/ listofepteppers.html, 2008. [11] H.P. vn Ditmrsh,. vn er Hoek, n.p. Kooi. Dynmi Epistemi Logi, volume 337 of Synthese Lirry. Springer, 2007. Hns vn Ditmrsh Computer Siene, University of Otgo, New Zeln & IRIT, Frne hns@s.otgo..nz Tim Frenh Shool of Computer Siene n Softwre Engineering, The University of estern Austrli, Austrli tim@sse.uw.eu.u