Complexity Results in Epistemic Planning

Proceedings of the Twenty-Fourth Interntionl Joint Conference on Artificil Intelligence (IJCAI 2015) Complexity Results in Epistemic Plnning Thoms Bolnder DTU Compute Tech. University of Denmrk Copenhgen, Denmrk too@dtu.dk Mrtin Holm Jensen DTU Compute Tech. University of Denmrk Copenhgen, Denmrk mrtin.holm.j@gmil.com Frncois Schwrzentruer IRISA ENS Rennes Rennes, Frnce frncois.schwrzentruer@ens-rennes.fr Astrct Epistemic plnning is very expressive frmework tht extends utomted plnning y the incorportion of dynmic epistemic logic (DEL). We provide complexity results on the pln existence prolem for multi-gent plnning tsks, focusing on purely epistemic ctions with propositionl preconditions. We show tht moving from epistemic preconditions to propositionl preconditions mkes it decidle, more precisely in EXPSPACE. The pln existence prolem is PSPACE-complete when the underlying grphs re trees nd NP-complete when they re chins (including singletons). We lso show PSPACE-hrdness of the pln verifiction prolem, which strengthens previous results on the complexity of DEL model checking. 1 Introduction An ll-pervding focus of rtificil intelligence (AI) is the development of rtionl, utonomous gents. An importnt trit of such n gent is tht it is le to exhiit gol-directed ehviour, nd this overrching im is wht is studied within the field of utomted plnning. At the sme time, such gol-directed ehviour will nturlly e confined to whtever model of the underlying domin is used. In utomted plnning the domin models employed re formulted using propositionl logic, ut in more complex settings (e.g. multigent domins) such models come up short due to the limited expressive power of propositionl logic. By extending (or replcing) this foundtionl uilding lock of utomted plnning we otin more expressive formlism for studying nd developing gol-directed gents, enling for instnce n gent to reson out other gents. For the ove resons utomted plnning hs recently seen n influx of formlisms tht re colloquilly referred to s epistemic plnning [Bolnder nd Andersen, 2011; Löwe et l., 2011; Aucher nd Bolnder, 2013; Yu et l., 2013; Andersen et l., 2012]. Common to these pproches is tht they tke dynmic epistemic logic (DEL) [Bltg et l., 1998] s the sic uilding lock of utomted plnning, which gretly surpsses propositionl logic in terms of expressive power. Briefly put, DEL is modl logic with which we cn reson out the dynmics of knowledge. In the singlegent cse, epistemic plnning cn cpture non-deterministic nd prtilly oservle domins [Andersen et l., 2012]. An even more interesting feture of DEL is the inherent ility to reson out multi-gent scenrios, lending itself perfectly to nturl descriptions of multi-gent plnning tsks. In [Bolnder nd Andersen, 2011] it is shown tht the pln existence prolem (i.e. deciding whether pln exists for multi-gent plnning tsk) is undecidle, nd this remins so even when fctul chnge is not llowed, tht is, when we only llow ctions tht chnges eliefs, not ontic fcts [Aucher nd Bolnder, 2013]. Allowing for fctul chnge, decidle frgment is otined y restricting epistemic ctions to only hve propositionl preconditions [Yu et l., 2013] (in the full frmework, preconditions of ctions cn e ritrry epistemic formuls). The computtionl complexity of this frgment elongs to (d + 1)-EXPTIME for gol whose modl depth is d [Muert, 2014]. In this work we consider exclusively the pln existence prolem for clsses of plnning tsks where preconditions re propositionl (s in most utomted plnning formlisms) nd ctions re non-fctul (chnging only eliefs). We show this prolem to e in EXPSPACE in the generl cse, ut lso identify frgments with tight complexity results. We do so y using the notion of epistemic ction stilistion [vn Benthem, 2003; Miller nd Moss, 2005; Sdzik, 2006], which llows us to put n upper ound on the numer of times n ction needs to e executed in pln. This numer depends crucilly on the structurl properties of the grph underlying the epistemic ction. To chieve our upper ound complexity results we generlise result of [Sdzik, 2006] on ction stilistion. We lso tckle lower ounds, therey showing cler computtionl seprtion etween these frgments. Our contriutions to the complexity of the pln existence prolem re summrised in Tle 1 (second column from the left), where we ve lso listed relted contriutions. The frgments we study hve oth conceptul nd technicl motivtion. Singleton epistemic ctions correspond to pulic nnouncements of propositionl fcts, chins nd trees to certin forms of privte nnouncements, nd grphs cpture ny propositionl epistemic ction. Possile pplictions of such plnning frgments could e.g. e plnning in gmes like Clue/Cluedo where ctions cn e seen s purely epistemic; or synthesis of protocols for secure communiction (where 2791

Underlying grphs of ctions SINGLETONS CHAINS TREES GRAPHS Non-fctul, propositionl preconditions NP-complete (Theorem 5.1) NP-complete (Theorem 5.2) PSPACE-complete (Theorem 5.3, Theorem 5.4) in EXPSPACE (Theorem 5.8) Types of epistemic ctions Fctul, propositionl preconditions PSPACE-hrd [Jensen, 2014] in NON- ELEMENTARY [Yu et l., 2013] Fctul, epistemic preconditions PSPACE-hrd [Jensen, 2014] Undecidle [Bolnder nd Andersen, 2011] Tle 1: Complexity results for the pln existence prolem. the gol specifies who re llowed to know wht). Techniclly our frgments increse in complexity s we loosen the restrictions put on the underlying grph. Some plnning tsks might therefore e simplified during preprocessing phse so tht etter upper ound cn e gurnteed. As cse in point, we cn preprocess ny plnning tsk nd replce ech grph ction with tree ction tht is equivlent up to predetermined modl depth k (y unrvelling). Letting k e the modl depth of the gol formul we otin n equivlent plnning tsk tht cn e solved using t most spce polynomil in k nd the size of the plnning tsk. In utomted plnning such preprocessing is often used to chieve sclle plnning systems. Our results lso llow us to prove tht the pln verifiction prolem ( suprolem of DEL model checking) is PSPACEhrd, even without non-deterministic union, therey improving the result of [Aucher nd Schwrzentruer, 2013]. Sections 2 nd 3 present the core notions from epistemic plnning. In Section 4 we improve the known upper ounds on the stilistion of epistemic ctions. This is put to use in Section 5 where we show novel results on the complexity of the pln existence prolem. Section 6 presents our improvement to the pln verifiction prolem, efore we conclude nd discuss future work in Section 7. 2 Bckground on Epistemic Plnning For the reminder of the pper we fix oth n infinitely countle set of tomic propositions P nd finite set of gents Ag. 2.1 Dynmic epistemic logic Definition 2.1 (Epistemic models nd sttes). An epistemic model is triple M = (W, R, V ) where the domin W is non-empty set of worlds; R : Ag 2 W W ssigns n epistemic (ccessiility) reltion to ech gent; nd V : P 2 W ssigns vlution to ech tomic proposition. We write R for R() nd wr v for (w, v) R. We often write W M for W, R M for R nd V M for V. For w W, the pir (M, w) is clled n epistemic stte whose ctul world is w. (M, w) is finite when W is finite. Epistemic sttes re typiclly denoted y symols such s s nd s 0. The lnguge of propositionl logic over P is referred to s L Prop, or sometimes simply the propositionl lnguge. Definition 2.2 (Propositionl ction models nd epistemic ctions). A propositionl ction model is triple A = (E, Q, pre) where E is non-empty nd finite set of events clled the domin of A; Q : Ag 2 E E ssigns n epistemic (ccessiility) reltion to ech gent; nd pre : E L Prop ssigns precondition of the propositionl lnguge to ech event. We write Q for Q() nd eq f for (e, f) Q. We often write E A for E, Q A for Q nd pre A for pre. For e E, the pir (A, e) is clled n epistemic ction whose ctul event is e. Epistemic ctions re typiclly denoted α, α, α 1, etc. Propositionl ction models re defined to fit exctly our line of investigtion here, though other presenttions consider preconditions of more complex lnguges nd postconditions tht llow for fctul (ontic) chnge [Bolnder nd Andersen, 2011; Yu et l., 2013]. The dynmic lnguge L D is generted y the BNF: ϕ := p ϕ (ϕ ϕ) ϕ α ϕ where Ag, p P nd α is n epistemic ction. Here denotes the knowledge (or, elief) modlity where ϕ reds s knows (or, elieves) ϕ, nd α is the dynmic modlity where α ϕ reds s α is pplicle nd ϕ holds fter executing α. The epistemic lnguge L E is the sulnguge of L D tht does not contin the dynmic modlity. As usul we use ϕ := ϕ, nd define y revition, nd the oolen connectives,,. Lstly, we define α 0 ϕ := ϕ nd α k ϕ := α ( α k 1 ϕ) for k > 0. Definition 2.3 (Semntics). Let (M, w) e n epistemic stte where M = (W, R, V ). For Ag, p P nd ϕ, ϕ L D we inductively define truth of formuls s follows, omitting the propositionl cses: (M, w) = ϕ (M, w) = α ϕ iff (M, v) = ϕ for ll wr v iff (M, w) = pre(e) nd (M A, (w, e)) = ϕ where α = (A, e) is n epistemic ction s.t. A = (E, Q, pre), nd the epistemic model M A = (W, R, V ) is defined vi the product updte opertor y: W = {(v, f) W E (M, v) = pre(f)}, R = {((v, f), (u, g)) W W vr u, fq g}, V (p) = {(v, f) W v V (p)} for p P. For ny epistemic stte s = (M, w) nd epistemic ction α = (A, e) stisfying (M, w) = pre A (e), we define s α = (M A, (w, e)). The epistemic stte s α represents the result of executing α in s. Note tht we hve s = α ϕ iff (M, w) = pre A (e) nd s α = ϕ. Two formuls ϕ, ϕ of L D re clled equivlent (written s ϕ ϕ ) when s = ϕ iff s = ϕ for every epistemic stte s. Exmple 2.4. Consider the epistemic stte s 1 of Figure 1. It represents sitution where p holds in the ctul world (w), ut where the two gents, nd, don t know this: s 1 = p p p. Consider now the epistemic ction α 1 = (A, e) of the sme figure. It represents privte nnouncement of p to gent, tht is, gent is told tht p holds (the ctul event, e), ut gent thinks tht nothing is 2792

s 1,,, w : p v : α 1, e: p f : (w, e): p (w, f): p, (v, f):,, s 1 α 1 Figure 1: (Top left) An epistemic stte s 1. We mrk ech world (circle) with its nme nd the tomic propositions tht re true. The ctul world is coloured lck. Edges show epistemic reltions of the gents. (Bottom left) An epistemic ction α 1. We use the sme conventions s for epistemic sttes, except n event (squre) is mrked y its nme nd its precondition. (Right) The epistemic model to the right is the result of execution of α 1 in s 1, tht is, s 1 α 1. hppening (event f). The dynmic modlity llows us to reson out the result of executing α 1 in s 1, so for instnce we hve s 1 = α 1 ( p p p): After gent hs een privtely informed out p, she will know it, ut will still not know p, nd will elieve tht doesn t either. This fct cn e verified y oserving tht p p p is true in the epistemic stte s 1 α 1 of Figure 1. 2.2 Pln existence prolem Definition 2.5 (Plnning tsks). An (epistemic) plnning tsk is triple T = (s 0, L, ϕ g ) where s 0 is finite epistemic stte clled the initil stte, L is finite set of epistemic ctions clled the ction lirry nd ϕ g L E is clled the gol. A pln for T is finite sequence α 1,..., α j of epistemic ctions from L s.t. s 0 = α 1 α j ϕ g. The sequence α 1,..., α j cn contin ny numer of repetitions, nd cn lso e empty. We sy tht T is solvle if there exists pln for T. The size of plnning tsk T = (s 0, L, ϕ g ) is given s follows. Following [Aucher nd Schwrzentruer, 2013], for ny α = (A, e) in L we define α = Ag E A 2 + e E pre(e) s the size of α, where pre(e) denotes the length of the (propositionl) formul pre(e). The size of n epistemic ction is lwys finite numer, since the domin of ny propositionl ction model nd Ag re oth finite. Let P P e the finite set of tomic propositions tht occur either in some precondition of n α L or in ϕ g. The size T is then T = P Ag W M 2 + α L α + ϕ g where s 0 = (M, w). Note tht pln is nothing more thn sequence of epistemic ctions leding to gol. It is not hrd to show tht this definition is equivlent to the definition of solution [Aucher nd Bolnder, 2013] nd n explntory dignosis [Yu et l., 2013], which re oth specil cses of solution to clssicl plnning tsk s defined in [Ghll et l., 2004] (for the reltion to clssicl plnning tsks, see [Aucher nd Bolnder, 2013]). Exmple 2.6. Consider gin Figure 1. We ll use α 2 to refer to the privte nnouncement of p to, otined simply y swpping the epistemic reltions of nd in α 1. Consider the plnning tsk T = (s, {α 1, α 2 }, ϕ g ) with ϕ g = p p p p. It is plnning tsk in which the only ville ctions re privte nnouncements of p to either or, nd the gol is for oth nd to know p, ut without knowing tht the other knows. A pln for T is α 1, α 2, since s = α 1 α 2 ϕ g. In other words, first nnouncing p privtely to nd then privtely to will chieve the gol of them oth knowing p without knowing tht ech other knows. Definition 2.7 (Pln existence prolem). Let X denote clss of plnning tsks. The pln existence prolem for X, clled PLANEX(X) is the following decision prolem: Given plnning tsk T X, does there exists pln for T 3 Bckground on Iterting Epistemic Actions To get to grips on the pln existence prolem, we now consider the result of iterting single epistemic ction. We then proceed to derive useful chrcteristion of exctly when plnning tsk is solvle. Definition 3.1 (n-ry product). Let α = (A, e) e n epistemic ction where A = (E, Q, pre). We denote y A n = (E n, Q n, pre n ) the n-ry product of A. We define E 0 = {e}, eq 0 e for ech Ag, nd pre 0 (e) =. For n > 0 we define E n = {(e 1,..., e n ) e i E for ll i = 1,..., n}, Q n = {((e 1,..., e n ), (f 1,..., f n )) e i Q f i for ll i = 1,..., n} for ech Ag, nd pre n ((e 1,..., e n )) = i=1,...,n pre(e i). The n-ry product of α is defined s α n = (A n, e n ), where e n denotes (e, e,..., e). } {{ } n This is not the stndrd definition of the n-ry product of n ction model, which insted goes vi definition of the product updte opertor on ction models. Definition 3.1 is equivlent to the stndrd definition when preconditions re of L Prop. The following lemm is derived from the xiomtiztion of [Bltg et l., 1998] (relying in prticulr on ction composition), nd is here stted for the cse of the n-ry product nd utilising tht preconditions re of L Prop. Lemm 3.2. For ny epistemic ction α nd ny ϕ L E we hve tht α n ϕ α n ϕ. This lemm expresses tht executing n epistemic ction n times is equivlent to executing its n-ry product once. 3.1 Bisimilrity nd Stilistion Concerning n-ry products of epistemic ctions, n interesting cse is when executing the n-ry product is equivlent to executing the (n + 1)-ry product. This puts n upper ound on the numer of times the ction needs to occur in pln since epistemic ctions with propositionl preconditions commute [Löwe et l., 2011]. To nlyse this, we introduce notions of isimultion nd n-isimultion on ction models (slightly reformulted from [Sdzik, 2006]). Definition 3.3 (Bisimilrity). Two epistemic ctions α = (A, e) nd α = (A, e ) re clled isimilr, written α α, if there exists (isimultion) reltion Z E A E A contining (e, e ) nd stisfying for every Ag: 2793

[tom] If (f, f ) Z then pre A (f) pre A (f ), [forth] If (f, f ) Z nd fq A g then there is g E A such tht f Q A g nd (g, g ) Z, nd [ck] If (f, f ) Z nd f Q A g then there is g E A such tht fq A g nd (g, g ) Z. Definition 3.4 (n-isimilrity). Let α = (A, e) nd α = (A, e ) e epistemic ctions. They re 0-isimilr, written α 0 α, if pre A (e) pre A (e ). For n > 0, they re n- isimilr, written α n α, if for every Ag: [tom] pre A (e) pre A (e ), [forth] If eq A f then there is n f E A such tht e Q A f nd (A, f) n 1 (A, f ), nd [ck] If e Q A f then there is n f E A such tht eq A f nd (A, f) n 1 (A, f ). The modl depth md(ϕ) of formul ϕ is defined s: md(p) = 0; md( ϕ) = md(ϕ); md(ϕ ψ) = mx{md(ϕ), md(ψ)}; md( ϕ) = 1 + md(ϕ); md( α ϕ) = md(ϕ). As epistemic ctions hve only propositionl preconditions, α -opertors do not count towrds the modl depth. This definition of modl depth, Lemm 3.5 nd Definition 3.6 re ll due to [Sdzik, 2006] (slightly reformulted). Lemm 3.5. Let α, α e two epistemic ctions nd ϕ L D. 1) If α α, then α ϕ α ϕ. 2) If md(ϕ) n nd α n α, then α ϕ α ϕ. Definition 3.6 (Stilistion). Let α e n epistemic ction. 1) α is -stilising t stge i if α i α i+k for ll k 0. 2) α is n -stilising t stge i if α i n α i+k for ll k 0. Exmple 3.7. The 2-ry product α 2 1 of α 1 of Figure 1 is: (e, e): p p (f, f):, (f, e): p (e, f): p It is esy to check tht α 1 α 2 1, using Z = {(e, (e, e)), (f, (f, f))}, This rgument cn e extended to show tht α 1 is indeed -stilising t stge 1. Since ny epistemic ction is finite, we hve: Lemm 3.8. If two epistemic ctions re n-isimilr for ll n, then they re isimilr. 3.2 Bounding the Numer of Itertions We re now redy to present our chrcteristion of when plnning tsk is solvle. We note tht Proposition 3.9 elow echoes the sentiment of [Yu et l., 2013, Theorem 5.15], in tht it sttes the conditions under which we cn restrict the serch spce when looking for pln. Proposition 3.9. Let T = (s 0, {α 1,..., α m }, ϕ g ) e plnning tsk nd B N. Suppose one of the following holds: 1) Every α i is -stilising t stge B, or 2) md(ϕ g ) = n nd every α i is n -stilising t stge B. Then T is solvle iff there exists k 1,..., k m B s.t. s 0 = α k1 1 αkm m ϕ g. Proof. Assume 2) holds (the cse of 1 is similr). Assume T is solvle, nd let α i1,..., ij e pln for T. Due to commuttivity of propositionl ction models so is ny permuttion of α i1,..., α ij [Yu et l., 2013]. We therefore hve s 0 = α 1 k 1 αm k m ϕg for some choice of k i 0. Using Lemm 3.2, it follows tht s 0 = α k 1 1 αk m ϕ g. We now let k i = min(k i, B) for ll i. By ssumption, md(ϕ g) = n nd so y definition md( α ϕ g ) = n for ny epistemic ction α. Comining this with the ssumption tht every α i is n - stilising t stge B k i, we pply 2) of Lemm 3.5 m times to conclude tht s 0 = α k1 1 αkm m ϕ g, s required. The proof of the other direction follows redily from Lemm 3.2 nd the definition of α k. Let T = {s 0, L, ϕ g } e plnning tsk with md(ϕ g ) = n. Given the proposition ove, to show tht T is solvle we only need to find the correct numer of times to iterte ech of the ctions in L, nd these numers never hve to exceed B for ctions tht re n -stilising t stge B. The following result, due to [Sdzik, 2006], shows tht such ound B exists for ny epistemic ction. Lemm 3.10. Let α = (A, e) e n epistemic ction nd n nturl numer. Then α is n -stilising t stge E A n. 4 Better Bounds for Action Stilistion In this section, we prove n originl contriution, Lemm 4.2, tht generlises Sdzik s Lemm 3.10 y giving etter ound for ction stilistion. The overll point is this: Sdzik gets n unnecessrily high upper ound on when n epistemic ction (A, e) stilises y considering it possile tht ny event cn hve up to E A successors. We get etter ound y counting pths. Definition 4.1 (Underlying grphs). Let (A, e) e n epistemic ction. We define Q A = Ag Q A. The underlying grph of (A, e) is the directed grph (A, Q A ) with root e. Let (A, e) denote n epistemic ction. Note tht (e, f) Q A iff there is n edge from e to f in A lelled y some gent. Stndrd grph-theoreticl notions crry over to epistemic ctions vi their underlying grphs. For instnce, we define pth of length n in (A, e) s pth of length n in the underlying grph, tht is, sequence (e 1, e 2,..., e n+1 ) of events such tht (e i, e i+1 ) Q A for ll i = 1,..., n (we llow n = 0 nd hence pths of length 0). A pth of length n is pth of length t most n. A mximl pth of length n is pth of length n tht is not strict prefix of ny other pth of length n. We use mpths n (e) to denote the numer of distinct mximl pths of length n rooted t e. If ll nodes hve successors, this numer is simply the numer of distinct pths of length n. Note tht mpths n (e) is lwys positive numer, s there is lwys t lest one pth rooted t e (even if e hs no outgoing edges, there is still the pth of length 0). Note lso tht for ny n > 0 nd ny event e hving t lest one successor in the underlying grph: 2794

mpths n (e) = eq A f mpths n 1(f). (1) In the epistemic ction α 1 of Figure 1 we hve mpths 2 (e) = 3, since there re three pths of length 2, (e, e, e), (e, e, f) nd (e, f, f), nd no shorter mximl pths. Lemm 4.2. Let α = (A, e 0 ) e n epistemic ction nd n ny nturl numer. Then α is n -stilising t stge mpths n (e 0 ). Proof. When f = (f 1,..., f m ) E Am nd e A, we use occ(e, f) to denote the numer of occurrences of e in f 1,..., f m. For instnce we hve occ(e, (e, e, f, f)) = 2. We now prove the following property P(n) y induction on n. P(n): If e E Ak+1 nd e E Ak only differ y some event e occurring t lest mpths n (e )+1 times in e nd t lest mpths n (e ) times in e, then (A k+1, e) n (A k, e ). Bse cse P(0): Since mpths 0 (e ) = 1, e nd e s descried ove must contin exctly the sme events (ut not necessrily with the sme numer of occurrences). By definition of the n-ry product of n epistemic ction we get pre Ak+1 (e) pre Ak (e ). This shows (A k+1, e) 0 (A k, e ). For the induction step, ssume tht P(n 1) holds. Given e nd e s descried in P(n), we need to show (A k+1, e) n (A k, e ). [tom] is proved s P(0). [forth]: Let nd f e chosen such tht eq Ak+1 f. We need to find f such tht e Q Ak f nd (A k+1, f) n 1 (A k, f ). Clim. There exists n f such tht e Q A f nd occ(f, f) mpths n 1 (f ) + 1. Proof of Clim. By contrdiction: Suppose occ(f, f) mpths n 1 (f) for ll f with e Q A f. Since eq Ak+1 f, the numer of occurrences of e in e is equl or less thn the numer of occurrences of Q -successors of e in f. Hence we get occ(e, e) e Q A f occ(f, f) e Q A f mpths n 1(f) (y ssumption) e Q A f mpths n 1(f) (y Q A = Ag Q A ) = mpths n (e ) (y eqution (1)). However, this directly contrdicts the ssumption tht e occurs t lest mpths n (e ) + 1 times in e, nd hence the proof of the clim is complete. Let f e s gurnteed y the clim. Now we uild f to e exctly like f, except we omit one of the occurrences of f (we do not hve to worry out the order of the elements of the vectors, since ny two vectors only differing in order re isimilr [Sdzik, 2006]). Since f nd f now only differ in f occurring t lest mpths n 1 (f ) + 1 times in f nd t lest mpths n 1 (f ) times in f, we cn use the induction hypothesis P(n 1) to conclude tht (A k+1, f) n 1 (A k, f ), s required. [ck]: This is the esy direction nd is omitted. Now we hve proved P(n) for ll n. Given n, from P(n) it follows tht (A k+1, e k+1 0 ) n (A k, e k 0) for ll k mpths n (e 0 ). And from this it immeditely follows tht (A, e 0 ) is n -stilising t stge mpths n (e 0 ). r : w 1 : p 1 s 0 w 2 : p 2 w 3 : p 3 w m : p n α i e i : p i Figure 2: Initil stte nd ctions used in Theorem 5.1. procedure PlnExists ( (s 0, {α 1,..., α m }, ϕ g ), B ) ) Guess vector (k 1,..., k m ) {0,..., B} m. ) Accept when s 0 = α 1 k1 α m km ϕ g. Figure 3: Non-deterministic lgorithm for the pln existence prolem. 5 Complexity of the Pln Existence Prolem 5.1 Singleton nd Chin Epistemic Actions We define SINGLETONS s the clss of plnning tsks (s 0, L, ϕ g ), where every α = (A, e) in L is singleton; i.e. E A contins single event. Theorem 5.1. PLANEX(SINGLETONS) is NP-complete. Proof. For ny singleton epistemic ction there is t most one mximl pth of length n for ll n. Hence, y Lemm 4.2 nd 3.8, such ctions re -stilising t stge 1. In NP: Follows from Theorem 5.2 elow s SINGLETONS is contined in CHAINS. NP-hrd: We give polynomil-time reduction from SAT. Let ϕ(p 1,..., p m ) e propositionl formul where p 1,..., p m re the tomic propositions in ϕ. We construct T = (s 0, {α 1,..., α m }, ϕ g ) s.t. s 0 nd ech α i re s in Figure 2 nd ϕ g = ϕ( p 1,..., p m ) is the formul ϕ in which ech occurrence of p i is replced y p i. For ny propositionl vlution ν, let s ν e the restriction of s 0 s.t. there is n -edge from r to w i in s ν iff ν = p i. This mens ν = p i iff s ν = p i, nd so from our construction of ϕ g we hve ν = ϕ iff s ν = ϕ g. Oserve now tht s 0 α i is exctly the restriction of s 0 so tht there is no - edge from r to w i, nd tht α i is the only ction ffecting this edge. Let ν(p i ) = 0 if ν = p i nd ν(p i ) = 1 otherwise. We now hve tht ϕ is stisfile iff there is ν s.t. ν = ϕ iff s ν = ϕ g iff s 0 = α 1 ν(p1) α m ν(pm) ϕ g iff T is solvle, where the lst equivlence follows from Proposition 3.9 since ν(p i ) {0, 1} nd ech α i is -stilising t stge 1. This shows tht ϕ is stisfile iff T is solvle. For n epistemic ction α = (A, e) we sy tht α is chin if its underlying grph (A, Q A ) is 1-ry tree whose unique lef my e Q A -reflexive. We define CHAINS s the clss of plnning tsks (s 0, L, ϕ g ) where every epistemic ction in L is chin. Theorem 5.2. PLANEX(CHAINS) is NP-complete. Proof. In NP: For ny chin epistemic ction there is t most one mximl pth of length n for ll n, hence ny such ction is -stilising t stge 1 using Lemms 4.2 nd 3.8. It therefore follows from Proposition 3.9 tht, for ny T CHAINS, PlnExists(T, 1) of Figure 3 is ccepting iff T is solvle. We must show step ) to run in polynomil 2795

time. Now if α is chin nd s n epistemic stte, then the numer of worlds rechle from the ctul world in s α is t most the numer of worlds in s. By only keeping the rechle worlds fter ech successive product updte, we get the required, s the gol is in L E. 1 NP-hrd: Follows from Theorem 5.1 s SINGLETONS is contined in CHAINS. 5.2 Tree Epistemic Actions We now turn to epistemic ctions whose underlying grph is ny tree. Formlly, n epistemic ction (A, e) is clled tree when the underlying grph (A, Q A ) is tree whose leves my e Q A -reflexive. We cll TREES the clss of plnning tsks (s 0, L, ϕ g ) where ll epistemic ctions in L re trees. Theorem 5.3. PLANEX(TREES) is in PSPACE. Proof. Consider ny tree ction α = (A, e) nd let l(α) denote its numer of leves. As α is tree, we get mpths n (e) l(α) for ny n. Using Lemm 4.2 nd 3.8, ny tree epistemic ction α is -stilising t stge l(α). From Proposition 3.9 we therefore hve, for ny T TREES, tht PlnExists(T, mx(l(α 1 ),..., l(α m ))) of Figure 3 is ccepting iff T is solvle. Step ) cn e done in spce polynomil in the size of the input [Aucher nd Schwrzentruer, 2013]. Hence, the pln existence prolem for TREES is in NPSPACE nd therefore in PSPACE y Svitch s Thm. We now sketch proof of PSPACE-hrdness of PLANEX(TREES), y giving polynomil-time reduction from the PSPACE-hrd prolem QSAT (stisfiility of quntified oolen formuls) to PLANEX(TREES). For ny quntified oolen formul Φ = Q 1 p 1 Q n p n ϕ [p 1,..., p n ] with Q i {, }, we define the plnning tsk T Φ = (s 0, {α 1,..., α n }, ϕ st ϕ ll ) where s 0 nd ech α i re s in Figure 4 (every edge implicitly lelled y ), ϕ st = O 1 O n ϕ[ 1,..., n ], nd ϕ ll = n+1 2n, where O i = if Q i = nd O i = if Q i =. Then T Φ is polynomil in Φ nd T Φ TREES. By Lemms 5.6 nd 5.7 elow we get T Φ is solvle iff Φ is true. Hence: Theorem 5.4. PLANEX(TREES) is PSPACE-hrd. s 0. w 0 : w 1 : w 2n+1 : α i c i 1 : c i i :. t i i : t i n 1 : t i n : t i n+1 : t i 2n+1 :. i 0 :.... i i 1 : f i i : f i n 1 : f i n : f i n+1 : f i 2n+1 : Figure 4: Initil stte nd ctions used in Theorem 5.4. 1 Oserve tht even if ech ction in α 1,..., α m is -stilising t stge 1, this is not sufficient condition for memership in NP s we must lso e le to verify the pln in polynomil time. (w 0, 1 0, 2 0 ) (w 1, t 1 1, 2 1 ) (w1, f 1 1, 2 1 ) (w 2, t 1 2, t2 2 ) (w2, t1 2, f 2 2 ) (w2, f 1 2, t2 2 ) (w2, f 1 2, f 2 2 ) Non c i i -chin c 1 1 -chin c 2 1, c2 2 -chin Figure 5: Binry decision tree simulted y s 0 α 1 α 2 (n = 2). The reduction is sed on the ide tht we cn simulte (complete) inry decision tree using s = s 0 α 1 α n. Ech world t depth n of s simultes vlution, using the convention tht p i is true iff there is mximl chin of length i in this world. By nesting elief modlities we cn check if such chin exists. Ech ction α i mkes two copies of every node etween depth i nd n, which is how we cn simulte every vlution. A world w t depth i n of s is clled n i-world. It cn now e verified tht ny i-world is of the form (w i, vi 1,..., vi i, i+1 i,..., n i ) where vj i {tj i, f j i }. See lso Figure 5. For ny i-world w, we define propositionl vlution ν w on {p 1,..., p i } y ν w = p j iff t j i occurs in w. We use w 0 = (w 0, 1 0,..., n 0 ) to denote the single 0-world in s (the ctul world of s ), nd define M so tht s = (M, w 0 ). Lemm 5.5. Let w e ny n-world. Then (M, w) = ϕ[ 1,..., n ] iff ν w = ϕ[p 1,..., p n ] is true. Proof sketch. Due to the c i 1,..., c i i chin in ech α i, we hve for ny n-world w nd i n tht (M, w) = i iff t i n occurs in w, from which the result redily follows. We sy tht n n-world w is ccepting if (M, w) = ϕ[ 1,..., n ], nd for i < n we sy tht the i- world w is ccepting if some (every) child w of w is ccepting nd O i = (O i = ). Lemm 5.6. T Φ is solvle iff w 0 is ccepting. Proof sketch. As cceptnce for i < n exctly corresponds to the O 1 O n prefix, we use Lemm 5.5 to show tht (M, w 0 ) = ϕ st iff w 0 is ccepting. Now we must show: 1) (M, w 0 ) = ϕ ll, nd then 2) T Φ is solvle iff α 1,..., α n is pln for T Φ. We omit proofs of oth 1) nd 2). Lemm 5.7. Φ is true iff w 0 is ccepting. Proof sketch. Let w denote ny i-world. Let ν w (p i ) = if ν w = p i nd ν w (p i ) = otherwise. We define Φ w = Q i+1 p i+1 Q n p n ϕ[ν w (p 1 ),..., ν w (p i ), p i+1,..., p n ]. By induction on k we now show: If k n nd w is n (n k)-world, then Φ w is true iff w is ccepting. For the se cse, k = 0 nd w is n n-world, hence ϕ[ν w (p 1 ),..., ν w (p n )] (= Φ w ) is true iff w is ccepting y Lemm 5.5. For the induction step we ssume tht for ny (n (k 1))-world w, Φ w is true iff w is ccepting. Let w e n (n k)-world. By construction, w hs two children v nd u. We cn then show tht Φ v nd Φ u re s Φ w, except the Q n k+1 p n k+1 prefix nd tht one sets p n k+2 true nd the other sets p n k+2 flse. Thus Φ w is true iff Q n k+1 = (or, Q n k+1 = ) nd Φ w is true for some (every) child w 2796

of w. Using the induction hypothesis, we get tht Φ w is true iff w is ccepting for some (every) child w of w. Hence, Φ w is true iff w is ccepting, y definition. This concludes the induction proof. For k = n it follows tht Φ w0 is true iff w 0 is ccepting. Since Φ = Φ w0 we re done. 5.3 Aritrry Epistemic Actions We cll GRAPHS the clss of plnning tsks (s 0, L, ϕ g ) where ll event models in L re ritrry grphs. In this cse, the originl result y Sdzik (Lemm 3.10) is sufficient. Theorem 5.8. PLANEX(GRAPHS) is in EXPSPACE. Proof. We consider (s 0, {α 1,..., α m }, ϕ g ) GRAPHS with α i = (A i, e i ) nd md(ϕ g ) = d. By Lemm 3.10, ech α i is d -stilising t stge E Ai d. It now follows from Proposition 3.9 tht PlnExists(T, mx{ E A1 d,..., E Am d }) of Figure 3 is ccepting iff T is solvle. The lgorithm runs in NEXPSPACE = EXPSPACE. 6 Complexity of the Pln Verifiction Prolem The pln verifiction prolem is defined s the following decision prolem: Given finite epistemic stte s 0 nd formul of the form α 1 α j ϕ g, does s 0 = α 1 α j ϕ g hold The pln verifiction prolem cn e seen s restriction of the model checking prolem in DEL. A similr reduction s for Theorem 5.4 gives tht: Theorem 6.1. The pln verifiction prolem (restricted to propositionl ction models tht re trees) is PSPACE-hrd. Model checking in DEL with the non-determinism opertor included in the lnguge hs lredy een proved PSPACE-hrd [Aucher nd Schwrzentruer, 2013]. Theorem 6.1 implies tht model checking in DEL is PSPACE-hrd even without this opertor. A similr result hs een independently proved in [vn de Pol et l., 2015]. 7 Future Work We remind the reder tht n overview of our contriutions re found in Tle 1 nd proceed to discuss future work. Since propostionl STRIPS plnning is PSPACE-complete [Bylnder, 1994], efficient plnning systems hve used relxed plnning tsks in order to efficiently compute precise heuristics. For instnce, the highly influentil Fst-Forwrd plnning system [Hoffmnn nd Neel, 2001] relxes plnning tsks y ignoring delete lists. Our contriutions here show tht restrictions on the grphs underlying epistemic ctions crucilly ffect computtionl complexity. This, in comintion with restrictions on preconditions nd postconditions (fctul chnge), provides pltform for investigting (trctle) relxtions of epistemic plnning tsks, nd hence for the development of efficient epistemic plnning systems. References [Andersen et l., 2012] Mikkel Birkegrd Andersen, Thoms Bolnder, nd Mrtin Holm Jensen. Conditionl epistemic plnning. In Luis Friñs del Cerro, Andres Herzig, nd Jérôme Mengin, editors, JELIA, volume 7519 of Lecture Notes in Computer Science, pges 94 106. Springer, 2012. 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Journl of Artificil Intelligence Reseserch, 14(1):253 302, My 2001. [Jensen, 2014] Mrtin Holm Jensen. Epistemic nd Doxstic Plnning. PhD thesis, Technicl University of Denmrk, 2014. DTU Compute PHD-2014. [Löwe et l., 2011] Benedikt Löwe, Eric Pcuit, nd Andres Witzel. Del plnning nd some trctle cses. In Hns vn Ditmrsch, Jérôme Lng, nd Shier Ju, editors, Logic, Rtionlity, nd Interction, volume 6953 of Lecture Notes in Computer Science, pges 179 192. Springer Berlin Heidelerg, 2011. [Muert, 2014] Bstien Muert. Logicl foundtions of gmes with imperfect informtion: uniform strtegies. PhD thesis, 2014. PhD thesis directed y Pinchint, Sophie nd Aucher, Guillume. Université de Rennes 1. [Miller nd Moss, 2005] Joseph S Miller nd Lwrence S Moss. The undecidility of iterted modl reltiviztion. Studi Logic, 79(3):373 407, 2005. [Sdzik, 2006] Tomsz Sdzik. Exploring the Iterted Updte Universe. ILLC Pulictions PP-2006-26, 2006. [vn Benthem, 2003] John vn Benthem. One is lonely numer: On the logic of communiction. 2003. [vn de Pol et l., 2015] Iris vn de Pol, Iris vn Rooij, nd Jku Szymnik. Prmeterized complexity results for model of theory of mind sed on dynmic epistemic logic. In TARK, 2015. [Yu et l., 2013] Qun Yu, Ximing Wen, nd Yongmei Liu. Multi-gent epistemic explntory dignosis vi resoning out ctions. In Frncesc Rossi, editor, IJCAI. IJ- CAI/AAAI, 2013. 2797