To Know is to Know the Value of a Variable

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1 To Know is to Know the Vlue of Vrible Alexndru Bltg 1 ILLC, University of Amsterdm Abstrct We develop n epistemic logic tht cn express knowledge of dependency between vribles (or complex terms). An epistemic dependency formul K t 1,...,t n t sys tht gent knows the vlue of term t conditionl on being given the vlues of terms t 1,...,t n. We dd dynmic opertors [!t 1,...,t n], cpturing the e ect of publicly (nd simultneously) nnouncing the vlues of terms t 1,...,t n. We prove completeness, decidbility nd finite model property. Keywords: knowing wht, knowledge de re, dynmic epistemic logic. 1 Introduction In this pper we build on the work of Plz [14,15], nd Wng nd Fn [24,25] on formlizing the notion of knowledge de re (knowledge of n object, knowledge wht ) over Kripke models 2. We understnd this s knowing the vlue of vrible. Here, vrible is wht in first-order modl logic is clled non-rigid designtor x, tking di erent vlues (in some fixed domin D) t di erent possible worlds. If we denote by w(x) the vlue of vrible x t world w, nd we denote by the epistemic ccessibility reltion of some gent, then Plz s semntics for knowledge de re is given by putting 3 : w = K x i 8v w (v(x) =w(x)). This is nturl nlogue of the usul semntics of knowledge tht in epistemic logic: n gent knows the vlue of x if tht vlue is the sme in ll her epistemic lterntives. When the rnge D of possible vlues of x is finite, then this opertor is obviously reduceble to the usul one, vi finite disjunction W d2d K (x = d). But in generl this is not possible. Plz [15] hd very simple xiomtiztion of this opertor (in combintion with the usul epistemic opertor K ' for knowing tht ), nd climed its completeness, bsed on reduction to stndrd epistemic logic. He lso extended 1 thelexndrubltg@gmil.com 2 In its turn, the work of Wng nd Fn builds on previous reserch in Security on knowledge of keys nd psswords, e.g. [9,11,23]. 3 We use the the sme symbol K for knowledge wht s the usul epistemic opertor for knowledge tht, nd we use vribles x, y,... to denote the non-rigid designtors. Plz, nd Wng nd Fn, use di erent nottion Kv for knowledge wht, nd denote the non-rigid designtors by constnts c. Inourfrmework,doingthiswouldbeveryconfusing, since we lso hve rigid designtors, which re nturlly denoted by constnts.

2 136 To Know is to Know the Vlue of Vrible this logic with public nnouncement opertors 4 [!'] (of which he is the min origintor [14]), nd used the resulting logic to tret the clssicl Sum nd Product puzzle. But he could not prove completeness of this extended logic. Wng nd Fn solved this problem by introducing conditionl version of the bove opertor K ' x ( conditionlly knowing wht ), with the intuitive mening tht gent could find the vlue of x if given the dditionl informtion tht ' ws the cse. The introduction of such conditionl opertor llows one to pre-encode the dynmics of [!'], following strtegy pioneered by vn Benthem [20], obtining Reduction Axioms tht llow us to reduce ny dynmic formul to sttic one. Their completeness proof ws very complex (in the multi-gent cse), going vi detour through first-order intentionl logic. A more nturl type of nnouncement in this context is the ction!x of publicly nnouncing the vlue of x. In recent tlk (Univ. of Amsterdm 2015), Wng stted s n open question the problem of finding complete xiomtiztion for logic tht combines the opertors for knowledge tht K, knowledge of vlue Kx, propositionl public nnouncements [! ] nd public nnouncements of vlues [!x]. This problem remined open until now, despite e orts in this direction by vn Eijck, Gttinger nd Wng. 5 In this pper we solve this problem, by introducing nother kind of conditionl version of the bove opertor. An epistemic dependency formul K x1,...,xn y sys tht n gent knows the vlue of some vrible y conditionl on being given the vlues of vribles x 1,...,x n. The semntics is the obvious generliztion of the bove cluse: if we use the bbrevition w(! x )=v(! x ) for the conjunction w(x 1 )=v(x 1 ) ^...w(x n )=v(x n ), then we put w = K x1,...,xn y i 8v w (w(! x )=v(! x ) ) v(y) =w(y)). In words: n gent knows y given x 1,...,x n if the vlue of y is the sme in ll the epistemic lterntives tht gree with the ctul world on the vlues of x 1,...,x n. This opertor hs connections with Dependence Logic 6 nd llows us to pre-encode the dynmics of the vlue-nnouncement opertor [!x]'. Besides the epistemic dependency formuls nd the dynmic public vluennouncement opertor, we introduce number of other forml innovtions, tht re useful for both technicl nd conceptul purposes. One is tht, in ddition to vribles, we lso llow constnts (i.e. rigid designtors) c, whose vlue is the sme in ll possible worlds, s well s more complex terms t (built 4 We use the dynmic-logic style nottion for this opertor tht is by now stndrd in Dynmic Epistemic Logic, which we regrd s nturl: this is dynmic modlity, cpturing wekest precondition of n ction exctly s in PDL,exceptthtthectionistheoneof publicly nnouncing '. Plzusesthemoreopquenottion' +. 5 After submitting the AiML bstrct, we becme wre of n unpublished drft by vn Eijck, Gttinger nd Wng, contining work in progress on prtil solution to this problem. The logic xiomtized there hs knowledge de re opertors Kx nd vlue nnouncements [!x], butitcnnot express the usul knowledge tht K, nor the usul (propositionl) public nnouncements [! ], soitisnot completesolutiontotheboveproblem. 6 See Section 4.

3 Bltg 137 from vribles nd constnts using functions). Moreover, we hve reltionl toms R(t 1,...,t n ) expressing reltionships between terms, nd in prticulr n equlity predicte t = t 0, which cptures identity of vlues nd plys n essentil role in our system. Although sttements K t cnnot in generl be reduced to knowledge tht sttements K ', formuls of the form x = c cn be used to provide locl reductions : semnticlly, t ech possible world w, K x is equivlent to K (x = c), where c is denotes the vlue w(x) of vrible x in world w. In our xiomtic system, this locl reduction tkes the shpe of our Knowledge De Re xiom, whose relevnt instnce in this cse is the vlidity (x = c) ) (K x, K (x = c)). In words: when the vlue of x is c, then knowing the vlue of x is the sme s knowing tht this vlue is c. Our Knowledge De Re xiom generlizes this to epistemic dependency formuls. Combined with our Existence of Vlue Rule (sying tht vribles lwys hve vlue), this llows us to prove complex properties of epistemic dependence (e.g. the well-known Armstrong xioms [2]) in simple wy, from bsic epistemic xioms. It lso llows us to provide rther simple completeness proof, bsed on vrition of the cnonicl model construction, in which constnts ct s witnesses for the vlues of vribles. Another technicl innovtion is tht we include specil type of vribles? ', storing the truth vlue of formul '. On the one hnd, this introduces nother lyer of (interesting) technicl complexity, since terms of the form? ' re even more non-rigid thn the generic vribles x, in tht they cn chnge their vlue while this vlue is being lernt. Indeed, while x keeps its vlue when tht vlue is publicly nnounced, terms? ' corresponding to Moore sentences ' (such s x = 0 but you don t know it ) my chnge their vlues fter being lernt. On the other hnd, the use of such fluctuting vribles llow us to simplify the syntx, by reducing the usul knowledge tht opertor to knowledge wht, vi the equivlence K ', (' ^ K?'). This is in the spirit of the Sch er quote bove: to know tht ' is to know the nswer to the question wht is the truth vlue of '? So, unlike Plz, nd Wng nd Fn, we do not need two epistemic opertors: there is only one kind of knowledge, nmely knowing the vlue of vrible. Similrly, propositionl nnouncements [!'] ) cn be reduced to lerning the vlue of vrible? '. So one could sy, without exggerting too much, tht ll knowledge is, or cn t lest be represented s, knowledge of the vlue of vrible. Hence, this pper s title: itself prphrse of Quine s fmous dictum. 7 This epistemicmodeling vrint seems less problemtic thn the originl, ontologicl version! And, in fct, our formlism suggests tht the unry knowledge opertor is just specil cse. The more generl version of our motto is: to know is to know the dependence between (vlues of) vribles. This fits well with the populr view of knowledge-cquisition s process of lerning correltions (with the gol of eventully trcking cusl reltionships in the ctul world). 7 To be is, purely nd simply, to be the vlue of vrible [16].

4 138 To Know is to Know the Vlue of Vrible 2 The Logic of Epistemic Dependency In the following, we ssume given finite set A of gents, nd four countble sets of symbols: set P of propositionl toms; setvrof vribles; setc of constnts, mong which there re two distinguished constnts 0 nd 1 (with 0 6= 1); set F of functionl symbols nd set R of reltionl symbols, together with n rity mp r : F[R!N, ssociting to ll symbols f 2 F, R 2 Rel nturl numbers r(f),r(r) 2 N. R includes n equlity symbol =, with r(=) = 2. Intuitively, the di erence between vribles nd constnts is tht constnts re rigid designtors, while vribles re non-rigid: so vrible cn tke di erent vlues in di erent possible worlds, while constnt denotes the sme objects in ll the worlds of given model. We use letters p, q,... to denote toms in P,lettersx, y,... to denote vribles in Vr, nd letters c, d,... to denote constnts in C. We denote by! x finite strings! x =(x 1,...,x k ) 2 Vr of vribles (of ny length k 0), nd similrly use! c to denote finite strings! c =(c1,...,c k ) 2 C of objects. We denote by the empty string. Syntx. TheLogic of Epistemic Dependency (LED) hs twofold syntx, consisting of set L = L(P, V r, C, F, r) of propositionl formuls ' nd set T = T (P, V r, C, F, R,r) of terms t, defined by double recursion: ' ::= p R(! t ) '! ' K! t t t ::= x c? ' f(! t ) where 2Are gents, x 2 Vr re vribles, c 2 C re constnts, t 2T re terms,! t re finite tuples of terms nd f 2F,R 2Rre symbols of rity equl to the length of! t. We bbrevite = (t, t 0 ) s t = t 0. Semntics. Amodel (for L nd T ) is structure M =(W, D, [0], [1],, k k, ( ), f, R ) 2A,f2F,R2R where: W is set of possible worlds; D is set of objects, contining t lest two designted objects [0] 6= [1]; W W re equivlence reltions, clled epistemic indistinguishbility reltions; k k is vlution function mpping ech tomic sentence p 2 P to set kpk W of possible worlds; ( ) : W (Vr[ C)! D is mp ssociting to ech world w 2 W nd ech vrible or constnt 2 Vr[ C some object w( ) 2 D, clled the vlue of t world w, nd stisfying the requirement tht the vlue of ech constnt is the sme in ll the worlds: i.e.,w(c) =w 0 (c) for ll c 2 C nd ll w, w 0 2 W ; nd for ll symbols f 2F,R2Rof rity r(f) =n, we re given n-ry mps f : D n! D nd n-ry reltions R D n,withthestndrd interprettion of equlity = s the digonl {(d, d) :d 2 D} of D. For the semntics, we simultneously define n extended vlution (the truth mp ) k'k M for ll formuls ', nd n extended vlue mp w(t) M for ll terms t nd ll worlds w 2 W. We will use the nottion w(! t ):= (w(t 1 ),...,w(t k )) for the string of vlues corresponding to ny given string of

5 Bltg 139 terms! t =(t 1,...,t k ) 2T. The truth mp is given for propositionl toms p 2 P by the vlution kpk, nd extended to other formuls by recursively putting: kr(! t )k = {w 2 W w(! t ) 2 R}; k'! k =(W \k'k) [k k; kk! t t 0 k = {w 2 W 8v 2 W (w v ^ w(! t )=v(! t ) ) w(t 0 )=v(t 0 ))}. The extended vlue mp w(t 0 ) is given by the vlue mp w( ) for vribles nd constnts 2 Vr[ C, nd extended to other terms by recursion: w(? ' ) = [1] i w 2k'k; w(? ' )=0i w 62 k'k; w(f(! t )) = f(w(! t )). Abbrevitions. We put > := (1 = 1);? := (1 = 0): ' := '!?; ' _ := '! ; ' ^ := ( ' _ ); ' $ := ('! ) ^ (! '); K! t ' := ' ^ K! t? ' ; hk! t i' := K! t '; K ' := K ' (where is the empty string); hk i' := K '; K ' := K ('! ). We lso put K!! t t 0 := V1ppleipplek K! t t 0 i, nd (! t =! t 0 ):= V 1ppleipplek t i = t 0 i,wherek is the length of! t 0. Ground Terms A ground term is term tht contins no vribles nd no propositionl formuls (hence, no?); in other words, ground terms re built only from constnts c, d,... 2 C by recursively pplying function symbols f,g,... Let us denote by T 0 the set of ll ground terms. Propositionl Substitution: For toms p 2 P nd formuls, the substitution of p with is n opertion mpping every formul ' 2Linto new formul '[p/ ] 2L, nd similrly mpping every tuple of term! t 2T into new tuple! t [p/ ], obtined by uniformly substituting p with s usul. 8 Vrible Substitution: For vribles x 2 V r nd terms t 2 T,the substitution of x with t is n opertion mpping every formul ' 2Linto new formul '[x/t] 2L, nd mpping every term t 0 2T into new term t 0 [x/t], obtined by uniformly substituting x with t in the usul wy. 9 Exmple 1. Alice nd Bob hve ech nturl number written on their foreheds. It is common knowledge tht Alice s number x is the immedite successor of Bob s number x b. Both re blindfolded, so nobody cn see the numbers. The model hs: Vr = {x,x b }, D = C = N is the set of nturl numbers; F = {+, } nd R = {=,>} contin the usul opertions nd reltions on N; thesetw of worlds consists of ll functions w : Vr! N, stisfying the given constrint w(x )=w(x b ) + 1; the epistemic reltions re given by the universl reltions: = b = W W. Note tht the sentence K x ^ K b x b ^ K (x >x b ) ^ K b (x >x b ) ^ K x b x ^ K x b x b is true in ll worlds. So nobody knows his/her number, but both know tht Alice s number is lrger, nd both could come to know the numbers if given only the other s number. 8 More precisely: p[p/ ] := ; q[p/ ] :=q; (R(t 1,...,t n))[p/ ] :=R(t 1 [p/ ],...,t n[p/ ]); ('! )[p/ ] :='[p/ ]! [p/ ]; K t 1,...,t n t[p/ ] :=K t 1[p/ ],...,t n[p/ ] t[p/ ]; [p/ ] := ; c[p/ ] :=c; x[p/ ] :=x;? '[p/ ] :=? '[p/ ] ; f(t 1,...,t n)[p/ ] :=f(t 1 [p/ ],...,t n[p/ ]). 9 I.e., [x/t] := ; c[x/t] :=c; x[x/t] :=t; y[x/t] :=y;? '[x/t] :=? '[x/t] ; f(t 1,...,t n)[x/t] := f(t 1 [x/t],...,t n[x/t]); p[x/t] := p; (R(t 1,...,t n)[x/t] := R(t 1 [x/t],...,t n[x/t]); ('! )[x/t] :='[x/t]! [x/t]; (K t 1,...,t n t 0 )[x/t] :=K t 1[x/t],...,t n[x/t] t 0 [x/t].

6 140 To Know is to Know the Vlue of Vrible Proof system. The proof system LED consists of the following: RULES: Propositionl Substitution: From ', infer'[p/ ]. Vrible Substitution: From ', infer'[x/t]. Modus Ponens Rule: From ' nd '!, infer. Necessittion: From ', inferk '. Existence-of-Vlue Rule (EV R): From x = c! ',infer ', provided tht c does not occur in '. AXIOMS: All the clssicl propositionl tutologies. All the S5 xioms for K. Knowledge De Re: (! x =! c ^ y = d )! K! x y $ K! x =! c y = d) Equlity Axioms: x = x x = y! y = x (x = y ^ y = z)! x = z! x =! y! f(! x )=f(! y ) Chrcteristic Functions: Knowledge of Functions: (x = y ^ R(! z,x,! w ))! R(! z,y,! w )? ' =1$ ',? ' =0$ ', K! x f(! x ) In fct, two of the xioms re redundnt: symmetry nd trnsitivity of = follow from the other xioms, but we chose to include them for convenience. We write ` if is provble in the proof system LED. For ny set of formuls nd ny formul, wewrite ` if there exist finitely mny formuls 1,..., n 2 (for some n 2 N) such tht ` ( 1 ^... n)!. We sy tht islogiclly closed if, for every formul 2L, ` implies 2. We sy tht is consistent if 6`?, nd tht formul' is consistent with if [{'} is consistent (equivlently: if 6` '). Lemm 2.1 For set hve tht: if ` then K! t ` K! t. [{ } ` i ` (! ). of formuls, put K! t :={K! t : 2 }. Then we

7 Bltg 141 Proposition 2.2 Let ' be formul nd z be vrible tht does not occur in the scope of ny K -opertor in '. Then the following is provble in LED: ` (x = y ^ '[z/x])! '[z/y]. NOTE The unrestricted version of the bove schem is not vlid! A counterexmple is obtined by tking for instnce ' to be the formul K (z = c). This is relted to the Phosphorus/Hesperus prdox. Proposition 2.3 ( Knowledge of Ground Terms nd Ground Identities). For ll ground terms t, t 0 2T 0, ll the instnces of the following schem re provble in LED: ` K t; ` t = t 0! K t = t 0. Proof. We prove the first clim by induction on t: For the bse step, let t := c be constnt. From the Knowledge De Re xiom, we get ` x = c! (K x $ K (x = c)). By substituting c for x nd using the first equlity xiom, we get ` K c $ K (c = c). But on the other hnd, by pplying Necessittion to the first equlity xiom, we hve ` K (c = c), nd hence we obtin ` K c. For the inductive step: consider term of the form f(! t ), where! t =(t 1,...,t n ) is tuple of ground terms. By the induction hypothesis, we cn ssume tht ` K t i for ll i =1,n. Using this nd the Knowledge De Re xiom, we derive `!t =!! c! K t =! c.! Combining this with ` K t =! c! K f(! t ) = f(! c ) (obtined by pplying Necessittion, Kripke s xiom nd Modus Ponens to the fourth equlity xiom), we obtin `!t =! c! K f(! t ) = f(! c. Combining this!t with the theorem ` = c ^ f(! c )=d! K! t =! c f(! t ) = d (obtined from the xiom ` K! t f(! t ) nd the Knowledge De Re xiom), we get tht!t ` = c ^ f(! c )=d! K f(! t )=d. This, together with the obvious theorem `!t = c! f(! t )=d!(! t = c ^ f(! c )=d) (n obvious consequence of the equlity xioms), gives us `!t = c! f(! t )=d! K f(! t )=d. Applying the the (EV R) rule, we get ` f(! t )=d! K f(! t )=d, whichby the Knowledge De Re xiom, yields ` f(! t )=d! K f(! t ). Applying gin the (EV R) rule, we obtin ` K f(! t ), s desired. As for the second clim: given the first clim, we hve ` K t nd ` K t 0. This, together with ( suitble substitution instnce of) the Knowledge De Re xiom nd the conjunctivity of knowledge, gives us ` (t = c ^ t 0 = c)! K (t = c ^ t 0 = c), nd hence (using equlity xioms nd the xioms of norml modl logic) ` (t = c ^ t 0 = c)! K t = t 0. Together with ` t = c! (t = t 0! t 0 = c) ( consequence of the equlity xioms), this yields ` t = c! (t = t 0! K t = t 0 ). By the (EV R) rule, we obtin ` t = t 0! K t = t 0, s desired. 2

8 142 To Know is to Know the Vlue of Vrible Proposition 2.4 All the following theorems re provble in LED: ` K x1,...,x k y! K x (1),...,x (k) y, for every permuttion : {1,...,k}!{1,...,k} ` (K! x!! y ^ K x,! y ` K! x!! y! K x,! z! z )! K! x! z! y ` K! x! y! K! x f(! y ) `!x =! c! (K! x '! K! x =! c ') ` K! x ('! )! (K! x '! K! x ) ` K! x '! ' ` K! x '! K! x K! x ' ` K! x '! K! x K! x ' Proof. We only prove the first two formuls, the other proofs re similr. For the first, we use the obvious propositionl vlidity ` (x 1 = c 1 ^...x k = c k )! (x (1) = c (1) ^...x (k) = c (k) ), together with two instnces of Knowledge De Re xiom: ` (x 1 = c 1 ^...x k = c k )! (K x1,...,x k y $ K x1=c1^...x k=c k y), nd ` (x 1 = c 1 ^...x k = c k )! (K x (1),...,x (k) y $ K x (1)=c (1)^...x (k) =c (k) y). From these we derive ` (x 1 = c 1 ^...x k = c k )! (K x1,...,x k y! K x (1),...,x (k) y), then pply repetedly the (EV R) rule to obtin the desired conclusion. For the second, we use three instnces of Knowledge De Re xiom: ` (! x =!!! c ^! y = d )! (K x!! y $ K x =! c!! y = d ), ` (! x =!! c ^! y = d ^! x =! e )! (K! x,! y!! z $ K x =!! c ^! y = d! z =! e ), nd ` (! x =! c ^! z =!! e )! (K x! z $ K! x =! c! z =! e ). From these, together with the usul properties of the norml propositionl opertor K (nd the fct tht K is just n bbrevition for K (! )), we obtin ` (! x =! c ^!y =! d ^ =! e )! ((K! x!! y ^K x,! y! z )! K! x! z ), then we pply the (EV R) rule. 2 One cn lso esily verify tht: Proposition 2.5 The following Necessittion-type rule for K! t LED: if` ' then ` K! t '. is derivble in Pseudo-modlities : necessittion/possibility forms. For ny finite string s 2 (L[(A T )), consisting of formuls 2Lnd/or pirs (,! t ) of gents 2And strings! t 2T of terms, we define pseudo-modlities [s] nd hsi, mpping ny formul 2Lto formuls [s] 2L(clled necessity form ) nd hsi 2L(clled possibility form ). The definition is by recursion, putting for necessity forms: [ ] := for the empty string ; [, s] :=! [s] ; nd [(,! t ),s] := K! t [s]. As for possibility forms, we put hsi := [s].

9 Bltg 143 Lemm 2.6 For every necessity form [s] there exists some formul 2L, such tht for ll 2L, we hve: ` [s] i `!. Moreover, the sme constnts nd vribles occur in s in s. Proof. If s =, then tke := >. Otherwise, [s] is just sequence of symbols of the form!... nd K! t..., followed t the end by. Strting from the left, we cn eliminte one by one ech knowledge symbol K! t... by pushing it into the premise, using the fct 10 tht: `! K! t holds i ` hk! t i! holds. At the end of this process, we obtin formul of the form!. It is esy to see tht depends only on s, not on, nd tht moreover contins the sme constnts nd vribles s s. 2 Lemm 2.7 Given s 2 (L[(A T )), t 2T, ' 2L, let c be constnt tht does not occur in s, t or '. Then the following rule is dmissible in LED: if ` [s](t = c! ') then ` [s]' Proof. Let be the formul ssocited to s by the previous Lemm: so for ll, ` [s] i `!. Suppose now tht we hve ` [s](t = c! '). Then we lso get `! (t = c! '), nd hence ` t = c! (! '). Let x be vrible not occurring in t, ' or s (nd hence, by the previous Lemm, not occurring in either). Using (some substitution instnce of one of) the Equlity Axioms, we obtin ` x = c! (t = x! (! ')). Since c does not to occur in s, t or ', by the previous Lemm it doesn t occur in either. By the (EV R) rule,we obtin ` t = x! (! '). Using the Vrible Substitution Rule (where we substitute t for x), we get ` t = t! (! '). But we lso hve ` t = t (by nother of the Equlity xioms), nd hence `! '. Using gin the previous Lemm, we obtin ` [s]'. 2 Theorem 2.8 The proof system LED is sound nd strongly complete (nd hence the logic LED is compct). Moreover, this logic hs the strong finite model property, nd hence it is decidble. The rest of this section is dedicting to the proof of this theorem. For ny countble set of constnts C, letl C be the lnguge of LED bsed only on constnts in C. AC-theory is consistent set of formuls in L C ; here, consistent mens consistent with respect to the proof system LED formulted for the lnguge L C.Amximl C-theory is C-theory tht is mximl (w.r.t. inclusion) mong ll C-theories. A C-witnessed theory is C-theory such tht, for every term t 2T C,strings2(L C [ (A TC )) nd formul ' 2L C, if ` [s](t = c! ') for ll c 2 C, then ` [s]'. Equivlently: if whenever hsi' is consistent with, then there exists some c 2 C s.t. hsi(t = c ^ ') 10 This is n instnce of the well-known fct tht in the xiomtic system S5, formul! 2 is theorem i the formul 3! is theorem.

10 144 To Know is to Know the Vlue of Vrible is consistent with. A mximl C-witnessed theory is C-witnessed theory which is not proper subset of ny other C-witnessed theory. For the completeness proof, we mke use of the following three esily verifible results: Lemm 2.9 If is C-theory nd 6`, then [{ } is lso C-theory. Moreover, if is C-witnessed, then [{ } is C-witnessed. Lemm 2.10 If n... is n incresing chin of C- theories, then S n2n n is C-theory. Moreover, if ll n re C-witnessed then S n2n n is C-witnessed. Lemm 2.11 A C-theory mximl C-witnessed theory. is C-witnessed mximl C-theory i it is The completeness proof goes now vi the following steps: cn be ex- Lemm 2.12 ( Lindenbum Lemm) Every C-witnessed theory tended to mximl C-witnessed theory T. Proof. Let 0, 1,..., n,... be n enumertion of formuls in L C.Wedefine n incresing chin n... of C-witnessed theories: first, put 0 := ; then, given the witnessed C-theory n, put n+1 := n if ` n, nd put n+1 := n [{ n } otherwise (if n 6` n ). Finlly, we put T := S n2n n. By Lemm 2.10, thisisc-witnessed theory. Moreover, it is lso mximl C-theory (since every formul consistent with T is in T ), so it is mximl C-witnessed theory. 2 Lemm 2.13 ( Extension Lemm) Let C be set of constnts, nd let C 0 = {c 0,c 1,...,c n,...} be countble set of fresh constnts, i.e. s.t. C \ C 0 = ;. Put C = C [ C 0. Then every C-theory cn be extended to C-witnessed theory, nd hence (by Lindenbum Lemm) to mximl C-witnessed theory T. Proof. Let 1,..., n... be n enumertion of ll the triplets of the form n =(s n,t n, n) consisting of ny necessity form s n 2 (L C [ (A T C )), ny term t n 2T C nd formul n 2L C. For every such triplet n =(s n,t n, n), put C 0 (n) =:{c 0 2 C 0 : c 0 occurs in either s n or t n or n}. Note tht C 0 (n) is lwys finite. We now construct n incresing chin n...of C-theories, stisfying the following three properties: (1) 0 = ; (2) for every n 2 N, the set Cn 0 := {c 0 2 C 0 : c 0 occurs in n } is finite; (3) for every triplet n = (s n,t n, n) in the bove enumertion, if n 6` hs n i n, then 9m 2 N s.t. hs n i(t n = c 0 m ^ n) 2 n+1. The construction is by recursion. For n = 0, we put 0 :=, which tkes cre of condition (1) bove. At step n+1, let n be C-theory stisfying cluse (1) bove, nd let n =(s n,t n, n)bethen-th triplet in the bove enumertion. We hve two cses: () if we hve n ` [s n ] n,then we put n+1 := n ; (b) in cse tht we hve n 6` [s n ] n, then we choose m

11 Bltg 145 to be the lest nturl number bigger 11 thn the indices of ll the constnts in C 0 (n)[cn, 0 nd put n+1 := n [{hs n i(t n = c 0 m^ n)}. To show tht this gives us C-theory, notice tht c 0 m doesn t occur in s n,t n, n or n. If n+1 were inconsistent, then we d hve n ` [s n ](t n = c 0 m! n ), so 9 1,..., k 2 n s.t. ` 1! ( n! [s n ](t n = c 0 m! n )) is theorem in LED. But c 0 m 62 C 0 (n) [ Cn,soc 0 0 m doesn t occur in s n,t n, n, 1,..., n (or in ny other formul of n). By Lemm 2.7, we hve tht ` 1! ( n! [s n ] n ) is lso theorem in LED, nd hence tht n ` [s] n, contrry to our ssumption (in cse b). So in both cses n+1 is C-theory. It is lso esy to see tht it stisfies condition (2): in cse () we hve Cn+1 0 = Cn 0 (finite by the inductive ssumption); in cse (b) we hve Cn+1 0 = Cn[C 0 0 (n)[{c 0 m} (still finite). Finlly, it is obvious tht condition (3) is stisfied. Given now this incresing sequence = 0 n of C-theories stisfying (1)-(3) bove, tke := S n2n n. By Lemm 2.10, is C-theory, nd it obviously includes = 0. Condition (3) bove implies tht is Cwitnessed. 2 Together, the lst three results imply tht, in order to show completeness, it is enough to show tht, for ny countble set C of constnts, every mximl C-witnessed theory hs model. Wenowproceedtoprovethis. From now on, we fix the set of constnts C, nd we ssume given mximl C-witnessed theory T 0. For ech term t 2T = T C, we cn define n equivlence reltion t on mximl C-witnessed theories T,T 0, by putting: T t T 0 i 8c 2 C((t = c) 2 T, (t = c) 2 T 0 ). Put := T t2t 0 t (where recll tht T 0 is the set of ground terms). It is obvious tht is lso n equivlence reltion on mximl C-witnessed theories. In ddition, we cn define nother equivlence reltion on the set of constnts C by putting: c c 0 i (c = c 0 ) 2 T 0. For ny constnt c 2 C, letus denote by [c] :={c 0 2 C : c c 0 } the equivlence clss of c modulo. Cnonicl Model The cnonicl model for T 0 is model M = (,D,[0], [1],, k k, ( ), f, R ) 2A,f2F,R2R for the lnguge L C, defined s follows: the stte spce is := {T L C : T mximl witnessed L C -theory with T T 0 }; the set of objects is D := {[c] : c 2 C}, where [c] is the equivlence clss of c modulo, nd the equivlence clsses [0] nd [1] re the two designted objects; the epistemic reltions re: T T 0 i 8' 2L C (K ' 2 T ) ' 2 T 0 ). For f 2F, we put f([c 1 ],...,[c n ]) := [c] for c 1,...,c n,c 2 C with (f(c 1,...,c n )=c) 2 0 ; nd for R 2 R, we put R := {([c 1 ],...,[c n ]) : R(c 1,...,c n ) 2 T 0. The vlution is kpk := {T 2 W : p 2 T }. Thevlue T ( ) of 2 Vr[ C t world T 2 is given by T (c) :=[c] for c 2 C, nd T (x) :=[c], for x 2 Vrnd c 2 C with (x = c) 2 T. It is esy to check tht these definitions re independent of the choice of representtives, so M is indeed well-defined model for L C. 11 Such number exists, due to the inductive ssumption (2) bove.

12 146 To Know is to Know the Vlue of Vrible Lemm 2.14 ( Intersection Lemm) For ll gents 2And finite strings of terms! t =(t 1,...,t n ), we hve:! t = \ t1 \ tn. Proof. The left-to-right inclusion: the LED-theorem ` K! x!! y! K x,! z! y (proven in Proposition 2.4) yields by substitution ` K '! K! t ' nd ` K ti '! K! t ', from which we obtin! t nd! t ti (for ll i =1,n). For the converse inclusion: suppose T,S 2 stisfy T S nd T ti S for ll i =1,n. To show tht T! t S,letK! t ' 2 T. We need to show tht ' 2 S: for this, notice tht, since T is C-witnessed (nd, for ech i =1,n there must exist constnts c i 2 C such tht T is consistent with t i = c i.since T is mximl, it follows tht (t i = c i ) 2 T, nd moreover tht (! t =! c ) 2 T. By pplying the theorem `!t =! c! (K! t '! K! t =! c '), we obtin tht ' 2 T,i.e.K (! t =! c! ') 2 T. This together with T S, gives us K! t =! c tht (! t =! c! ') 2 S. But from (! t =! c ) 2 T nd T ti S for ll i =1,n, we derive tht (! t =! c ) 2 T, nd hence by closure of (the mximl theory T ) under modus ponens, we obtin tht ' 2 T. 2 As consequence of Lemm 2.14, ll! t re equivlence reltions. Lemm 2.15 ( Dimond Lemm) Let T 2, nd let,! t,' be such tht K! t ' 62 T. Then there exists some theory S 2 such tht T! t S but ' 62 S. Proof. Let := { : K! t 2 T }. We will show the following Clim: Theset [ { '} is C-witnessed theory. To prove this clim, we first need to show tht this set is consistent. Suppose not; then there exist 1,..., n 2 (hence,k! t i 2 T for ll i = 1,n) such tht ` ( 1 ^... n )! ' is theorem. But then we lso hve tht ` (K! t 1^...K! t n )! K! t ' nd (K! t 1^...K! t n ) 2 T,henceK! t ' 2 T, in contrdiction with our ssumption (tht K! t ' 62 T ). Next, to show tht [ { '} is C-witnessed, suppose tht, for some triple (s 0,t 0,' 0 ), we hve [ { '} `[s 0 ](t 0 = c! 0 ) for ll c 2 C. By previous lemm, this gives tht ` ( '! [s 0 ](t 0 = c! 0 )) for ll c, nd by nother lemm we obtin tht K! t ` K! t ( '! [s 0 ](t 0 = c! 0 )) (where recll tht K! t ={K! t : 2 }). But note tht K T, nd hence we get T ` K! t ( '! [s 0 ](t 0 = c! 0 )) for ll c 2 C. Since T 2 isc-witnessed, it follows (by pplying the C-witnessing condition to the necessittion form ((,! t ), ', s 0 )) tht T ` K! t ( '! [s 0 ] 0 ), nd hence by mximlity tht K! t ( '! [s 0 ] 0 ) 2 T,hence( '![s 0 ] 0 ) 2. From this we obtin tht [ { '} `[s 0 ] 0, thus proving our Clim bove. Given the Clim, we cn now use the Extension Lemm (in combintion with Lindenbum Lemm) to extend the set [{ '} to mximl C-witnessed theory S. It is esy to see tht we hve S T T 0,henceS 2. We obviously hve ( ') 2 S, so by consistency ' 62 S. Finlly, S gives us tht T! t S. 2

13 Bltg 147 Lemm 2.16 ( Knowledge de Re Lemm) Let T 2, nd let,! t,t 0 be such tht K! t t 0 62 T. Then there exists some theory S 2 such tht T! t S but T 6 t0 S. Proof. Since T is mximl C-witnessed theory, there exist! c 2 C, c 2 C such tht! t =! c,t 0 = c 0 2 T. By using the theorem ` (! t =! c ^ t 0 = c 0 )! (K! t t 0 $ K (! t =! c! t 0 = c 0 )) (which is substitution instnce of the Knowledge de Re xiom) nd the ssumption tht K! t t 0 62 T, we obtin tht K (! t =! c! t 0 = c 0 ) 62 T. By the Dimond Lemm, there exists some S 2 such tht T! t S but (! t =! c! t 0 = c 0 ) 62 S. By the mximlity of S, we get tht (! t =! c ^ t 0 6= c 0 ) 2 S, nd hence tht T ti S for ll i =1,n but T 6 t 0 S. Using lso T! t S nd the Intersection Lemm bove, we conclude tht T! t S (nd T 6 t 0 S), s desired. 2 Lemm 2.17 ( Truth Lemm) Let M =(,D,, k k, ( ), f ) 2A,f2F be the cnonicl model for (some theory) T 0. Then for every formul ' nd every term t, we hve: (1) T 2k'k M i ' 2 T,nd (2) T (t) =[c] i (t = c) 2 T. Proof. We prove both clims by simultneous induction on the complexity 12 of formuls ' nd terms t: To prove (1): for tomic formuls p 2 P, (1) is trivil. For reltionl toms R(! t ), let! [c] 2 D be s.t. T (! t )=! [c], so by the induction hypothesis for (2) we hve (! t =! c ) 2 T. Then we hve the following sequence of equivlencies: T 2 kr(! t )k M i T (! t ) 2 R i! [c] 2 R i R(! [c]) 2 T 0 i (using T T 0 ) R(! [c]) 2 T i R(! [t]) 2 T (where t the lst step we used the fct tht (! t =! c ) 2 T nd the equlity xioms). For implictionl formuls!, this goes s usul, using the properties of mximlly consistent theories. For epistemic formuls K! t t 0,with! t =(t 1,...,t n ): to prove the left-to-right impliction, suppose tht T 2kK! t t 0 k M but (K! t t 0 ) 62 T.Bythe Knowledge de Re Lemm, there exists some S 2 such tht T! t S but T 6 t0 S. By the Intersection Lemm, we obtin tht T S nd T ti S for ll i 2{1,...,n}; i.e. for every c 2 C nd i 2{1,...,n}, we hve tht: (t i = c) 2 T, (t i = c) 2 S. By the induction hypothesis for clim (2), this implies tht T (! t )=S(! t ). From this, together with T S nd the fct tht T 2kK! t t 0 k M (s well s the semntic cluse for knowledge de re), we obtin tht T (t 0 )=S(t 0 ). Applying gin the induction hypothesis for (2), we get tht 12 Our notion of complexity is function comp : L C [T C! N, defined recursively by putting: comp(p) = comp(c) = comp(x) = 0, comp(r(t 1,...,t n)) = 1 + mx(comp(t 1 ),...,comp(t n)), comp(! ) = 1 + mx(comp( ),comp( )), comp(k t 1,...,t n t 0 )=1+mx(comp(t 1 ),...,comp(t n),comp(t 0 )), comp(? )=1+comp( ), comp(f(t 1,...,t n)) = 1 + mx(comp(t 1 ),...,comp(t n)).

14 148 To Know is to Know the Vlue of Vrible (t 0 = c) 2 T, (t 0 = c) 2 S holds for ll c 2 C, i.e. T t0 S contrry to our ssumption bove. To show the converse: suppose tht (K! t t 0 ) 2 T.SinceTis C-witnessed there must exist! c 2 C,c 0 2 C such tht (! t =! c ), (t 0 = c 0 ) 2 T. By the induction hypothesis for (2), we hve T (t i )=[c i ] for ll i 2{1,...,n}, nd lso T (t 0 )=[c 0 ]. To prove now tht T 2k'k M,letS 2 be such tht T S nd T (! t )=S(! t ). It is enough to prove tht T (t 0 )=S(t 0 ). Using the Knowledge de Re xiom, nd the fct tht (K! t t 0 ) 2 T, we obtin tht (K! t =! c t 0 = c 0 ) 2 T,i.e. K (! t =! c! t 0 = c 0 ) 2 T. Since T S,wemust hve (! t =! c! t 0 = c 0 ) 2 S. From T (! t )=S(! t ) nd T (t i )=[c i ] for ll i, we get tht S(t i )=[c i ] for ll i, nd so by the induction hypothesis for (2) we hve (! t =! c ) 2 S. This, together with (! t =! c! t 0 = c 0 ) 2 S, gives us tht (t 0 = c 0 ) 2 S. Applying gin the induction hypothesis for the second clim, we obtin tht S(t 0 )=[c 0 ]=T(t 0 ), s desired. To show (2): it is trivilly true for vribles x 2 Vrnd constnts c 0 2 C. For terms of the form? ' : we know tht by definition T (? ' ) = [1] holds i T 2 k'k M holds, i.e. (by the induction hypothesis for (1)) i ' 2 T i (? ' $ 1) 2 T (by the Chrcteristic Functions Axiom). A similr rgument shows tht T (? ' ) = [0] i (? ' $ 0) 2 T. Since [0] nd [1] re the only possible vlues of T (? ' ), we obtin the desired conclusion. For terms of the form f(t 1,...,t n ), let c 1,...,c n 2 C be s. t. T (t 1 )=[c 1 ],...,T(t n )=[c n ], i.e. (t 1 = c 1 ),...,(t n = c n ) 2 T. (By the definition of the cnonicl vlue function, such constnts must exist.) We hve T (f(t 1,...,t n )=f(t(t 1 ),...,T(t n )) = f([c 1 ],...,[c n ]) = [f(c 1,...,c n )]. Hence we hve tht: T (f(t 1,...,t n )=[c] holds i f(c 1,...,c n )] = [c] holds, i.e. i (f(c 1,...,c n )=c) 2 T. 2 In prticulr, T 0 is stisfied t world T 0 in M: this finishes our proof of strong completeness. The decidbility proof goes vi the following two steps: STEP 1: Reduction of LED vlidities to vlidities in less expressive lnguge. Let LED 0 be the lnguge with the following syntx ' ::= p R(! t ) '! ' K ' t ::= x c f(! t ) In other words: we only llow terms tht do not contin chrcteristic functions? nd we only llow the usul (propositionl) modlities. The semntics is the obvious one, with ll constructs interpreted s in GK nd with the epistemic modlities interpreted in the usul wy (using the reltions ). In fct, for technicl resons it is convenient to lso look t the extended lnguge LED 1 obtined by dding the usul epistemic modlities to LED: ' ::= p R(! t ) '! ' K! t t K ' t ::= x c? ' f(! t )

15 Bltg 149 (Once gin, the semntics is the obvious one). It is cler tht LED 1 nd LED 0 re co-expressive, sincek ' is equivlent to ' ^ K? '. In contrst, LED 0 is less expressive lnguge thn LED (nd hence thn LED 1 ): Counterexmple. We show tht the formul K x is not equivlent to ny formul in LED 0. Suppose, towrds contrdiction tht K x is equivlent to some formul 0 in LED 0. Let C 0 be the finite set of constnts occurring in 0, nd C 1 := C 0 [{0, 1}. Tke model M 1 with two distinct worlds W 1 = {w, w 0 }, four distinct objects D = {[0], [1],d,d 0 }, f( ) = [0] for ll functions nd rguments, = W 1 W 1, kpk = ; for ll p, nd w(x) =d, w 0 (x) =d 0 for ll vribles x. Tke nother model M 2 with only one world W 2 = {w 00 }, sme D, f nd kpk s for M 1, but with = W 2 W 2 nd w 00 (x) = d for ll vribles x. It is esy to see tht the worlds w, w 0 nd w 00 stisfy exctly the sme formuls in the lnguge of LED 0 bsed only on constnts in C 0. Hence, these three worlds re equivlent wrt the truth vlue of 0. However, K x is true t w 00, while being flse t w nd w 0. This contrdicts the equivlence between K x nd 0. So the modlities for knowledge of vlue relly increse the expressivity of our lnguge. Nevertheless, we cn prove tht every vlidity of LED 1 trnsltes to vlidity of LED 0 : Proposition 2.18 ( Vlidity Reduction ) There exists computble mp from the lnguge LED 1 to the lnguge LED 0, such tht, for every formul ' of LED 1, we hve: ' is vlid i (') is vlid. Proof. The proof is by induction, using nother notion of complexity tht counts only the number of nested de re modlities nd nested? symbols. 13 Note tht every term t of LED 1 cn be rewritten s t = t 0 [x 1 /? 1,...,x n /? n ], for some term t 0 of LED 0 s well s some vribles x 1,...,x n nd formuls 1,..., n (in LED 1 ), with ( i ) < (t). For! c 2 {0, 1} n, we introduce the nottions t 0 [! c ]:=t 0 [x 1 /c 1,...,x n /c n ], nd! c (! ):= V i=1,n c i( i ), where c( ) := for c = 1, nd c( ) = for c = 0. Now for ny tuple of terms! t = (t 1,...,t m ) of LKG 1, let t i 0 (with i 2 {1,...,m}) be the corresponding terms in LED 0, with vribles x i 1,...,x i n i nd formuls 1,..., ni i i (for i 2 {1,...,m}), s.t. t i = t i 0[x i i 1/? 1,...,x i n i /? i ni ] holds for ll i 2 {1,...,m}. Then we put (R(t 1,...,t m ) 0 := W! c 1 2{0,1} n 1... W V!! c m 2{0,1} nm i2{1,...,m} c i (! i ) ^ R(t 1 0[! c 1 ],...,t m 0 [! c m ]). Clim A: R(! t ) is logiclly equivlent to (R(! t )) 0. (The proof is n esy verifiction.) 13 More precisely, we recursively put: (p) = (x) = (c) =0, ('! ) =mx( ('), ( )), (K ') = ( ') = ('), (K t 1,...,t n t) =1+mx( (t 1 ),..., (t n), (t)), (? ')=1+ ('), (R(t 1,...,t n)) = (f(t 1,...,f n)) = mx( (t 1 ),..., (t n)).

16 150 To Know is to Know the Vlue of Vrible Given this clim, let ' be ny formul in LED 1. We cn obviously bring it to conjunctive norml form, i.e. estblish vlidity = ', V W i j ij, where the formuls ij re of one the following bsic forms p, p, R(! t ), R(! t ), K! t t 0, K! t t 0, K, or K. Let T ' be the set of ll terms occurring in this norml form, nd let F be n injective mp tht ssocites to ech term t 2T ' some fresh constnt F (t) 2 C \T ' (such tht F (t) 6= F (t 0 ) for t 6= t 0 ). For ny string! t =(t 1,...,t n ) of terms in T ',putf(! t ):= (F (t 1 ),...,F(t n )). We now ssocite to ech of the bove bsic formuls ij ij some corresponding formuls 0, s follows: ij 0 := ij if ij is the form ij p or K ; 0 := (R(! t )) 0 (s defined bove) if ij is of the form R(! t ); ij!t 0 := K! = F ( t ) ) t 0 = F (t 0 ) if ij is of the form K! t t 0 ; nd finlly ( ij ) 0 := 0,if ij = with of one of the forms p, R(! t ), K ' or K! t t 0. We ssocite now to our formul ' bove new formul ' 0 of lower - complexity, by putting 0 1 ' 0 ^ t2t It is now esy to verify the following: t = F (t) A ) ^ _ Clim B: If' is not in LED 0,then (' 0 ) < ('). Finlly, we cn prove the key step of our Vlidity Reduction : Clim C: ' is vlid i ' 0 is vlid. V Proof of Clim C: Note the vlidity = t2t t = F (t) ) ( ij, ij 0 ). (This is obvious when ij is of the form p, p, K or K ; it follows from Clim A when ij is of the form R(! t ) or R(! t ); nd it follows from the Knowledge de Re Axiom when ij is of the form K! t t 0 or K! t t 0.) Using the norml form of ', we obtin the following vlidity: 0 1 ( ) ^ t = F (t) A ) (', ^ _ ij 0 ). t2t ' i j To prove now one direction of Clim C, ssume tht ' is vlid. Using (*), V it follows tht t2t t = F (t) ) V W ij i j 0 is vlid, i.e. ' 0 is vlid. For the other direction: ssume tht ' 0 is vlid, nd let M =(W, D, [0], [1],, k k, ( ), f, R ) 2A,f2F,R2R be model nd w 0 2 W be ny world. We cn chnge this to di erent model M 0 =(W, D, [0], [1],, k k, ( ) 0, f, R ) 2A,f2F,R2R, where we chnged only the vlue mp (nd only t w 0 )byputtingw 0 (c) 0 = w 0 (t) wheneverf (t) =c with t 2T ', nd w(c) 0 = w(c) inrest. SinceF is injective, this gives us well-defined vlue mp. The chnge doesn t ect the vlues of the terms t 2T ' (since they don t contin ny of the fresh constnts whose vlue ws chnged), so we hve w 0 (t) 0 = w 0 (t) =w 0 (F (t)) 0 for ll these i j ij 0.

17 Bltg 151 terms, hence w 0 2 V t2t t = F (t) M 0. Using (*) nd the fct tht ' 0 is vlid, it follows tht w 0 2 'k M 0.But' contins none of the constnts whose vlues were chnged, so its truth vlue ws not ected by the chnge, i.e. we lso hve w 0 2 'k M.SinceMnd w 0 re rbitrry, we conclude tht ' is vlid. Applying repetedly the lst two Clims, we get n immedite proof of Proposition 2.18, by induction on ('). 2 Thus, we hve reduced the problem of proving FMP for LED to the corresponding problem for the simpler lnguge LED 0. STEP 2: Finite Model Property for LED 0 Proposition 2.19 The logic LED 0 hs (strong) finite model property: every stisfible formul ' 0 is stisfible in finite model. Proof. Let ' 0 be stisfible formul in lnguge L = L(P, V r, C, F, R,r) for LED 0, nd let M =(W, D, [0], [1],, k k, ( ), f, R ) 2A,f2F,R2R be model nd w 0 2 W be world such tht w 0 2k' 0 k M. Tke L[T be the smllest set of formuls nd terms in LED 0 tht contins ' 0, 0 nd 1, nd is closed under subterms nd subformuls. 14 It is esy to see tht is finite. Let us put T := T \, L := L\, P := P \, Vr := Vr\, C := C \. We now define n equivlence reltion = on W by putting: w = v i 8' 2 (w 2k'k M, v 2k'k M ). For ny w 2 W we denote by w := {v 2 W : w = v} the =-equivlence clss of w, nd we put W := { w : w 2 W } for the set of ll =-equivlence clsses. Note tht W is finite. Fix now some rbitrry well-ordering < of W. For every clss w 2 W,we denote by w 0 the first element of the clss w (wrt <). Let D 0 := {w(t) : w 2 W,t 2T }[{[0], [1]}. Note tht D 0 is finite. If D \ D 0 6= ;, then choose some d 0 2 D \ D 0, nd put D := D 0 [{d 0 }.IfhoweverD\ D 0 = ;, thenput D := D 0 = D. Note tht in both cses D is finite subset of D. We now define filtrted model M =(W,D, [0], [1],, k k, ( ), f, R ) 2A,f2F,R2R. by tking W nd D s bove, nd putting [0] := [0]; [1] := [1]; w v i 8K 2 (w 2kK k M, v 2kK k M ); kpk := { w : w 2kpk M } for p 2 P, nd kpk := ; for p 2 P \ P ; w ( ) :=w( ) for 2 Vr [ C, nd w ( ) :=d 0 for 2 (Vr[ ) \ (Vr [ C ); f (d 1,...,d n ):=f(d 1,...,d n )if there exists term f(t 1,...,t n ) 2 nd world w 2W s.t. w (t i )=d i for ll i 2{1,...,n}; nd f (d 1,...,d n ):=d 0 otherwise; finlly, R := R \ (D D ). Note tht f : D! D is well-defined function, = is the digonl {(d, d) :d 2 D }, nd M is indeed finite model. Clim D ( Term Lemm ): For every term t 2T nd w 2 W,wehve w (t) = w(t). 14 More precisely, hs to stisfy the following closure conditions: (1) if ('! ) 2 then ', 2 ; (2) if K ' 2 then' 2 ; (3) if f(t 1,...,t n) 2 thent 1,...,t n 2 ; (4) if R(t 1,...,t n) 2 thent 1,...,t n 2.

18 152 To Know is to Know the Vlue of Vrible Proof of Clim D: Proof by induction: the bse cse is by definition; for the inductive step: w (f(t 1,...,t n ) = f ( w (t 1 ),..., w (t n )) = f (w(t 1 ),...,w(t n )) = f(w(t 1 ),...,w(t n )) = w(f(t 1,...,t n )), where we used the induction hypothesis, the definition of f nd f(t 1,...,t n ) 2T. Clim E ( Filtrtion Lemm ) For ll formuls 2, we hve: w 2k k M i w 2k k M, for every w 2 W. Proof of Clim E: Proof by induction on. All steps go s in the clssicl proof of the Filtrtion Lemm (for the logic S5), except for the reltionl toms R(t 1,...,t n ) 2, for which we hve t 1,...,t n 2T, nd thus Clim D cn be pplied. So we hve the sequence of equivlencies: w 2kR(t 1,...,t n k M i ( w (t 1 ),..., w (t n )) 2 R i (by definition of R )( w (t 1 ),..., w (t n )) 2 R i (by Clim D) (w(t 1 ),...,w(t n )) 2 R i w 2kR(t 1,...,t n )k M. This finishes our proof tht LED 0 hs FMP. 2 Putting together Step 1 nd Step 2, we conclude tht LED lso hs FMP, nd thus (being lso xiomtizble) it is decidble. 3 Lerning the Vlue of Vrible We now extend LED with public vlue-nnouncement opertors h!! t i for every tuple! t 2T. These opertors ct on both formuls nd terms. The syntx of Public Announcement Logic of Epistemic Dependency (PALED) is given by: ' ::= p R(! t ) '! ' K! t t h!! t i' t ::= x c? ' f(! t ) h!! t it where x re vribles, c re constnts, t re term in T,! t re finite tuples of terms, nd f nd R re symbols of rity equl to the length of! t. The opertions of propositionl substitution nd vrible substitution cn be extended in the obvious wy to the new formuls nd terms. 15 Semntics. Our notion of model is the sme s for LED. For every model M, we define n extended vlution (truth mp) k'k, n extended vlue mp w(t) M, nd the updte M!t of model M with ny finite string of terms! t 2T. The truth mp nd the extended vlue mp re defined s for LED, except tht we dd the cluses: kh!! t i' k M = k'k M! t, nd w(h!! t it 0 ) M = w(t 0 ) M! t. For the updte M!t := (W, M!, k k, ( ), f, R ), we leve ll the components t the sme, except for chnging the epistemic reltions s follows: M! t = {(w, s) 2 W W w s, w(! t ) M = s(! t ) M }. 15 Formlly, we put: (h!t 1,...,t ni')[p/ ] := h!t 1 [p/ ],...,t n[p/ ]i('[p/ ]); (h!t 1,...,t nit 0 )[p/ ] := h!t 1 [p/ ],...,t n[p/ ]i(t 0 [p/ ]); (h!t 1,...,t ni')[x/t] := h!t 1 [x/t],...,t n[x/t]i('[x/t]); (h!t 1,...,t nit 0 )[x/t] :=h!t 1 [x/t],...,t n[x/t]i(t 0 [x/t]).

19 Bltg 153 So ll gents jointly lern the vlues of! t, nd nothing else chnges. Exmple 2 The formul h!x i(k x b ^ K b x b ) is true in (ll worlds of) the model from Exmple 1 bove. This cn be verified by performing n updte!x, which removes epistemic rrows between worlds hving di erent vlues for x nd checking tht K x b ^ K b x b holds in the updted model. So, fter the vlue of Alice s number is nnounced, everybody will know Bob s number. Propositionl Public Announcements. The stndrd (propositionl) public nnouncement formuls from P AL cn be defined s bbrevitions in our syntx, by putting: h! i := ^h!(? )i, nd [! ] :=!h!(? )i. Proof system. We obtin complete system for P ALED by restricting the Substitution Rules to sttic contexts, nd dding Necessittion Rule for nnouncements, s well s Reduction Axioms. More precisely: (i) Restricted Propositionl Substitution: From ', infer'[p/ ], provided tht p doesn t occur in the scope of ny dynmic opertor in '. (ii) Restricted Vrible Substitution: From ', infer '[x/t], provided tht x doesn t occur in the scope of ny dynmic opertor in '. (iii) All the other xioms nd rules of the system LED. (iv) Necessittion Rule for Announcements: From ` ' infer `h!! t i'. (v) Propositionl Reduction Axioms 16 : (vi) Term Reduction Axioms: h!! t ip $ p h!! t ir(t 1,...,t n ) $ R(h!! t it 1,...,h!! t it n ) h!! t i('! ) $ h!! t i'!h!! t i h!! t ik t1,...,tn t 0 $ K! t,h!! t it 1,...,h!! t it n h!! t it 0 h!! t ic = c h!! t ix = x h!! t i? ' =? h!! t i' h!! t if(t 1,...,t n ) = f(h!! t it 1,...h!! t it n ) Applying the Reduction Axiom itertively, we cn eliminte ll dynmic opertors in the usul wy, nd thus prove: 16 Note tht the third reduction xiom is the xiom K for nnouncements. This is usully stted for universl modlities [ ], butthesecoincidewiththeexistentilonesinourcse, since vlue nnouncements re deterministic ctions (whose trnsition reltions re functions). Combining this xiom with the Necessittion Rule for nnouncements, one cn show tht formuls obtined by prefixing provbly equivlent formuls with dynmic opertors re provbly equivlent. This is needed to pply Reduction Axioms repetedly in order to grdully reduce (nd eventully eliminte) nested nnouncement opertors, e.g. h! t 1 ih! t 2 i.

20 154 To Know is to Know the Vlue of Vrible Theorem 3.1 The proof system P ALED is sound nd wekly complete for the logic P ALED. Moreover, P ALED hs the sme expressivity s LED. 4 Comprison with other work Our epistemic dependency formuls re closely connected to Dependency Logic [18,19]. Note tht K x1,...,xn y expresses only locl dependency (t the ctul world): this is reflected in the fct tht this ttitude is not introspective (i.e. K x1,...,xn y does not imply K K x1,...,xn y). However, its introspective version K K x1,...,xn y gives more globl dependency (cross ll the epistemicllypossible worlds), thus cpturing knowledge of the dependency. It is esy to see tht w = K K x1,...,xn y is equivlent to the ssertion tht the dependence tom =(x 1,...,x n,y) holds t the tem {v : w v} comprising the set of (vrible ssignments ssocited to ll) epistemic lterntives of w. But note tht LED is decidble, in contrst to most vrints of Dependence Logic! Our logic hs lso interesting reltions with the so-clled erotetic logics [8,12,26], including inquisitive logics [10]. First, s Hintikk [13], Sch er [17], Aloni nd others [1] rgued, ll types of knowledge (knowing tht, knowing wht, knowing who, knowing how, knowing whether, knowing which) re specil cses of knowing the nswer to question: All knowledge involves question. To know is to know the nswer ([17]: 401). Second, every vrible (mpping worlds to set D of vlues) induces prtition of the stte spce; so vribles could be used to represent ny prtitionl question. Knowing the nswer to question is the sme s knowing the vlue of the corresponding mp. 17 Our epistemic dependency formuls cpture n epistemic version of interrogtive impliction, s studied in inquisitive logics. But the vrible representtion gives us more informtion: if we identify the vlues in D with bstrct nswers, then we cn compre nswers for di erent questions, nd thus formlize phenomen such s knowing the nswer without knowing the question, tht cnnot be delt with in stndrd inquisitive semntics. We believe tht n ccount of questions s functions from worlds to (sets of) nswers gives better model for interrogtives then the usul inquisitive representtion. In contrst to both Inquisitive Logic nd Dependency Logic, our pproch preserves the clssiclity of propositionl clculus, nd re-interprets the nonclssicl fetures in terms of modl-epistemic opertors. As in Dynmic Epistemic Logic [14,7,5,22,20], nd its most nturl interrogtive versions [3,21,6,4], our semntics is modl in the usul sense, with formuls evluted t worlds, rther thn t sets of worlds. We think of this s n dvntge of our pproch, suggesting tht questions (nd vrible dependency) cn be understood without denying the clssicl logicl principles. To know is to know the nswer. But the logic of Aristotle, Boole nd Frege is not ded just yet. 17 As for the non-prtitionl questions (hving non-unique complete nswers), s considered in Inquisitive Semntics, they could be represented in our frmework s functions from worlds to sets of nswers in P(D). It would be interesting to study the epistemic logic of knowing n nswer (rther thn the nswer) in this generlized frmework.