Clausal Resolution for Modal Logics of Confluence

Transcription

1 Clusl Resolution for Modl Logics of Confluence Cláudi Nlon 1, João Mrcos 2, nd Clre Dixon 3 1 Deprtment of Computer Science, University of Brsíli C.P CEP: Brsíli DF Brzil nlon@unb.br 2 LoLITA nd Dept. of Informtics nd Applied Mthemtics, UFRN, Brzil jmrcos@dimp.ufrn.br 3 Deprtment of Computer Science, University of Liverpool Liverpool, L69 3BX United Kingdom CLDixon@liverpool.c.uk rxiv: v1 [cs.lo] 1 My 2014 Abstrct. We present clusl resolution-bsed method for norml multimodl logics of confluence, whose Kripke semntics re bsed on frmes chrcterised by pproprite instnces of the Church-Rosser property. Here we restrict ttention to eight fmilies of such logics. We show how the inference rules relted to the norml logics of confluence cn be systemticlly obtined from the prmetrised xioms tht chrcterise such systems. We discuss soundness, completeness, nd termintion of the method. In prticulr, completeness cn be modulrly proved by showing tht the conclusions of ech newly dded inference rule ensures tht the corresponding conditions on frmes hold. Some exmples re given in order to illustrte the use of the method. 1 Introduction Keywords: norml modl logics, combined logics, resolution method Modl logics re often introduced s extensions of clssicl logic with two dditionl unry opertors: nd, whose menings vry with the field of ppliction to which they re tilored to pply. In the most common interprettion, formule p nd p re red s p is necessry nd p is possible, respectively. Evlution of modl formul depends upon n orgnised collection of scenrios known s possible worlds. Different modl logics ssume different ccessibility reltions between such worlds. Worlds nd their ccessibility reltions define so-clled Kripke frme. The evlution of formul hinges on such structure: given n pproprite ccessibility reltion nd world w, formul p is stisfied t w if p is true t ll worlds ccessible from w; formul p is stisfied t w if p is true t some world ccessible from w. In norml modl logics extending the clssicl propositionl logic, the schem (ϕ ψ) ( ϕ ψ) (the distribution xiom K), where ϕ nd ψ re wellformed formule nd stnds for clssicl impliction, is vlid, nd the schemtic Pre-print version of the pper ccepted to IJCAR C. Nlon ws prtilly supported by CAPES Foundtion BEX 8712/11-5. J. Mrcos ws prtilly supported by CNPq nd by the EU-FP7 Mrie Curie project PIRSES- GA

2 2 Cláudi Nlon, João Mrcos, nd Clre Dixon rule ϕ/ ϕ (the necessittion rule Nec) preserves vlidity. The wekest of these logics, nmed K (1), is semnticlly chrcterised by the clss of Kripke frmes with no restrictions imposed on the ccessibility reltion. In the multimodl version, nmed K (n), Kripke frmes re directed multigrphs nd modl opertors re equipped with indexes over set of gents, given by A n = {1, 2,..., n}, for some positive integer n. Accordingly, in this cse clssicl logic is extended with opertors 1, 2,..., n, where formul s p, with A n, my be red s gent considers p to be necessry. The modl opertor is the dul of, being introduced s n bbrevition for, where stnds for clssicl negtion. The logic K (n) cn be seen s the fusion of n copies of K (1) nd its xiomtistion is given by the union of the xioms for clssicl propositionl logic with the xiomtic schemt K, nmely (ϕ ψ) ( ϕ ψ), for ech A n ; nd the set of inference rules is given by modus ponens nd the rule schemt Nec, nmely ϕ/ ϕ, for ech A n. The bsic norml multimodl logic K (n) nd its extensions hve been widely used to represent nd reson bout complex systems. Some of the interesting extensions include the norml multimodl logics bsed on K nd (the combintion of) xioms s, for instnce, T ( ϕ ϕ), D ( ϕ ϕ), 4 ( ϕ ϕ), 5 ( ϕ ϕ), nd B ( ϕ ϕ). For exmple, the description logic ALC, which is employed for resoning bout ontologies, is syntctic vrint of K (1) [22]; the epistemic logic, denoted by S5 (n), which is used in deling with problems rnging from multi-gency to communiction protocols [21,11], cn be xiomtised by combining K, T, nd 5. The ddition of those xioms (or their combintions) to K (n) imposes some restrictions on the clss of models where formule re vlid. Thus, formul vlid in logic contining T is vlid only if it is vlid in frme where the ccessibility reltion for ech gent is reflexive. The other xioms, D, 4, 5, nd B, demnd the ccessibility reltion for ech gent to be, respectively, seril, trnsitive, Eucliden, nd symmetric. A logic of confluence K p,q,r,s (n) of the form G p,q,r,s is modl system xiomtised by K (n) plus xioms p q ϕ r s ϕ where A n, ϕ is well-formed formul, p, q, r, s N, where 0 ϕ = def ϕ nd i+1 ϕ = def i ϕ, nd where 0 ϕ = def ϕ nd i+1 ϕ = def i ϕ, for i N (the superscript is often omitted if equl to 1). Such xiomtic schemt were notbly studied by Lemmon [16]. Using Modl Correspondence Theory, it cn be shown tht the frme condition on logic where n instnce of G p,q,r,s is vlid corresponds to generlised dimond-like structure representing the Church-Rosser property (the philosophicl literture sometimes clls such property incestul [8]), s illustrted in Fig. 1 [6]. To be more precise, let W be nonempty set of worlds nd let R W W be the ccessibility reltion of gent A n. By wr 0 w we men tht w = w, nd wr i+1 w mens tht there is some world w such tht wr w nd w R i w. Thus, wr i w holds if there is n i-long R -pth from w to w ; lterntively, to ssert tht, we my lso write (w, w ) R i. Given these definitions, the condition on frmes tht corresponds to the xiom G p,q,r,s is described by w 0, w 1, w 2 (w 0 R p w 1 w 0 R r w 2 w 3 (w 1 R q w 3 w 2 R s w 3 )), where w 0, w 1, w 2, w 3 W.

3 Clusl Resolution for Modl Logics of Confluence 3 w 1 p steps q steps w 0 w 3 r steps s steps w 2 Fig. 1. Church-Rosser property for frmes where G p,q,r,s = p q ϕ r s ϕ is vlid. Mny well-known modl xiomtic systems re identified with prticulr logics of confluence. For instnce, T (n) corresponds to K 0,1,0,0 (n), norml modl logic in which the xiom ϕ ϕ is vlid, for ll A n nd ny formul ϕ. The xiom 4 my be written s G 0,1,2,0, tht is, 1 ϕ 2 ϕ. The Gech xiom G1 is given by G 1,1,1,1 ( ϕ ϕ). Formule in K 1,1,1,1 (n) re stisfible if, nd only if, they re stisfible in model with n reltions stisfying the so-clled dimond property, nd nlogous clims hold for instnce concerning formule of T (n) nd models whose reltions re ll reflexive, nd formule of 4 (n) nd models whose reltions re ll trnsitive. Logics of confluence re interesting not only becuse they encompss gret number of norml modl logics s prticulr exmples, but lso in view of their ttrctive computtionl behviour. Indeed, if we think of multimodl frmes s bstrct rewriting systems, for instnce, nd think of modl lnguges s wy of obtining n internl nd locl perspective on such frmes, then ech given notion of confluence ensures tht certin different pths of trnsformtion will eventully led to the sme result. Hving decidble proof procedure for logic underlying such clss of frmes helps in estblishing direct form of verifying the properties of the structures tht they represent. As contribution towrds uniform pproch to the development of proof methods for logics of confluence, in this work we del with the logics where p, q, r, s {0, 1}. Tble 1 shows the relevnt xiomtic schemt, some stndrd nmes by which they re known, nd the corresponding conditions on frmes. The xiom G 0,1,1,1 seems not to be nmed in the literture; the corresponding property follows the nming convention given in [5, pg. 127]. Note tht G 0,0,0,0, G 0,1,1,0, nd G 1,0,0,1 re instnces of clssicl tutologies nd re thus not included in Tble 1. Also, given the dulity between nd, G p,q,r,s is semnticlly equivlent to G r,s,p,q. Thus, there re in fct eight fmilies of multimodl logics relted to the xioms G p,q,r,s, where p, q, r, s {0, 1}. We present clusl resolution-bsed method for solving the stisfibility problem in logics xiomtised by K plus G p,q,r,s, where p, q, r, s {0, 1}. The resolution cl-

4 4 Cláudi Nlon, João Mrcos, nd Clre Dixon (p,q,r,s) Nme Axioms Property Condition on Frmes (0, 0, 1, 1) B ϕ ϕ symmetric w, w (wr w w R w) (1, 1, 0, 0) ϕ ϕ (0, 0, 1, 0) Bn ϕ ϕ modlly bnl w, w (wr w w = w ) (1, 0, 0, 0) ϕ ϕ (0, 1, 0, 1) D ϕ ϕ seril w w (wr w ) (1, 0, 1, 0) F ϕ ϕ functionl w, w, w ((wr w wr w ) w = w ) (0, 0, 0, 1) T ϕ ϕ reflexive w(wr w) (0, 1, 0, 0) ϕ ϕ (1, 0, 1, 1) 5 ϕ ϕ Eucliden w, w, w ((wr w wr w ) w R w ) (1, 1, 1, 0) ϕ ϕ (1, 1, 1, 1) G1 ϕ ϕ convergent w, w, w ((wr w wr w ) w (w R w w R w )) (0, 1, 1, 1) G 0,1,1,1 ϕ ϕ 0,1,1,1-convergent w, w (wr w w (wr w w R w )) (1, 1, 0, 1) ϕ ϕ Tble 1. Axioms nd corresponding conditions on frmes. culus is bsed on tht of [17], which dels with the logicl frgment corresponding to K (n). The new inference rules to del with xioms of the form G p,q,r,s dd relevnt informtion to the set of cluses: the conclusion of ech inference rule ensures tht properties relted to the corresponding conditions on frmes hold, tht is, the newly dded cluses cpture the required properties of model. We discuss soundness, completeness, nd termintion. Full proofs cn be found in [18]. 2 The Norml Modl Logic K (n) The set WFF K(n) of well-formed formule of the logic K (n) is constructed from denumerble set of propositionl symbols, P = {p, q, p, q, p 1, q 1,...}, the negtion symbol, the conjunction symbol, the propositionl constnt true, nd unry connective for ech gent in the finite set of gents A n = {1,..., n}. When n = 1, we often omit the index, tht is, ϕ stnds for 1 ϕ. As usul, is introduced s n bbrevition for. A literl is either propositionl symbol or its negtion; the set of literls is denoted by L. By l we will denote the complement of the literl l L, tht is, l denotes p if l is the propositionl symbol p, nd l denotes p if l is the literl p. A modl literl is either l or l, where l L nd A n. We present the semntics of K (n), s usul, in terms of Kripke frmes. Definition 1. A Kripke frme S for n gents over P is given by tuple (W, w 0, R 1, R 2,..., R n ), where W is set of possible worlds (or sttes) with

5 Clusl Resolution for Modl Logics of Confluence 5 distinguished world w 0, nd ech R is binry reltion on W. A Kripke model M = (S, π) equips Kripke frme S with function π : W (P {true, flse}) tht plys the role of n interprettion tht ssocites to ech stte w W truthssignment to propositionl symbols. The so-clled ccessibility reltion R is binry reltion tht cptures the notion of reltive possibility from the viewpoint of gent : A pir (w, w ) is in R if gent considers world w possible, given the informtion vilble to her in world w. We write M, w = ϕ (resp. M, w = ϕ) to sy tht ϕ is stisfied (resp. not stisfied) t the world w in the Kripke model M. Definition 2. Stisfction of formul t given world w of model M is set by: M, w = true M, w = p if, nd only if, π(w)(p) = true, where p P M, w = ϕ if, nd only if, M, w = ϕ M, w = (ϕ ψ) if, nd only if, M, w = ϕ nd M, w = ψ M, w = ϕ if, nd only if M, w = ϕ, for ll w such tht wr w The formule flse, (ϕ ψ), (ϕ ψ), nd ϕ re introduced s the usul bbrevitions for true, ( ϕ ψ), ( ϕ ψ), nd ϕ, respectively. Formule re interpreted with respect to the distinguished world w 0, tht is, stisfibility is defined with respect to pointed-models. A formul ϕ is sid to be stisfied in the model M = (S, π) of the Kripke frme S = (W, w 0, R 1,..., R n ) if M, w 0 = ϕ; the formul ϕ is stisfible in Kripke frme S if there is model M of S such tht M, w 0 = ϕ; nd ϕ is sid to be vlid in clss C of Kripke frmes if it is stisfied in ny model of ny Kripke frme belonging to the clss C. 3 Resolution for K (n) In [17], sound, complete, nd terminting resolution-bsed method for K (n), which in this pper we cll RES K, is introduced. As the proof-method for logics of confluence presented here relies on RES K, in order to keep the present pper self-contined, we reproduce the corresponding inference rules here nd refer the reder to [17] for detiled ccount of the method. The pproch tken in the resolution-bsed method for K (n) is clusl: formul to be tested for (un)stisfibility is first trnslted into norml form, explined in Section 3.1, nd then the inference rules given in Section 3.2 re pplied until either contrdiction is found or no new cluses cn be generted. 3.1 A Norml Form for K (n) Formule in the lnguge of K (n) cn be trnsformed into norml form clled Seprted Norml Form for Norml Logics (SNF). As the semntics is given with respect to pointed-model, we dd nullry connective strt in order to represent the world from which we strt resoning. Formlly, given model M = (W, w 0, R 1,..., R n, π), we hve tht M, w = strt if, nd only if, w = w 0. A formul in SNF is represented by

6 6 Cláudi Nlon, João Mrcos, nd Clre Dixon conjunction of cluses, which re true t ll rechble sttes, tht is, they hve the generl form i A i, where A i is cluse nd, the universl opertor, is chrcterised by (the gretest fixed point of) ϕ ϕ A n ϕ, for formul ϕ. Observe tht stisfction of ϕ imposes tht ϕ must hold t the ctul world w nd t every world rechble from w, where rechbility is defined in the usul (grph-theoretic) wy. Cluses hve one of the following forms: Initil cluse strt Literl cluse true Positive -cluse l l r b=1 r b=1 Negtive -cluse l l where l, l, l b L. Positive nd negtive -cluses re together known s modl - cluses; the index my be omitted if it is cler from the context. The trnsltion to SNF uses rewriting of clssicl opertors nd the renming technique [20], where complex subformule re replced by new propositionl symbols nd the truth of these new symbols is linked to the formule tht they replced in ll sttes. Given formul ϕ, the trnsltion procedure is pplied to (strt t 0 ) (t 0 ϕ), where t 0 is new propositionl symbol. The universl opertor, which surrounds ll cluses, ensures tht the cluses generted by the trnsltion of formul re true t ll rechble worlds. Clssicl rewriting is used to remove some clssicl opertors from ϕ (e.g. (t ψ 1 ψ 2 ) is rewritten s (t ψ 1 ) (t ψ 2 )). Renming is used to replce complex subformule in disjunctions (e.g. if ψ 2 is not literl, (t ψ 1 ψ 2 ) is rewritten s (t ψ 1 t 1 ) (t 1 ψ 2 ), where t 1 is new propositionl symbol) or in the scope of modl opertors (e.g. if ψ is not literl, (t ψ) is rewritten s (t t 1 ) (t 1 ψ), where t 1 is new propositionl symbol). We refer the reder to [17] for detils on the trnsformtion rules tht define the trnsltion to SNF, their correctness, nd exmples of their ppliction. l b l b 3.2 Inference Rules for K (n) In the following, l, l, l i, l i L (i N) nd D, D re disjunctions of literls. Literl Resolution. This is clssicl resolution pplied to the clssicl propositionl frgment of the combined logic. An initil cluse my be resolved with either literl cluse or nother initil cluse (rules IRES1 nd IRES2). Literl cluses my be resolved together (LRES). [IRES1] (true D l) (strt D l) (strt D D ) [IRES2] (strt D l) (strt D l) (strt D D ) [LRES] (true D l) (true D l) (true D D )

7 Clusl Resolution for Modl Logics of Confluence 7 Modl Resolution. These rules re pplied between cluses which refer to the sme context, tht is, they must refer to the sme gent. For instnce, we my resolve two or more -cluses (rules MRES nd NEC2); or severl -cluses nd literl cluse (rules NEC1 nd NEC3). The modl inference rules re: [MRES] (l 1 l) (l 2 l) (true l 1 l 2) [NEC1] (l 1 l 1). (l m l m) (l l) (true l 1... l m l) (true l 1... l m l ) [NEC2] (l 1 l 1) (l 2 l 1) (l 3 l 2) (true l 1 l 2 l 3) [NEC3] (l 1 l 1). (l m l m) (l l) (true l 1... l m) (true l 1... l m l ) The rule MRES is syntctic vrition of clssicl resolution, s formul nd its negtion cnnot be true t the sme stte. The rule NEC1 corresponds to necessittion (pplied to ( l 1... l m l), which is equivlent to the literl cluse in the premises) nd severl pplictions of clssicl resolution. The rule NEC2 is specil cse of NEC1, s the prent cluses cn be resolved with the tutology true l 1 l 1 l 2. The rule NEC3 is similr to NEC1, however the negtive modl cluse is not resolved with the literl cluse in the premises. Insted, the negtive modl cluse requires tht resolution tkes plce between literls on the right-hnd side of positive modl cluses nd the literl cluse. The resolvents in the inference rules NEC1 NEC3 impose tht the literls on the left-hnd side of the modl cluses in the premises re not ll stisfied whenever their conjunction leds to contrdiction in successor stte. Given the syntctic forms of cluses, the three rules re needed for completeness [17]. Note tht for NEC1, we my hve m = 0; for NEC2 the number of premises is fixed; nd tht for NEC3, if m = 0, then the literl cluse in the premises is true flse, which cnnot be stisfied in ny model. Thus, NEC3 is not pplied when m = 0. We define derivtion s sequence of sets of cluses T 0, T 1,..., where T i results from dding to T i 1 the resolvent obtined by n ppliction of n inference rule of RES K to cluses in T i 1. A derivtion termintes if, nd only if, either contrdiction, in the form of (strt flse) or (true flse), is derived or no new cluses cn be derived by further ppliction of the resolution rules of RES K. We ssume stndrd simplifiction from clssicl logic to keep the cluses s simple s possible. For exmple, D l l on the right-hnd side of cluse would be rewritten s D l. Exmple 1. We wish to check whether the formul 1 2 ( b) 1 ( 2 2 b) is vlid in K (2). The trnsltion of its negtion into the norml form is given by cluses (1) (9) below. Then the inference rules re pplied until flse is generted. In order to improve redbility, the universl opertor is suppressed. The full refuttion follows.

8 8 Cláudi Nlon, João Mrcos, nd Clre Dixon 1. strt t 1 2. t 1 1 t 2 3. t 2 2 t 3 4. true t 3 5. true t 3 b 6. t 1 1 t 4 7. true t 4 t 5 t 6 8. t t 6 2 b 10. true t 2 t 5 [NEC1, 3, 8, 4] 11. true t 2 t 4 t 6 [LRES, 10, 7] 12. true t 2 t 6 [NEC1, 3, 9, 5] 13. true t 2 t 4 [LRES, 12, 11] 14. true t 1 [NEC1, 2, 6, 13] 15. strt flse [IRES1, 14, 1] Cluses (10) nd (12) re obtined by pplictions of NEC1 to cluses in the context of gent 2. Cluse (14) is obtined by n ppliction of the sme rule, but in the context of gent 1. Cluses (11) nd (13) result from pplictions of resolution to the propositionl prt of the lnguge shred by both gents. Cluse (15) shows tht contrdiction ws found t the initil stte. Therefore, the originl formul is vlid. 4 Clusl Resolution for Logics of Confluence The inference rules of RES K, given in Section 3.2, re resolution-bsed: whenever set of (sub)formule is identified s contrdictory, the resolvents require tht they re not ll stisfied together. The extr inference rules for K p,q,r,s (n), with p, q, r, s {0, 1}, which we re bout to present, hve different flvour: whenever we cn identify tht the set of cluses imply tht p q ψ holds, we dd some new cluses tht ensure tht r s ψ lso holds. If this is not the cse, tht is, if the set of cluses implies tht r s ψ holds, then contrdiction is found by pplying the inference rules for K (n). Becuse of the prticulr norml form we use here, there re, in fct, two generl forms for the inference rules for K p,q,r,s (n), given in Tble 2 (where l, l re literls nd C is conjunction of literls). [RES p,1,r,s ] (l l ) ( p l r s l ) [RES p,0,r,s ] (C p l ) (C r s l ) Tble 2. Inference Rules for G p,q,r,s Soundness is checked by showing tht the trnsformtion of formul ϕ WFF K(n) into its norml form is stisfibility-preserving nd tht the ppliction of the inference rules re lso stisfibility-preserving. Stisfibility-preserving results for the trnsformtion into SNF re provided in [17]. To extend the soundness results so s to cover the new inference rules, note tht the conclusions of the inference rules in Tble 2 re derived using the semntics of the universl opertor nd the distribution xiom, K. For RES p,1,r,s, we hve tht the premise (l l ) is semnticlly equivlent to ( l l). By the definition of the universl opertor, we

9 Clusl Resolution for Modl Logics of Confluence 9 obtin ( p ( l l)). Applying the distribution xiom K to this cluse results in ( p l p l)), which is semnticlly equivlent to ( p l p l ). As is n bbrevition for nd becuse p l implies r s l in K p,1,r,s (n), by clssicl resoning, we hve tht ( p l p l ) implies ( p l r s l ), the conclusion of RES p,1,r,s. Soundness of the inference rule RES p,0,r,s cn be proved in similr wy. As the conclusions of the bove inference rules my contin complex formule, they might need to be rewritten into the norml form. Thus, we lso need to dd cluses corresponding to the norml form of p l nd s l, which occur in the conclusions of the inference rules. Let ϕ be formul nd let τ(ϕ) be the set of cluses resulting from the trnsltion of ϕ into the norml form. Let L(τ(ϕ)) be the set of literls tht might occur in the cluse set, tht is, for ll p P such tht p occurs in τ(ϕ), we hve tht both p nd p re in L(τ(ϕ)). The set of definition cluses is given by (pos,l l) ( pos,l l) for ll l L(τ(ϕ)), where pos,l is new propositionl definition symbol used for renming the negtive modl literl l, tht is, the definition cluses correspond to the norml form of pos,l l. Note tht we hve definition cluses for every propositionl symbol nd its negtion, e.g. for propositionl symbol p τ(ϕ), we hve the definition cluses (pos,p p), ( pos,p p), (pos, p p), nd ( pos, p p), for every A n occurring in τ(ϕ). We ssume the set of definition cluses to be vilble whenever those symbols re used. It is lso importnt to note tht those new definition symbols nd the respective definition cluses cn ll be introduced t the beginning of the ppliction of the resolution method becuse we do not need definition cluses pplied to definition symbols in the proofs, s given in the completeness proof [18]. As no new propositionl symbols re introduced by the inference rules, there is finite number of cluses tht might be expressed (modulo simplifiction) nd, therefore, the clusl resolution method for ech modl logic of confluence is terminting. As discussed bove nd from the results in [17], we cn estblish the soundness of the proof method. Theorem 1. The resolution-bsed clculi for logics of confluence re sound. Proof (Sketch). The trnsformtion into the norml form is stisfibility preserving [17]. Given set T of cluses nd model M tht stisfies T, we cn construct model M for the union of T nd the definition cluses, where M nd M my differ only in the vlution of the definition symbols. By setting properly the vlutions in M, we hve tht M, w = pos,p if nd only if M, w = p, for ny w W. Soundness of the inference rules for RES K is lso given in [17]. Soundness of RES p,1,r,s follow from the xiomtistion of K p,q,r,s RES p,0,r,s (n). Tble 3 shows the inference rules for ech specific instnce of G p,q,r,s, where p, q, r, s {0, 1}, l, l L, nd D is disjunction of literls. As G p,q,r,s is semnticlly equivlent to G r,s,p,q, the inference rules for both systems re grouped together. Some of the inference rules in Tble 3 re obtined directly from Tble 2. For nd

10 10 Cláudi Nlon, João Mrcos, nd Clre Dixon Logic Inference Rules T [RES 0,0,0,1 ] (true D l) ( D l) [RES 0,1,0,0 ] (l l ) (true l l ) Bn [RES 0,0,1,0 ] (true D l) ( D l) [RES 1,0,0,0 ] (l l ) (true l l ) B [RES 0,0,1,1 ] (true D l) ( D pos,l ) [RES 1,1,0,0 ] (l l ) ( l l) D [RES 0,1,0,1 ] (l l ) (l l ) Logic G 0,1,1,1 Inference Rules [RES 0,1,1,1 ] (l l ) (l pos,l ) [RES 1,1,0,1 ] (l l ) (pos,l l ) F [RES 1,0,1,0 ] (l l ) (l l ) 5 [RES 1,0,1,1 ] (l l ) (l pos,l ) [RES 1,1,1,0 ] (l l ) (pos,l l ) G1 [RES 1,1,1,1 ] (l l ) (pos,l pos,l ) Tble 3. Inference Rules for severl instnces of G p,q,r,s is vlid, hs the form (l l )/ ( 0 l 0 0 l ) in Tble 2; in Tble 3, the conclusion is rewritten in its norml form, tht is, (true l l ). For other systems, the form of the inference rules re slightly different from wht would be obtined from direct ppliction of the generl inference rules in Tble 2. This is the cse, for instnce, for the instnce, the rule for reflexive systems, i.e. where the xiom G 0,1,0,0 inference rules for symmetric systems, tht is, those systems where the xiom G 1,1,0,0 is vlid. From Tble 2, in symmetric systems, for premise of the form (l l ), the conclusion is given by ( l l ), which is trnslted into the norml form s (true pos,l l ). We hve chosen, however, to trnslte the conclusion s ( l l), which is semnticlly equivlent to the conclusion obtined by the generl inference rule, but voids the use of definition symbols. The inference rules given in Tble 2 provide systemtic wy of designing the inference rules for ech specific modl logic of confluence. We note, however, tht we do not lwys need both inference rules in order to chieve complete proof method for prticulr logic. In the completeness proofs provided in [18], we show for instnce tht the inference rules which introduce modlities in their conclusions from literl cluses (tht is, the inference rules RES 0,0,r,s ) re not needed for completeness. We lso show tht we need just one specific inference rule for logics in which G 0,1,1,1 nd 5 re vlid: RES 0,1,1,1 nd RES 1,0,1,1, respectively. Given formul ϕ in K p,q,r,s (n), with p, q, r, s {0, 1}, the resolution method for K (n), given in Section 3, nd the inference rule RES p,q,r,s re pplied to τ(ϕ) nd the set of definition cluses. The extr inference rules for K p,q,r,s (n) do not need to be pplied to cluses if such ppliction genertes new nested definition symbols,

11 Clusl Resolution for Modl Logics of Confluence 11 tht is, we do not need definition cluses for definition symbols. For instnce, the ppliction of RES 1,1,1,1 to cluse of the form (l pos,l ) would result in (pos,l pos,pos,l ). Although it is not incorrect to pply the inference rules to such cluse, this might cuse the method not to terminte. We cn show, however, tht the ppliction of inference rules to cluses which would result in nested literls is not needed for completeness, s the restrictions imposed by those symbols re lredy ensured by existing definition symbols nd relevnt inference rules (see Theorem 3 below). This ensures tht no new definition symbols re introduced by the proof method. Completeness is proved by showing tht, for ech specific logic of confluence, if given set of cluses is unstisfible, there is refuttion produced by the method presented here. The proof is by induction on the number of nodes of grph, known s behviour grph, built from set of cluses. The grph construction is similr to the construction of cnonicl model, followed by filtrtions bsed on the set of formule (or cluses), often used to check completeness for proof methods in modl logics (see [3], for instnce, for definitions nd exmples). Intuitively, nodes in the grph correspond to sttes nd re defined s mximlly consistent sets of literls nd modl literls occurring in the set of cluses, including those literls introduced by definition cluses. Tht is, for ny literl l occurring in the set of cluses, including definition cluses, nd gents A n, node contins either l or l; nd either l or l. The set of edges correspond to the gents ccessibility reltions. Edges or nodes tht do not stisfy the set of cluses re deleted from the grph. Such deletions correspond to pplictions of one or more of the inference rules. We prove tht n empty behviour grph corresponds to n unstisfible set of cluses nd tht, in this cse, there is refuttion using the inference rules for RES K, given in Section 3, nd the inference rules for the specific logic of confluence, presented in Tble 3. Theorem 2. Let T be n unstisfible set of cluses in G p,q,r,s, with p, q, r, s {0, 1}. A contrdiction cn be derived by pplying the resolution rules for RES K, presented in Section 3, nd Tble 3. Proof (Sketch). We construct behviour grph nd show tht the ppliction of rules in Tble 3 removes nodes nd edges where the corresponding frme condition does not hold. The full proof is provided in [18]. Theorem 3. The resolution-bsed clculi for logics of confluence terminte. Proof (Sketch). From the completeness proof, the introduction of literl such s pos,pos,l for n gent nd literl l is not needed. We cn show tht the restrictions imposed by such cluses, together with the resolution rules for ech specific logicl system, re enough to ensure tht the corresponding frme condition lredy holds. As the proof method does not introduce new literls in the cluse set, there is only finite number of cluses tht cn be expressed. Therefore, the proof method is terminting. Exmple 2. We show tht ϕ = def p 1 1 p, which is n instnce of B 1, is vlid formul in symmetric systems. As symmetry is implied by reflexivity nd Euclidenness, insted of using RES 1,1,0,0 1, we combine the inference rules for both T 1 nd 5 1. Cluses (1) (4) correspond to the trnsltion of the negtion of ϕ into the norml form. Cluses (5) (8) re the definition cluses used in the proof.

12 12 Cláudi Nlon, João Mrcos, nd Clre Dixon 1. strt t 0 2. true t 0 p 3. t 0 1 t 1 4. t 1 1 p 5. pos 1,t1 1 t 1 [Def. pos 1,t1 ] 6. pos 1,t1 1 t 1 [Def. pos 1,t1 ] 7. pos 1,p 1 p [Def. pos 1,p] 8. pos 1,p 1 p [Def. pos 1,p] 9. true t 0 pos 1,t1 [MRES, 5, 3] 10. true t 1 pos 1,p [MRES, 7, 4] 11. pos 1,p 1 pos 1,p [RES 1,0,1,1 1, 7] 12. true pos 1,p pos 1,t1 [NEC1, 11, 6, 10] 13. true p pos 1,p [RES 0,1,0,0 1, 8] 14. true p pos 1,t1 [LRES, 13, 12] 15. true t 0 p [LRES, 14, 9] 16. true t 0 [LRES, 15, 2] 17. strt flse [IRES1, 16, 1] Cluse (11) results from pplying the Eucliden inference rule to cluse (7). Cluse (13) results from pplying the reflexive inference rule to (8). The remining cluses re derived by the resolution clculus for K (1). As contrdiction is found, given by cluse (17), the set of cluses is unstisfible nd the originl formul ϕ is vlid. 5 Closing Remrks We hve presented sound, complete, nd terminting proof method for logics of confluence, tht is, norml multimodl systems where xioms of the form G p,q,r,s = p q ϕ r s ϕ where p, q, r, s {0, 1}, re vlid. The xioms G p,q,r,s provide generl form for xioms widely used in logicl formlisms pplied to representtion nd resoning within Computer Science. We hve proved completeness of the proof method presented in this pper for eight fmilies of logics nd their fusions. The inference rules for prticulr instnces of these logics cn be systemticlly obtined nd the resulting clculus cn be implemented by dding to the existing prover for K (n) [24] the cluses dependent on the cluse-set. Efficiency, of course, depends on severl spects. Firstly, for certin clsses of problems, dedicted proof methods might be more efficient. For instnce, if the stisfibility problem for prticulr logic is in NP (s in the cse of S5 (1) ), then our procedure my be less efficient s the stisfibility problem for K (1) is lredy PSPACE-complete [15]. Secondly, efficiency might depend on the inference rules chosen to produce proofs for specific logic. For instnce, for S5 (n), the user cn choose the inference rules relted to reflexivity nd Euclidenness, or choose the inference rules relted to serility, symmetry, nd Euclidenness. The number of inference rules used to test the unstisfibility of set of cluses for prticulr logic might ffect the number of cluses generted by the resolution method s well s the size of the proof. As in the cse of derived inference rules in other proof methods, using more inference rules might led to shorter proofs. Thirdly, s in the cse of the resolution-bsed method for propositionl logic, efficiency might be ffected by strtegies used to serch for proof. Future work includes the design of strtegies for RES K (n) nd for specific logics of confluence. Fourthly, efficiency might lso depend on the form of the input problem. For instnce, comprisons

13 Clusl Resolution for Modl Logics of Confluence 13 between tbleux methods nd resolution methods [14,13] hve shown tht there is no overll better pproch: for some problems resolution proof methods behve better, for others tbleux bsed methods behve better. Providing resolution-bsed method for the logics xiomtised by K nd G p,q,r,s gives the user choice for utomted tools tht cn be used depending on the type of the input formule. There re quite few dedicted methods for the logics presented in this pper. In generl, however, those methods do not provide systemtic wy of deling with logics bsed on similr xioms or their extensions. Therefore, we restrict ttention here to methods relted to logics of confluence. Tbleux methods for logics of confluence where the mono-modl xioms T, D, B, 4, 5, De (for density, the converse of 4), nd G re vlid, cn be found in [7,9]. For ech of those xioms, tbleu inference rule is given. The inference rules cn then be combined in order to provide proof methods for modl logics under S5 (1). Whilst the tbleux procedures in [7,9] re designed for mono-modl logics they seem to be extndble to multimodl logics s long s there re no interctions between modlities. Those procedures do not cover ll the logics investigted in this pper. In [2], lbelled tbleux re given for the mono-modl logics xiomtised by K nd xioms G p,q,r,s where q = s = 0 implies p = r = 0. This restriction voids the introduction of inference rules relted to the identity predicte, but lso excludes, for instnce, functionl nd modlly bnl systems, which re treted by the method introduced in the present pper. In [4], hybrid logic tbleux methods for logics of confluence re given: the inference rules crete nodes, lbelled by nominls. The nominls re used in order to eliminte the Skolem function relted to the existentil quntifier in the first-order sentence corresponding to the xiom G p,q,r,s. This proof method provides tbleu rules for ll instnces of the xiom. Soundness nd completeness re discussed, but termintion of the method is not delt with nd it is not cler wht re the bounds for creting new nodes in the generl cse. In [12], sound, complete, nd terminting disply clculi for tense logics nd some of its extensions, including those with the xiom G p,q,r,s, re presented. It hs been shown tht these clculi hve the property of seprtion, tht is, they provide complete proof methods for the component frgments. The pper investigtes the reltion between the disply clculi nd deep inference systems (where the sequent rules cn be pplied t ny node of proof tree). By finding pproprite propgtion rules for the fusion of tense logic with either S4 (1), S5 (1), or functionl systems, completeness of serch strtegies re presented. However, propgtion rules for the xiom of convergence, G1, or for the combintion of pth xioms (i.e. xioms of the form i ϕ j ϕ) with serility re not given. Also relted, in [1], prefixed tbleux procedures for confluence logics tht vlidte the multimodl version of the xiom b ϕ c d ϕ, where ϕ is formul, re given. Note tht the logics in [1] re systems with instnces of the xiom G 1,1,1,1,b,c,d, tht is, logic which llows the interction of the gents, b, c, d A n, nd might led to undecidble systems. To the best of our knowledge, there re no resolution-bsed proof methods for logics of confluence. However, resolution-bsed methods for modl logics, bsed on trnsltion into first-order logic, hve been proposed for severl modl logics. A survey on trnsltion-bsed pproches for non-trnsitive modl logics (i.e. modl logics tht do not include the xiom 4) cn be found in [19]. The trnsltion-bsed pproch hs the

14 14 Cláudi Nlon, João Mrcos, nd Clre Dixon cler dvntge of being esily implemented, mking use of well-estblished theoremprovers, nd deling with ny logic tht cn be embedded into first-order, should it be decidble or not. However, first-order provers cnnot del esily with logics tht embed some properties which re covered by prticulr xioms of confluence (e.g. functionlity). In order to void such problemtic frgments within first-order logic, the xiomtic trnsltion principle for modl logic, introduced in [23], besides using the stndrd trnsltion of modl formule into first-order, tkes n xiomtistion for prticulr modl logic nd introduces set of first-order modl xioms in the form of schem cluses. As n exmple, dpted from [23], in order to prove tht p is stisfible in KT4 (n), for ech modl subformul (i.e. p nd p) nd for ech considered xiom (i.e. T nd 4), one schem cluse is dded, resulting in: Q (x) R(x, y) Q p (y) Q p (x) Q p (y) p Q (x) R(x, y) Q p (y) Q p (x) Qp(y) p where the predicte Q ϕ (x) cn be red s ϕ holds t world x nd R is the predicte symbol to express the ccessibility reltion for gent. Note tht the cluses on the left re relted to trnsitivity (4) nd the two cluses on the right re relted to reflexivity (T). The xiomtic trnsltion pproch is similr to the pproch tken in the present pper nd in [17] s the schem cluses provide wy of tlking bout properties of the ccessibility reltion. As in our cse, soundness follows esily from the properties of the trnsltion. Termintion follows from the fct tht only finite number of schem cluses re needed. However, s in the cse of the proof method presented here, generl completeness of the method is difficult to be proved nd it is given only for prticulr fmilies of logics. In [10], trnsltion-bsed pproch for properties which cn be expressed by regulr grmmr logics (including trnsitivity nd Euclideness) is given. Completeness of the method hs been proved for some fmilies of logics. In the present pper, we hve restricted ttention to the cse where p, q, r, s {0, 1}, but we believe tht the proof method cn be extended in uniform wy for deling with the unstisfibility problem for ny vlues of p, q, r, nd s, by dding inference rules of the following form: [RES p,q,r,s ] (l p r l ) (l r s l ) which requires serch for cluses tht correspond to the norml form of the premise nd the introduction of s mny new definition symbols s the number of modlities occurring in the conclusion. The inference rule RES p,q,r,s is obviously sound, but we hve yet to identify the restrictions on the number of new propositionl symbols introduced by the method in order to ensure termintion. Future work includes this extension, the complexity nlysis, the implementtion of the proof method, nd prcticl comprisons with other methods. References 1. M. Bldoni, L. Giordno, nd A. Mrtelli. A tbleu clculus for multimodl logics nd some (un)decidbility results. In Proc. of TABLEAUX-98, pges Springer-Verlg, 1998.

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