The Meaning of Negative Premises in Transition System Specifications II

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1 The Mening of egtive Premises in Trnsition System Specifictions II R.J. vn Gleek Computer Science Deprtment, Stnford University Stnford, CA , USA. This pper reviews severl methods to ssocite trnsition reltions to trnsition system specifictions with negtive premises in Plotkin s structurl opertionl style. Besides forml comprison on generlity nd reltive consistency, the methods re lso evluted on their tste in determining which specifictions re meningful nd which re not. Additionlly, this pper contriutes proof theoretic chrcteristion of the well-founded semntics for logic progrms. 1 Trnsition system specifictions & Introduction In this pper V nd A re two sets of vriles nd ctions. Mny concepts tht will pper re prmeterised y the choice of V nd A, ut s in this pper this choice is fixed, corresponding index is suppressed. Definition 1 (Signtures). A function declrtion is pir (f, n) of function symol f V nd n rity n I. A function declrtion (c, 0) is lso clled constnt declrtion. A signture is set of function declrtions. The set T(Σ) of terms over signture Σ is defined recursively y: V T(Σ), if (f, n) Σ nd t 1,..., t n T(Σ) then f(t 1,..., t n ) T(Σ). A term c() is often revited s c. A Σ-sustitution σ is prtil function from V to T(Σ). If σ is sustitution nd S ny syntctic oject (uilt from terms), then S[σ] denotes the oject otined from S y replcing, for x in the domin of σ, every occurrence of x in S y σ(x). In tht cse S[σ] is clled sustitution instnce of S. S is sid to e closed if it contins no vriles. The set of closed terms is denoted T(Σ). Definition 2 (Trnsition system specifictions). Let Σ e signture. A positive Σ-literl is n expression t t nd negtive Σ-literl n expression t or t t with t, t T(Σ) nd A. For t, t T(Σ) the literls t t nd t, s well s t t nd t t, re sid to deny ech other. A trnsition rule over Σ is n expression of the form H α with H set of Σ-literls (the premises or ntecedents of the the rule) nd α Σ-literl (the conclusion). A rule H α with H = is lso written α. An ction rule is trnsition rule with positive conclusion. A trnsition system specifiction (TSS) is pir (Σ, R) with Σ signture nd R set of ction rules over Σ. This mild revision of Stnford report STA-CS-T-95-16, with dded emphsis on 3-vlued interprettions. An extended strct ppered in F. Meyer uf der Heide & B. Monien, editors: Automt, Lnguges nd Progrmming, Proc. 23 th Interntionl Colloquium, ICALP 96, Pderorn, Germny, LCS 1099, Springer, 1996, pp This work ws supported y OR under grnt numer J The revision ws written while the uthor ws employed t the tionl ICT Austrli, nd t IRIA, Sophi Antipolis, Frnce. 1

2 A TSS is stndrd if its rules hve no premises of the form t t, nd positive if ll premises of its rules re positive. The first systemtic study of trnsition system specifictions with negtive premises ppers in Bloom, Istril & Meyer [2]. The concept of (positive) TSS presented ove ws introduced in Groote & Vndrger [10]; the negtive premises t were dded in Groote [9]. The notion generlises the GSOS rule systems of [2] nd constitutes the first formlistion of Plotkin s Structurl Opertionl Semntics (SOS) [11] tht is sufficiently generl to cover most of its pplictions. The premises t t re dded here, minly for technicl resons. The following definition tells when trnsition is provle from TSS. It generlises the stndrd definition (see e.g. [10]) y (lso) llowing the derivtion of trnsition rules. The derivtion of trnsition t t corresponds to the derivtion of the trnsition rule with H =. The H t t cse H corresponds to the derivtion of t t under the ssumptions H. Definition 3 (Proof ). Let P = (Σ, R) e TSS. A proof of trnsition rule H α from P is well-founded, upwrdly rnching tree of which the nodes re lelled y Σ-literls, such tht: the root is lelled y α, nd if β is the lel of node q nd K is the set of lels of the nodes directly ove q, then either K = nd β H, or K β is sustitution instnce of rule from R. If proof of H α from P exists, then H α is provle from P, nottion P H α. A closed negtive literl α is refutle if P β for literl β denying α. Definition 4 (Trnsition reltion). Let Σ e signture. A trnsition reltion over Σ is reltion T T(Σ) A T(Σ). Elements (t,, t ) of trnsition reltion re written s t t. Thus trnsition reltion over Σ cn e regrded s set of closed positive Σ-literls (trnsitions). A closed literl α holds in trnsition reltion T, nottion T = α, if α is positive nd α T or α = (t t ) nd (t t ) T or α = (t ) nd (t t ) T for no t T(Σ). Write T = H, for H set of closed literls, if T = α for ll α H. Write T = p, for p closed proof, if T = α for ll literls α tht pper s node-lels in p. The min purpose of TSS (Σ, R) is to specify trnsition reltion over Σ. A positive TSS specifies trnsition reltion in strightforwrd wy s the set of ll provle trnsitions. But s pointed out in Groote [9], it is much less trivil to ssocite trnsition reltion to TSS with negtive premises. Severl solutions re proposed in [9] nd Bol & Groote [3]. Here I will present these solutions from somewht different point of view, nd lso review few others. P 1 c c c c c c The TSS P 1 cn e regrded s n exmple of TSS tht does not specify well-defined trnsition reltion (under ny plusile definition of specify ). 1 So unless systemtic wy cn e found to ssocite mening to TSSs like P 1, one hs to ccept tht some TSSs re meningless. Hence there re two questions to nswer: Which TSSs re meningful, (1) nd which trnsition reltions do they specify? (2) 1 All my exmples P i consider TSSs (Σ, R) in which Σ consists of the single constnt c only. 2

3 In this pper I present 11 possile nswers to these questions, ech consisting of clss of TSSs nd mpping from this clss to trnsition reltions. Two such solutions re consistent if they gree which trnsition reltion to ttch to TSS in the intersection of their domins. Solution S extends S if the clss of meningful TSSs ccording to S extends tht of S nd the two re consistent, i.e. seen s prtil functions S is included in S. I will compre the 11 solutions on consistency nd extension, nd evlute them on their tste in determining which specifictions re meningful nd which re not. A trnsition reltion cn e seen s function T : T(Σ) A T(Σ) {present, sent}, telling which potentil trnsitions re present in T nd which re sent. A 3-vlued trnsition reltion T : T(Σ) A T(Σ) {present, undetermined, sent} extends this concept y leving the vlue of certin trnsitions undetermined. Although there turns out to e no stisfctory wy to ssocite (2-vlued) trnsition reltion to every TSS, I present two stisfctory methods to ssocite 3- vlued trnsition reltion to every TSS. One of these is the well-founded semntics of Vn Gelder, Ross & Schlipf [6]; the other my e new. I contriute proof theoretic chrcteristions of these 3-vlued solutions. The most generl completely cceptle nswer to (1) when insisting on 2-vlued trnsition reltions, is, in my opinion: the TSSs whose well-founded semntics is 2-vlued. Logic progrmming The prolems nlysed in [9] in ssociting trnsition reltions to TSSs with negtive premises hd een encountered long efore in logic progrmming, nd most of the solutions reviewed in the present pper stem from logic progrmming s well. However, the proof theoretic pproch to Solutions 7 nd I, s well s Solutions 6, 8, 9 nd II nd some comprtive oservtions, re, s fr s I know, new here. The connection with logic progrmming my e est understood y introducing proposition system specifictions (PSSs). These re otined y replcing the set A of ctions y set of predicte declrtions (p, n) with p V predicte symol (different from ny function symol) nd n I. A literl is then n expression p(t 1,..., t n ) or p(t 1,..., t n ) with t i T(Σ). A PSS is now defined in terms of literls in sme wy s TSS. A proposition is closed positive literl, nd proposition reltion or closed theory set of propositions. The prolem of ssociting proposition reltion to PSS is of similr nture s ssociting trnsition reltion to TSS, nd in fct ll concepts nd results mentioned in this pper pply eqully well to oth situtions. If I would not consider TSSs involving literls of the form t, TSS would e specil cse of PSS, nmely the cse where ll predictes re inry, nd it would mke sense to present the pper in terms of PSSs. The min reson for not doing so is to do justice to the rôle of literls t in denying literls of the form t t. However, every TSS cn e encoded s PSS nd vice vers, in such wy tht ll concepts of this pper re preserved under the trnsltions. In order to encode PSS s specil kind of TSS, first of ll n n-ry predicte p cn e expressed in terms of n n-ry function f p nd the unry predicte holds, nmely y defining holds(f p (t 1,..., t n )) s p(t 1,..., t n ). ext, if p is unry predicte then p(t) cn e encoded s the p trnsition t 0, with 0 constnt introduced specilly for this purpose (cf. Verhoef [14]). A TSS cn e encoded s PSS y considering to e inry predicte for ny A, or, s in Bol & Groote [3], s single ternry predicte with A interpreted s term. A negtive literl t t denotes (t t ) nd t cn e seen s n revition of the (infinite) conjunction of t t for t T(Σ). These trnsltions preserve ll concepts of this pper. In order to void the infinite conjunction, Bol & Groote introduce the unry version of (or ctully the inry version of ) s seprte predicte, linked to the inry (ternry) version y the 3

4 rule x y x, implicitly present in every TSS. As shown in nomly A.3 in [3] this trnsltion does not preserve Solution 2 (lest model). However, it does preserve the other concepts. A logic progrm is just PSS oeying some finiteness conditions. Hence everything I sy out TSSs pplies to logic progrmming too. Consequently, this pper cn in prt e regrded s n overview of topic within logic progrmming, ut voiding the logic progrmming jrgon. However, I do not touch issues tht re relevnt in logic progrmming, ut not mnifestly so for trnsition system specifictions. For these, nd mny more references, see Apt & Bol [1]. 2 Model theoretic solutions Vlued solutions Solution 1 (Positive). A first nd rther conservtive nswer to (1) nd (2) is to tke the clss of positive TSSs s the meningful ones, nd ssocite with ech positive TSS the trnsition reltion consisting of the provle trnsitions. Before proposing more generl solutions, I will first recll two criteri from Bloom, Istril & Meyer [2] nd Bol & Groote [3] tht cn e imposed on solutions. Definition 5 (Supported model) [2, 3]. A trnsition reltion T grees with TSS P if: T = t t H there is closed sustitution instnce of rule of P with T = H. t t T is model of P if holds; T is supported y P if holds. The first nd most indisputle criterion imposed on trnsition reltion T specified y TSS P is tht it is model of P. This is clled eing sound for P in [2]. This criterion sys tht the rules of P, interpreted s implictions in first-order or conditionl logic, should evlute to true sttements out T. The second criterion, of eing supported, sys tht T does not contin ny trnsitions for which it hs no plusile justifiction to contin them. In [2] eing supported is clled witnessing. ote tht the universl trnsition reltion on T(Σ) is model of ny TSS. It is however rrely the intended one, nd the criterion of eing supported is good tool to rule it out. ext I check tht Solution 1 stisfies oth criteri. Proposition 1 Let P e positive TSS nd T the set of trnsitions provle from P. Then T is supported model of P. Moreover T is the lest model of P. Proof: Tht T is supported model of P is n immedite consequence of the definition of provility. Furthermore, let T e ny model of P, then y induction on the length of proofs it follows tht T T. Strting from Proposition 1 there re t lest three wys to generlise Solution 1 to TSSs with negtive premises. One cn generlise either the concept of proof, or the lest model property, or the lest supported model property of positive TSSs. Strting with the lst two possiilities, oserve tht in generl no lest model nd no lest supported model exists. A counterexmple is given y the TSS P 1 (given erlier), which hs two miniml models, {c c} nd {c c}, oth of which re supported. Solution 2 (Lest). A TSS is meningful iff it hs lest model (this eing its specified trnsition reltion). 4

5 Solution 3 (Lest supported). A TSS is meningful iff it hs lest supported model. These two solutions turn out to hve incomprle domins, in the sense tht neither one extends the other. The TSS P 2 elow hs {c c} s its lest model, ut hs no supported models. On the other hnd P 3 hs two miniml models, nmely {c c} nd {c c}, of which only the ltter one is supported. This is its lest supported model. P 2 c c c P 3 c c c Oviously Solution 1 is extended y oth solutions ove. However, Solutions 2 nd 3 turn out to e inconsistent with ech other. P 4 hs oth lest model nd lest supported model, ut they re not the sme. c c c c c c c P 4 P c c c c 5 c c c c Solution 2 is not very productive, ecuse it fils to ssign mening to the perfectly resonle TSS P 3. Moreover, it cn e criticised for yielding unsupported trnsition reltions, s in the cse of P 2. However, in P 4 the lest model {c c} ppers to e etter choice thn the lest supported model {c c, c c}, s the support for trnsition c c is not overwhelming. Thus, to my tste, Solution 3 is somewht unnturl. In Bloom, Istril & Meyer [2] the following solution is pplied. Solution 4 (Unique supported). A TSS is meningful iff it hs unique supported model. The positive TSS P 5 ove hs two supported models, nd {c c}, nd hence shows tht Solution 4 does not extend Solution 1. Although for the kind of TSSs considered in [2] (the GSOS rule systems) this solution coincides with ll cceptle solutions mentioned in this pper, in generl it suffers from the sme drwck s Solution 3. The lest supported model of P 4 is even the unique supported model of this TSS. My conclusion is tht the criterion of eing supported is too wek to e of ny use in this context. This conclusion ws lso reched y Fges [5] in the setting of logic progrmming, who proposes to strengthen this criterion. Being supported cn e rephrsed s sying tht trnsition my only e present if there is nonempty proof of its presence, strting from trnsitions tht re lso present. However, these premises in the proof my include the trnsition under derivtion, therey llowing for loops, s in the cse of P 4. ow the ide ehind well-supported model is tht the sence of trnsition my e ssumed priori, s long s this ssumption is consistent, ut the presence of trnsition needs to e proven without ssuming the presence of (other) trnsitions. Thus trnsition my only e present if it dmits vlid proof, strting from negtive literls only. Definition 6 (Well-supported). 2 A trnsition reltion T is well-supported y TSS P if: T = t t there is closed proof p, with T = p, of trnsition rule without positive premises. t t ote tht is trivil, nd well-supported trnsition reltion is surely supported. 2 The originl version of this pper, which ppered s Stnford report STA-CS-T-95-16, contined n incorrect definition of well-supportedness (ut leding to the sme notion of well-supported model). As oserved y Jn Rutten, Proposition 3 in tht version, stting tht well-supported trnsition reltions re supported, ws flse. With the new Definition 6 this proposition ecomes trivil nd is therefore omitted. The mistke hd no other d consequences. 5

6 My concept of well-supportedness cn esily e seen to coincide with the one of Fges [5]. It is closely relted to the erlier concept of stility, developed y Gelfond & Lifschitz [7] in logic progrmming, nd dpted for TSSs y Bol & Groote [3]. Definition 7 (Stle trnsition reltion). A trnsition reltion T is stle for TSS P if: T = t t there is set of closed negtive literls with P nd T =. t t Proposition 2 T is stle for P iff it is well-supported model of P. Proof: if : follows immeditely from the well-support of T, nd follows from the soundness of T y trivil induction on the length of proofs. H only if : Suppose there is closed sustitution instnce of rule of P with T = H. t t Assuming tht T is stle, for ny t i i t i H there must e closed trnsition rule without positive premises with P i t i i t i the negtive literls in H. Then, y comintion of proof-frgments, t t i t i i t i nd T = i. Let e the union of ll those i s nd t t is closed trnsition rule without positive premises with P nd T =. Hence T = t t. Tht T is well-supported now follows y trivil induction on the length of proofs, tking into ccount tht proof of closed trnsition rule cn esily e turned into closed proof. In [3] stility ws defined in terms of n opertor Strip on TSSs without vriles. If P is such TSS nd T trnsition reltion, Strip(P, T ) is otined from P y removing from P ll rules with negtive premises tht do not hold in T, nd removing from the remining rules the negtive premises tht do hold (Definition 4.1 in [3]). This yields positive TSS, whose ssocited trnsition reltion is denoted Strip(P,T ). ow T is sid to e stle for P if T = Strip(P,T ). This definition is extended to TSSs P with vriles y identifying such TSS with the TSS of ll closed sustitution instnces of rules in P. Proposition 3 The concept of stility of Definition 7 coincides with tht from [3]. Proof: Let P e TSS nd P e the TSS consisting of ll closed sustitution instnces of rules in P. ote tht T is stle for P in the sense of Definition 7 iff it is for P. The construction of Strip entils tht Strip(P, T ) t t iff P for set of closed t t negtive literls with T =. It follows immeditely tht oth definitions re equivlent. The following two solutions re dpttions of Solutions 3 nd 4, were the requirement of eing supported hs een replced y tht of eing well-supported. The second is tken from [3]. Solution 5 (Stle). A TSS is meningful iff it hs lest stle trnsition reltion. Solution 5 (Stle). A TSS is meningful iff it hs unique stle trnsition reltion. The prticulr numering of these two solutions is justified y the following. Proposition 4 Let T 1 e model of TSS P nd T 2 e well-supported y P. If T 1 T 2 then T 1 = T 2. It follows (from the specil cse tht T 1 nd T 2 re oth stle) tht TSS hs lest stle trnsition reltion iff it hs unique stle trnsition reltion. 6

7 Proof: As T 1 T 2 one hs from which it follows tht T 1 = t t T 2 = t t T 2 = t t T 1 = t t nd T 2 = t T 1 = t. (3) ow suppose T 2 = t with P t t t. Then there is closed trnsition rule t t nd T 2 =. By (3) one hs T 1 = nd hence T 1 = t t. without positive premises Solution 5 improves Solutions 3 nd 4 y rejecting the TSS P 4 s meningless. It lso improves Solution 2 y rejecting the TSS P 2 (whose lest model ws not supported). Surprisingly however, Solution 5 not only differs from the erlier solutions y eing more fstidious; it lso provides mening to perfectly cceptle TSSs tht were left meningless y Solutions 2, 3 nd 4. P 6 c c c c c c c An exmple is the TSS P 6. There is clerly no stisfying wy to otin c c. Hence c nd consequently c c. {c c} is indeed the unique stle trnsition reltion of this TSS. However, P 6 hs two miniml models, oth of which re supported, nmely {c c} nd {c c}. Proposition 5 Solution 5 (stle) is consistent with Solution 2 (lest) nd 3 (lest supported). Proof: If TSS hs oth (lest) well-supported model nd lest [supported] model, the two must e equl y Proposition 4. As the set of trnsitions provle from positive TSS is y definition well-supported, Solution 5 (stle) extends Solution 1 (positive). Hence the reltions etween the solutions seen so fr re s indicted in Figure 1 elow. An rrow indictes n extension. The reltion indictes consistency nd incomprle domins (neither one extends the other). There re no more extension nd consistency reltions thn indicted in the figure (tking into ccount tht positive unique supported follows from the informtion displyed). All counterexmples pper erlier in this section. lest model (2) positive (1) unique stle (5) unique supported (4) lest stle (5) lest supported (3) Figure 1: Reltions etween Solutions 1 5 It is interesting to see how the vrious solutions del with circulr rules, such s c c c c, nd rules like c. The support-sed solutions (3 nd 4) my use circulr rule to otin trnsition c c tht would e unsupported otherwise (Exmple P 4 ). This is my min rgument to reject these 7

8 solutions. In ddition they my (or my not) reject TSSs s meningless ecuse of the presence of such rule (Exmple P 6 ). On the other hnd, Solutions 2 nd 5 politely ignore these rules. To my tste, there re two cceptle ttitudes towrds circulr rules: to ignore them completely (s done y Solutions 1, 2 nd 5), or to reject ny TSS with such rule for eing miguous, unless there is independent evidence for trnsition c c. A strong rgument in fvour of the first pproch is the existence of useful rules of which only certin sustitution instnces re circulr (cf. [3]). A solution tht cters to the second option will e proposed in Section 3. Solution 2 cn tret rule c s equivlent to c c (nmely if there re no other closed c c terms thn c, cf. P 2 ), which gives rise to unsupported trnsition reltions. Solutions 3, 4 nd 5 do not go so fr, ut use such rule to choose etween two otherwise eqully ttrctive trnsition reltions. This is illustrted y the TSS P 7, which determines the trnsition reltion {c c} ccording to ech of the solutions 2, 3, 4 nd 5. P 7 c c c c c c c c c P 8 c c c c c Ignoring rules like c c c is uncceptle, s this would yield unsound trnsition reltions (nonmodels). But it could e rgued tht ny TSS with such rule should e rejected s meningless, unless there is independent evidence for trnsition c t, s in P 8. This would rule out P 7. Solutions tht cter to this tste will e proposed in Section Vlued solutions 3-vlued interprettions of logicl progrms re considered, mong others, in Vn Gelder, Ross & Schlipf [6] nd Przymusinski [13]. The sme cn e done for TSSs. The mening of TSS is then not given y trnsition reltion, i.e. prtition of T(Σ) A T(Σ) into the the trnsitions tht hold nd those tht don t, ut prtition of T(Σ) A T(Σ) into three sets: true, flse nd unknown. Such 3-vlued interprettion cn e given s set of closed inry Σ-literls, not contining literls tht deny ech other. Here literl is inry if it hs the form t t or t t. Definition 8 (3-Vlued trnsition reltion). Let Σ e signture. A 3-vlued trnsition reltion over Σ is set T of closed inry Σ-literls, not contining literls tht deny ech other. A closed literl α holds in T, nottion T = α, if α is inry nd α T or α = (t ) nd (t t ) T for ll t T(Σ). Write T = H, for H set of closed literls, if T = α for ll α H. Write CT for the positive literls in T, the trnsitions tht certinly hold, nd P T for {t t (t t ) T }, the trnsitions tht possily hold. Using this convention, 3-vlued trnsition reltion T cn lterntively e presented s pir / \CT, P T \ / of trnsition reltions s in Definition 4, stisfying CT P T. A 3-vlued trnsition reltion / \CT, P T \ / is sid to e 2-vlued if CT = P T. In Przymusinski [13], of the concept of stle trnsition reltion (or well-supported model) is generlised to 3-vlued interprettions. Definition 9 (3-Vlued stility). A 3-vlued trnsition reltion T is stle for TSS P if: T = t nd t there is set of closed negtive literls with P nd T =, t t T = t t for ech set of closed negtive literls stisfying P, t t one hs T = α for literl α denying literl in. 8

9 By Definitions 4 nd 8, for positive literls α one hs T = α CT = α wheres for negtive literls α one hs T = α P T = α. Hence Definition 9 cn e reformulted s follows: Proposition 6 A 3-vlued trnsition reltion / \CT, P T \ / is stle for TSS P iff: CT = t t there is set of closed negtive literls with P nd P T =, t t nd P T = t t there is set of closed negtive literls with P nd CT =. t t ote tht for negtive literl α, CT = α mens tht α possily holds (no denying literl certinly holds), wheres P T = α mens tht α certinly holds (no denying literl possily holds). With this in mind, Proposition 6 explins Definition 9 s vlid generlistion of Definition 7. The definition in [13] cn e shown to mount to the sme concept. A stle trnsition reltion s in Definition 7 cn e regrded s stle 3-vlued trnsition reltion / \CT, P T \ / with CT = P T. On 3-vlued trnsition reltions the inclusion reltion is clled the informtion ordering; T T holds when in T the truth or flsity of more trnsitions is known. This is the cse iff CT CT nd P T P T. Przymusinski [13] showed tht every logic progrm dmits 3-vlued stle trnsition reltion, nd the sme cn e sid for TSSs. There is even lest one w.r.t. the informtion ordering. He lso showed tht the lest 3-vlued stle model coincides with the well-founded semntics of n ritrry TSS (logicl progrm) proposed erlier y Vn Gelder, Ross & Schlipf [6]. See Section 4 for vrint of the pproch of [6]. Assuming tht A = {, }, the TSS P 1 hs three 3-vlued stle trnsition reltions, nmely {c c, c c}, {c c, c c} nd. The first two re 2-vlued. For resons of symmetry the ltter, which is lso the lest, is most suited s the intended mening of this TSS. This is its well-founded semntics. Hence the following solution. 3-Vlued Solution I (Well-founded semntics) Any TSS is meningful. informtion-lest 3-vlued stle trnsition reltion. Its mening is its The existence of this reltion will e demonstrted in the next section. The exmple P 1 shows tht there need not e lest 3-vlued stle trnsition reltion w.r.t. the truth ordering, defined y requiring CT CT nd P T P T. 3-Vlued Solution I is not numered with the other solutions, s it does not provides 2-vlued trnsition reltions. However, 2-vlued trnsition reltions cn e otined y restricting ttention to those TSSs for which the lest 3-vlued stle trnsition reltion / \CT, P T \ / stisfies CT = P T. Alterntively, just the component CT (or just P T ) of the lest 3-vlued stle trnsition reltion / \CT, P T \ / could e tken to e the mening of TSS. These possiilities will e explored in the next section. Finlly I propose nother 3-vlued nswer to (1) nd (2), sed on generlistion of the notion of supported model. Definition 10 (3-Vlued supported model). A 3-vlued trnsition reltion T is supported model of TSS P if: T = t t H there is closed sustitution instnce of rule of P with T = H, t t nd T = t t for ech closed sustitution instnce H of rule of P, t t one hs T = α for literl α denying literl in H. A supported model s in Definition 5 cn e regrded s 3-vlued supported model tht hppens to e 2-vlued. In the next section I will show tht every TSS dmits n informtion-lest 3-vlued supported model. 3-Vlued Solution II (Lest 3-vlued supported model). Any TSS is meningful. Its mening is its informtion-lest 3-vlued supported model. 9

10 Solutions I nd II gree on the tretment of P 1, P 2, P 3, P 7 nd P 8. Assuming tht A = {, }, the trnsition reltion ssocited to P 1 nd P 7 is, mening tht oth potentil trnsitions re undetermined. The mening of P 2 is c c, i.e. the -trnsition is undetermined. The mening of P 3 nd P 8 is {c c, c c}; here oth potentil trnsitions re determined. According to Solution I the mening of P 5 is {c c, c c} wheres Solution 2 yields {c c}, leving the -trnsition undetermined. Likewise, Solution II ssocites the empty trnsition reltion to P 4, leving oth trnsitions undetermined, wheres Solution I yields {c c}. 3 Proof theoretic solutions In this section I will propose solutions sed on generlistion of the concept of proof. ote tht in proof two kinds of steps re llowed, itemised with in Definition 3. The first step just llows hypotheses to enter, in cse one wnts to prove trnsition rule. This step cn not e used when merely proving trnsitions. The essence of the notion is the second step. This step reflects the postulte tht the desired trnsition reltion must e model of the given TSS. As consequence those nd only those trnsitions re provle tht pper in ny model. When generlising the notion of proof to derive negtive literls it mkes sense to import more postultes out the desired trnsition reltion. ote tht, y Definitions 5 nd 7, (2-vlued) model T of TSS P is supported iff T = t t for ech closed sustitution instnce H of rule of P, t t one hs T = α for literl α denying literl in H. nd well-supported (or stle) iff T = t t for ech set of closed negtive literls stisfying P one hs T = α for literl α denying literl in. Therefore I propose the following two concepts of provility., t t Definition 11 (Supported proof ). A supported proof of closed literl α from TSS P = (Σ, R) is well-founded, upwrdly rnching tree of which the nodes re lelled y Σ-literls, such tht: the root is lelled y α, nd if β is the lel of node q nd K is the set of lels of the nodes directly ove q, then β is positive nd K β is sustitution instnce of rule from R, or β is negtive nd for ech closed sustitution instnce of rule of P whose conclusion denies β, literl in K denies one of its premises. α is s-provle, nottion P s α, if supported proof of α from P exists. A literl is s-refutle if denying literl is s-provle. Definition 12 (Well-supported proof ). A well-supported proof of closed literl α from TSS P = (Σ, R) is well-founded, upwrdly rnching tree of which the nodes re lelled y Σ-literls, such tht: the root is lelled y α, nd if β is the lel of node q nd K is the set of lels of the nodes directly ove q, then β is positive nd K β is sustitution instnce of rule from R, or β is negtive nd for every set of negtive closed literls such tht P γ for γ closed literl denying β, literl in K denies one in. 10

11 α is ws-provle, nottion P ws α, if well-supported proof of α from P exists. A literl is ws-refutle if denying literl is ws-provle. ote tht these proof-steps estlish the vlidity of β when K is the set of literls estlished erlier. The lst step from Definition 12 llows one to infer t t whenever it is mnifestly impossile to infer t t (ecuse every conceivle proof of t t involves premise tht hs lredy een refuted), or t whenever for ny term t it is mnifestly impossile to infer t t. This prctice is sometimes referred to s negtion s filure [4]. Definition 11 llows such n inference only if the impossiility to derive t t cn e detected y exmining ll possile proofs tht consist of one step only. This corresponds with the notion of negtion s finite filure of Clrk [4]. The extension of these notions (especilly ws ) from closed to open literls α, or to trnsition rules H α, is somewht prolemtic, nd not needed in this pper. The following my shed more light on s nd ws. From here onwrds, sttements hold with or without the text enclosed in squre rckets. Also, proof s in Definition 3 will e referred to s positive proof. Proposition 7 Let P e TSS. Then P s t [t H ] iff every closed sustitution instnce of t t rule of P hs n s-refutle premise. Moreover P ws t [t ] iff every set of closed negtive literls with P contins n ws-refutle literl. t t Proof: Firly trivil. Proposition 8 For P TSS nd α closed literl one hs P α P s α P ws α. Proof: The first sttement is trivil. The second will e estlished with induction on the structure of s -proof of α. Let K α e the lst step in such proof. As P s K y mens of strict suproofs, it follows y induction tht P ws K. Here I write P x K for K set of literls if P x β for ll β K. If α is positive, P ws α follows immeditely from the definitions of s- nd ws-provility. Thus suppose α is negtive. Let {α i } i I e the set of negtive literls in K, nd let K i for i I e the collection of lst proof-steps in ws -proofs of the α i. Let L = i I K i (K {α i } i I ). Then clerly P ws L, so it suffices to show tht for every set of negtive closed literls such tht P γ for γ literl denying α, literl in L denies one in. Consider -proof p of γ with set of negtive literls nd γ denies α. By the definition of s, p contins literl δ tht denies literl β in K. This literl is the lel of node right ove the root. In cse δ occurs in, β is positive nd therefore occurs in L. In cse δ, β must e negtive nd hence e α i for certin i I. Becuse K i α i is vlid step in ws -proof nd P δ with δ denying α i, literl in must deny one in K i L. Proposition 9 Let qusi-proof e defined s in Definition 3, ut without the requirement of well-foundedness. If in TSS P ny qusi-proof is well-founded, then P s α P ws α. Proof: Suppose P ws α. Let K α e the lst step in ws-proof of α. Applying induction on such proofs, I my ssume P s K. In cse α is positive the desired result P s α follows immeditely, so suppose it is not. Let β e literl tht denies α nd let H β e closed sustitution instnce of rule of P. This instnce constitutes positive one-step proof p of H β from P. I hve to show tht H contins n s-refutle literl. Suppose y contrdiction tht is does not. Then, y Proposition 7, for every positive literl γ H there must closed sustitution instnce Hγ γ of rule of P, without s-refutle premises. Adding these rules to p yields lrger proof p of rule with H = {γ H γ positive} H γ {γ H γ negtive}. Iterting this procedure y pplying H β 11 α i

12 the sme resoning to H etc. yields qusi-proof of sttement β with set of s-irrefutle closed negtive literls. By ssumption this qusi-proof must e proof. By the ws-provility of α it follows tht must contin literl tht is denied y literl from K, nd hence s-refutle. This yields contrdiction. Definition 13 (Consistency, soundness nd completeness). For P TSS nd α closed literl, write P = s α [resp. P = 3s α] if T = α for ny [3-vlued] supported model T of P nd P = ws α [resp. P = 3ws α] if T = α for ny [3-vlued] well-supported model T of P. A notion x is clled consistent if there is no TSS deriving two literls tht deny ech other. sound w.r.t. = x if for ny TSS P nd closed literl α, P x α P = x α. complete w.r.t. = x if for ny TSS P nd closed literl α, P x α P = x α. Proposition 10 ws is consistent. Proof: Let s sy tht two proofs p nd q deny ech other if their roots re lelled with literls tht deny ech other. By induction on their structure I estlish tht no two proofs from the sme TSS P deny ech other. So let p nd q e two ws -proofs from P nd ssume tht no two proper suproofs deny ech other. By contrdiction suppose the roots of p nd q re lelled with t t nd t (or t t ) respectively. ote tht the ottom prt of p is positive proof of rule t t, where contins only negtive literls. Let K e the set of literls lelling nodes directly ove the root of q. Then from the lst step of q it follows tht (nd thus p) contins negtive literl tht denies one in K, thus yielding proper suproofs of p nd q tht deny ech other. As P α P s α P ws α, if follows tht lso s nd re consistent. Proposition 11 ws is sound w.r.t. = ws nd = 3ws. Likewise s is sound w.r.t. = s nd = 3s. Proof: Let P e TSS nd T [3-vlued] well-supported model of P. With strightforwrd induction on the structure of proofs if follows tht P ws α T = α. The other prt goes likewise. Lemm 1 If P is TSS nd t closed literl, then P x t iff P x t t for ny term t T(Σ). Proof: This follows immeditely from the oservtion tht closed literl γ denies t denies t t for some t T(Σ). iff it The following theorem implies tht ny TSS hs lest 3-vlued [well-]supported model w.r.t. the informtion ordering. This justifies 3-vlued Solutions I nd II mentioned erlier. Moreover, it provides proof theoretic chrcteristion of these solutions. Theorem 1 For ny TSS P, the set of closed inry literls [w]s-provle from P constitutes 3-vlued [well-]supported model of P. It is even the lest one w.r.t. the informtion ordering. Proof: By Proposition 10 the set T of closed inry literls [w]s-provle from P constitutes 3-vlued trnsition reltion. Using Lemm 1, it is strightforwrd to check tht T stisfies the required equtions. It follows from the soundness of [w]s w.r.t. = 3[w]s (Proposition 11) tht T is included in ny other 3-vlued [well-]supported model of P. 12

13 Corollry 1 ws is complete w.r.t. = 3ws. Likewise s is complete w.r.t. = 3s. Proof: If P = 3[w]s α then y definition α certinly holds in ll [well]-supported models of P. Thus α certinly holds in the lest such model w.r.t. the informtion ordering, which is the one of Theorem 1. This implies P [w]s α. However, s nd ws re not complete w.r.t. = [w]s. A trivil counterexmple concerns TSSs like P 2 tht hve no [well-]supported models. P 2 = [w]s α for ny α, which y Proposition 10 is not the cse for [w]s. A more interesting counterexmple concerns the TSS P 7, which hs only one [well-]supported model, nmely {c c}. In spite of this, P 7 [w]s c c nd P 7 [w]s c. As rgued in the previous section, when insisting on 2-vlued solutions there is point in excluding P 7 from the meningful TSSs, since there is insufficient evidence for the trnsition c c. Here the incompleteness of [w]s w.r.t. = [w]s comes s lessing rther thn shortcoming. The 3-vlued solutions I nd II re two stisfctory methods to ssocite 3-vlued trnsition reltion to ny TSS. I hve given oth model theoretic nd proof theoretic chrcteristions of these solutions. In the reminder of this section I continue the serch for 2-vlued solutions. In this context, in line with question (1), I will cll TSS meningless if it hs no stisfctory 2-vlued interprettion. 3.1 Solutions sed on completeness I will now introduce the concept of complete TSS: one in which ny trnsition is either provle or refutle. Just s in the theory of logic there is distinction etween the completeness of logic (e.g. first-order) nd the completeness of prticulr theory (e.g. rithmetic), here the completeness of TSS is something different from the completeness of proof-method x. Let x e s or ws. Definition 14 (Completeness of TSS). A TSS P is x-complete if for ny trnsition t either P x t t or P x t t. By complete I will men ws-complete. t ote tht TSS is [w]s-complete iff its lest (nd only) 3-vlued [well-]supported model is 2-vlued. Solution 6 (Complete with support). A TSS is meningful iff it is s-complete. The ssocited trnsition reltion consists of the s-provle trnsitions. Solution 7 (Complete). A TSS is meningful iff it is (ws-)complete. The ssocited trnsition reltion consists of the ws-provle trnsitions. In Bol & Groote [3] method clled reduction for ssociting trnsition reltion with TSS ws proposed, inspired y the well-founded models of Vn Gelder, Ross & Schlipf [6] in logic progrmming. In Section 4 I show tht this solution coincides with Solution 7. Solution 7 cn therefore e regrded s proof theoreticl chrcteristion of the ides from [6, 3]. Solution 6 my e new. The TSS P 6 is complete, ut not complete with support. P 3 is even complete with support. The following proposition sys tht stndrd TSS (i.e. without premises t t ) is complete if every closed negtive stndrd literl cn e proved or refuted. Proposition 12 A stndrd TSS P is complete iff for ny closed literl t for some closed term t or P ws t. either P ws t t 13

14 Proof: only if : Immeditely y Lemm 1. if : Suppose P ws t t. In tht cse ny set = {t i i i I} such tht P t t contin literl t with P ws t. By ssumption, for such literl there is t with P ws t t. It follows from Definition 12, tking K to e the set of ll trnsitions t t (one for ech possile choice of ), tht P ws t t. As literls t t do not pper in the premises of rules in stndrd TSS, their occurrence in well-supported proof-tree cn e limited to the root. Thus Proposition 12 sys tht the concept of complete TSS cn e introduced without considering such literls t ll. The reson tht these were introduced nevertheless, is tht Proposition 12 does not pply to completeness with support. A counterexmple is given y the TSS Q. Q t t 1 t t 2 R t t 1 t t 2 t t 2 t t 2 Q s t t 2 nd Q s t t 2, thus this TSS is incomplete with support. However, for ny closed literl u, either Q s u u for some term u or Q s u. Moreover, even for the derivtion of stndrd literls, nonstndrd literls my e essentil in supported proofs. The vlidity of R s t for instnce, cn only e estlished y proof tree contining t t 2. Proposition 13 The set of [w]s-provle trnsitions of [w]s-complete TSS P is model of P. Proof: Let P e n x-complete TSS nd T the set of x-provle trnsitions. Suppose is t t closed sustitution instnce of rule in P, nd T = H. By Definition 4 (of T = H) P x β for ech positive premise β in H, nd P x γ for no trnsition γ denying negtive premise in H. Thus, y completeness nd Lemm 1, P x β for ny β in H. Hence P x t t. Proposition 14 The set of [w]s-provle trnsitions of ny TSS is well-supported. Proof: Let P e TSS nd T the set of x-provle trnsitions. Suppose T = t t, i.e. P x t t with t nd t closed terms. Tke [well-]supported proof of this trnsition from P, nd delete ll rnches ove node lelled with negtive literl. This yields positive proof p of rule with set of closed negtive literls. For ny literl α in p one hs P t t x α. If α is positive, this immeditely gives α T. If α is negtive, then, y the consistency of x, P x β for no closed literl β denying α. This implies T = α, nd hence T = p. Proposition 15 Solution 6 [7] is strictly extended y Solution 4 [5]. Proof: Suppose P is [w]s-complete. By Propositions 13 nd 14 the [w]s-provle trnsitions constitute [well-]supported model of P, nd y Proposition 11 this is the only such model. Strictness follows from the TSS P 7, which hs unique [well-]supported model, ut is left meningless y Solutions 6 nd Advntges of the proof theoretic solutions ow I will turn to the dvntges of the proof theoretic solutions over the model theoretic ones. At the end of Section 2 I discussed the rôle of rules like c nd c c nd suggested tht ny TSS c c c c contining the former rule should e rejected s meningless, unless there is independent evidence for trnsition c t. As shown y counterexmple P 7 ll model theoretic solutions fil this test. The next proposition shows tht the proof theoretic solutions ehve etter in this respect. 14 H must

15 Proposition 16 Let P, P e TSSs tht only differ in rule c tht is in P ut not in P. Then c c P is [w]s-complete only if P is [w]s-complete nd proves the sme literls s P, including c t for some term t. Proof: Suppose P is complete. It cnnot e tht P [w]s c, since in tht cse one could derive P [w]s c c, contrdicting Proposition 10 (consistency). Thus the lel c does not pper in ny proof of literl from P. It follows tht ny literl provle from P is lredy provle from P. By Lemm 1, since P [w]s c, P [w]s c t for some term t. I lso recommended two cceptle ttitudes towrds rule like c c. Below I show tht Solution 7 c c ignores such rules completely (which is one option), wheres Solution 6 rejects TSS with such rule, unless there is independent evidence for trnsition c c (the other option). Proposition 17 Let P, P e TSSs tht only differ in rule c c tht is in P ut not in P. Then c c P is ws-complete iff P is ws-complete. If P is ws-complete it proves the sme literls s P. c c c Proof: Any ppliction of c cn e eliminted from positive or well-supported proof. Proposition 18 Let P, P e TSSs tht only differ in rule c c tht is in P ut not in P. Then c c P is s-complete only if P is s-complete nd proves the sme literls s P, including c c. Proof: Suppose P is complete. c c c It is esy to eliminte pplictions of the rule c from ny supported proof, so ny literl provle from P is lso provle from P. Hence P is complete. Due to the rule c c it is impossile to prove c c from P. Thus P s c c. c c 3.3 Solutions sed on soundness The reminder of this section is devoted to 2-vlued generlistions of the proof theoretic solutions. The first ide is to define the trnsition reltion ssocited to TSS P just s in Solutions 6 nd 7, tht is s the set of [w]s-provle trnsitions, ut without requiring tht P is [w]s-complete. This mounts to tking s the mening of P the component CT of its lest [well-]supported model / \CT, P T \ /. In generl this my yield unsound trnsition reltions (non-models), which is not cceptle. This hppens in the cse of P 1, P 2, P 4 nd P 7. Thus the following restriction is needed. Solution 8 (Sound with support). A TSS is meningful if the set of s-provle trnsitions (this eing the ssocited trnsition reltion) constitutes model. Solution 8 A TSS is meningful if the set of ws-provle trnsitions constitutes model. ote tht y Proposition 14 the trnsition reltion determined y such TSS is even stle. Proposition 19 Solution 8 coincides with Solution 7. Proof: It follows immeditely from Proposition 13 tht complete TSS is lso meningful in the sense of Solution 8. ow let P e TSS tht is meningful in the sense of Solution 8 nd T the set of ws-provle trnsitions. Suppose P ws t t for certin t, t T(Σ). Then T = t t. By the soundness of T every set of closed negtive literls such tht P must contin t t literl δ with T = δ. The ltter mens P ws ɛ for trnsition ɛ denying δ. Collecting ll such ɛ s (one for every choice of ) in set K yields well-supported proof of t t. 15

16 Proposition 20 Solution 8 is extended y Solutions 3 (lest supported) nd 8 (= 7, complete), nd extends Solutions 1 (positive) nd 6 (complete with support). Proof: By Proposition 14 TSS tht is sound with support determines trnsition reltion tht is supported model. By Proposition 11 (the soundness of s w.r.t. = s ), this trnsition reltion is included in ny supported model. Therefore it constitutes the lest. By definition, the trnsition reltion T 1 determined y TSS P tht is sound with support is model of P. The set T 2 of trnsitions tht re ws-provle from P is well-supported, y Proposition 14. By Proposition 8 T 1 T 2 nd Proposition 4 yields T 1 = T 2. If follows tht T 2 is model too. If TSS P is positive, then P s t t iff P t trnsitions form model. The lst sttement follows immeditely from Proposition Solutions sed on irrefutility t. By Proposition 1 the (s-)provle A second (nd lst) ide is to define the trnsition reltion T ssocited to TSS P s the set of x-irrefutle trnsitions, i.e. T = {t t P x t t }, in which x is s or ws. This mounts to tking s the mening of P the component P T of its lest [well-]supported model / \CT, P T \ /. This is consistent with Solutions 6 nd 7, s for x-complete TSSs one hs P x t t P x t t. Proposition 21 The set of x-irrefutle trnsitions of ny TSS constitutes model. H Proof: Let P e TSS nd e closed sustitution instnce of rule of P. Let T e the t t set of x-irrefutle trnsitions nd suppose T = t t, i.e. P x t t. I hve to prove tht T = H. In cse x = s it follows from Proposition 7 tht H contins n x-refutle literl. I estlish the sme in cse x = ws. Suppose tht P ws u u for ech positive premise α = (u u ) in H. Then for ech such α there is set α of ws-irrefutle negtive closed literls with P α α. Let e otined from H y replcing α y α for ech positive α in H. Then contins negtive literls only. Since P ws t t nd P, must contin ws-refutle literl. This literl must e in H, t t which hd to e estlished. In cse the x-refutle literl in H is positive, sy u u, one hs P x u u, which implies T = u u c. In cse it is negtive, sy v, one hs v c T(Σ) : P x v v, which y the consistency of x implies v c : P x v v, which implies v : T = v c v c nd thus T = v. c In cse of literl v v just leve out the existentil quntifictions. For the moment I restrict ttention to solutions yielding well-supported trnsition reltions. Solution 9 A TSS is meningful if the set of s-irrefutle trnsitions (this eing the ssocited trnsition reltion) is well-supported. Solution 9 A TSS is meningful if the set of ws-irrefutle trnsitions is well-supported. ote tht y Proposition 21 the trnsition reltion determined y such TSS is even stle. Proposition 22 Solution 9 coincides with Solution 6 nd 9 with 7. 16

17 Proof: It follows immeditely from Proposition 14 tht the set of x-irrefutle trnsitions of n x-complete TSS is well-supported. ow let P e TSS whose set T of x-irrefutle trnsitions is well-supported. Suppose P x t t for certin t, t T(Σ). Then T = t t. By the stility of T there is set of closed negtive literls such tht P nd T =. The ltter mens c t t T = v v c for ny literl v v in, which mens P c x v v. By Definition 4 nd c Lemm 1 the sme holds for literls v in. Therefore P x t t. 3.5 Attching 2-vlued mening to ll trnsition system specifictions In this section I will ssocite 2-vlued trnsition reltion to ritrry TSSs. As illustrted y P 1 nd P 2, such trnsition reltion cn not lwys e supported model. I will insist on soundness (eing model), nd thus hve to give up support. Hence mong the model theoretic solutions only Solution 2 (lest model) cn provide inspirtion. Let me first decide wht to do with P 1. Since the ssocited trnsition reltion should e model, it must contin either c c or c c. For resons of symmetry I cnnot choose etween these trnsitions, so the only wy out is to include oth. There is no reson to include ny more trnsitions. Hence the trnsition reltion ssocited to P 1 should e {c c, c c}. The simplest model theoretic solution I thought of tht gives this result is to define T 1 s the union of ll miniml models of TSS. In mny cses this will e the desired trnsition reltion, ut it cn hppen tht T 1 is not model. In tht cse T 2 is defined s the union of ll miniml models contining T 1, nd iterting this procedure until it stilises gives the ssocited trnsition reltion. However, in generl this solution yields more trnsitions then I would like to see. The trnsition reltion ssocited to P 3 for instnce would e {c c, c c}, wheres {c c} ppers to e sufficient. The sme would hold fter ddition of second premise c to the only rule in P 3. In cse there re other closed terms esides c the ssocited trnsition reltion will e even lrger. Therefore I will not pursue this ide further, nd turn to the proof theoretic solutions insted. The reson for preferring trnsition c c over c c in P 3 is not tht c c is provle fter ddition of the premise c it is not ut tht c c is refutle. Therefore I consider: Solution 9 (Irrefutle). Any TSS is meningful. The ssocited trnsition reltion consists of the ws-irrefutle trnsitions. In the cse of P 1 this yields the desired result {c c, c c} nd likewise P 2, P 3 nd P 4 yield {c c}. The trnsition reltion of P 7 is the sme s the one of P 1. This indictes tht Solution 9 is inconsistent with Solutions 2 5. I don t consider this to e prolem, s the model theoretic lloction of trnsition reltion to P 7 ws not very convincing. A vrint of Solution 9 is to ssocite to TSS the set of its s-irrefutle trnsitions. This solution is inconsistent with Solution 1 (positive) s the trnsition reltion of P 5 would consist of c c. ote tht this trnsition reltion is supported. In order to rule out this nomly one would hve to restrict the meningful TSSs to the ones for which the ssocited trnsition reltion is well-supported, which yields Solution 9, tht hs een shown to coincide with Solution 6. Another vrint is to stick to the ws-irrefutle trnsitions, ut require those to form supported model. ote tht dding rules x y for A to n ritrry TSS does not chnge the x y ssocited trnsition reltion ccording to Solution 9, ut mkes this reltion supported. Thus requiring the ssocited trnsition reltion to e supported is not much of restriction. Moreover, 17