A highly restricted temporal logic with a tractable decision procedure

This paper demonstrates the tractability of a highly restricted temporal logic, into which English sentences containing a variety of temporal prepositional constructions can be translated. This paper is intended as a preliminary study for a larger project of identifying tractable temporal logics corresponding to temporal constructions in natural language. This larger project also includes current work exploring the automatic translation of English sentences involving temporal prepositions (and related adverbials) into temporal logics. 1 Background and objectives This paper is intended as a preliminary study for the project of identifying tractable temporal logics corresponding to temporal constructions in natural language. This project relates to current work exploring the automatic translation of English sentences involving temporal prepositions (and related adverbials) into temporal logics (Pratt and Br ee [1]). On this translation scheme, sentences such as The ferry had departed by the time John telephoned Mary will arrive sometime between Monday and Tuesday David worked on the paper every day for four months Susan will not nish the report until 2 months after Mark returns are mapped to a modal temporal logic in which the sentential operators quantify over subintervals of the interval at which they are evaluated (see below for examples). This translation from English to temporal logic has already been implemented in a program running under LPA Prolog 4.5.1 In this paper we shall not discuss any details of the translation process; nor do we attempt to analyse the properties of the full logic into which that translation process maps English. Rather, we concentrate on establishing the tractability of a highly restricted subset of that logic. 2 The syntax We have an in nite stock P of proposition letters fp1; p2; : : :g representing event-types. The formulae of the language are constructed as follows: 1. If p is a proposition letter, p and : p are formulae. 2. If p is a proposition letter and is a formula, then !(p; ) and (p; ) are formulae. 3 The semantics Propositions p 2 P receive truth-values with respect to time-intervals. We allow open, closed and halfopen intervals; however, we insist that, if I is an interval, end(I) > start(I), where start(I) and end(I) are the greatest lower bounds and least upper bounds of I, respectively. This restriction is motivated by the assumption that the events described in this logic never take zero time. Formally, we take a structure to be a set M P where is the set of intervals of the real line and P is the set of proposition letters. Intuitively, the pair hp; Ji 2M just in case an event of the type represented by p occurs over the interval J , starting at the beginning of J and nishing at the end of J . A structure M assigns truth-values to formulae with respect to a closed interval I, according to the following rules: 1Available upon request from the author. 1 S1: M j=I (p) i hp; Ji 2M for some subinterval J I S2: M j=I : (p) i hp; Ji2= M for all subintervals J I S3: M j=I !(p; ) i hp; Ji 2M for some subinterval J I , such that hp;Ki2= M for all subintervals K I beginning earlier than J , and M j=G , where G is the closed interval [start(I); start(J)]. S4: M j=I (p; ) i hp; Ji 2M for some subinterval J I , such that hp;Ki2= M for all subintervals K I ending later than J , and M j=G , where G is the closed interval [end(J); end(I)]. Clearly, this logic is very limited. However, it allows us to express many temporal prepositionconstructs in English: The ferry had departed by the time John telephoned !([John telephone]; [the ferry depart]) Mary will arrive sometime between Monday and Tuesday !([Tuesday]; [Monday]; [Mary arrive]) Note that [Monday] and [Tuesday] are taken to be events which occur over intervals here. However, the logic is unable to express the uniqueness of such intervals within a period such as a week. That the symbol : cannot attach to whole formulae corresponds approximately to the way not interacts with temporal preposition phrases in English. Thus, for example, the sentence Mary will not leave until John telephones presupposes that John will telephone, and states that Mary will not leave until (the rst time) an event of that type occurs. Taking the liberty of merging the presupposition and truth-conditions, we might represent the meaning of this sentence by means of a formula with the negation on the inside: !([John telephone];: [Mary leave]). 4 The decision procedure We present a procedure for taking a nite set of sentences in the above language and determining whether a model can be constructed for them. The basic data-structures in this decision procedure are so-called left and right proto-models. We say that a set of formulae is left-handed if its members are of the forms !(p1; ), p and : p; and we say that is right-handed if its members are of the forms (p1; ), p and : p. A left proto-model is a partial model of a left-handed set of formulae, and a right proto-model is a partial model of a righthanded set of formulae. Proto-models are partial in that the intervals with respect to which proposition letters are made true are only partially ordered. Left proto-models in general contain right proto-models as constituents; right proto-models in general contain left proto-models as constituents. Since left and right proto-models are mirror images, we describe only left proto-models below. Think of a left proto-model L as attaching to a closed interval I, within which all of the events mentioned in L have to occur. L has four components with the following intuitive interpretations: required(L): a list of propositions which must be instantiated at some subinterval of I forbidden(L): a list of propositions which may not be instantiated at any subinterval of I events(L): a list of events, where an event is de ned below order(L): a partial ordering on the set of end points fstart(E)jE 2 events(L)g [ fend(E)jE 2 events(L)g. 2 Jp S So Rp = right proto( p) I Gp Hp Ep Figure 1: Schematic representation of a left proto-model Think of an event E in a left proto-model L as an occasion within the basic interval I of L at which some atomic proposition p is true. Such an event will typically be generated by a formula of the form !(p; ) or of the form p and has the following components: proposition(E): the proposition which is true in the event right proto(E): a right proto-model encoding events that must occur between the start of the basic interval I and the start of the event E. For every proposition letter p, a proto-model L contains at most one event whose proposition is p; we denote this event by Ep. Fig. 1 depicts an event Ep in a left proto-model L. We take the event Ep to occur over an interval Jp, and the right proto-model Rp belonging to Ep to attach to a closed interval Hp just prior to Jp. Gp denotes the closed interval from the start of I to the start of Jp. The function left proto model( ) takes a left-handed set of formulae and builds a left protomodel. The function uses the following notation. If is left-handed we write: blob arrows( ) = fpj !(r1; !(r2; : : : !(p; ) : : :)) 2 some r1, r2, . . . , g p = f j !(r1; !(r2; : : : !(p; ) : : :)) 2 some r1, r2, . . . , s.t. not of the form !(q; )g : If is right-handed we write: blob arrows( ) = fpj (r1; (r2; : : : (p; ) : : :)) 2 some r1, r2, . . . , g p = f j (r1; (r2; : : : (p; ) : : :)) 2 some r1, r2, . . . , s.t. not of the form (q; )g : Finally, for any , we write: diamonds( ) = fpj p 2 g neg diamonds( ) = fpj: p 2 g The function is de ned as follows. 3 function left proto model( ) create the new proto-model L with: events(L) = banned(L) = required(L) = order(L) = ; A 1 for r in blob arrows( ) do 2 create a new right proto-model Rr = right proto model( r) 3 set banned(Rr) = banned(Rr) [ frg 4 create the new event Er with: 5 proposition(Er) = r 6 right proto(Er) = Rr 7 set events(L) = events(L) [ fErg 8 set required(L) = required(L) [ required(Rr) [ frg 9 endfor B 1 for r in diamonds( ) do 2 if there is no E 2 events(L) with proposition(E) = r then 3 create a new right proto-model Rr with: 4 events(L) = banned(L) = required(L) = order(L) = ; 5 create the new event Er with: 6 proposition(Er) = r 7 right proto(Er) = Rr 8 set events(L) = events(L) [ fErg 9 set required(L) = required(L) [ frg 10 endif 11 endfor C 1 for r in neg diamonds( ) do 2 set banned(L) = banned(L) [ frg 3 endfor D 1 for each formula of the form !(r1; !(r2; : : : !(rn; ) : : :)) in do 2 for i = 1 : : :n do 3 set order(L) = order(L) [ fstart(Eri ) end(Eri+1 )g 4 endfor 5 endfor E 1 for r in blob arrows( ) [ diamonds( ) do 2 for s in blob arrows( ) n frg do 3 if required(Rr) \ banned(Rs) 6= ; then 4 set order(L) = order(L) [ fstart(Er) > start(Es)g 5 endif 6 if r 2 banned(Rs) then 7 set order(L) = order(L) [ fend(Er) > start(Es)g 8 endif 9 endfor 10 endfor return L end left proto model The function right proto model( ) is identical, except that the inequalities are reversed, the pairs of words start and end and left and right are switched, and line D1 contains the mirror-image schema (r1; (r2; : : : (rn; ) : : :)). 4 The following de nition identi es those proto-models that can be extended to models. De nition 1 Let P be a (left or right) proto-model. We say that P is well-behaved if the following conditions hold. 1. required(P ) \ banned(P ) = ; 2. order(P ) is consistent (contains no cycles) 3. All the sub-proto-models belonging to the events of P are well-behaved We can determine the satis ability of a left-handed set of formulae as follows. First, we construct left proto model( ). If the result is a well-behaved proto-model, then has a model; otherwise, has no model. That this procedure really does test for consistency is proved in section 6. Similarly, to determine the satis ability of a right-handed set of formulae , we construct right proto model( ) and determine whether the result is a well-behaved proto-model. If is neither leftnor right-handed, we split into sets which are. Speci cally, we de ne: exists( ) = f 2 j is of the form pg neg exists( ) = f 2 j is of the form : pg lefts( ) = f 2 j is of the form !(p; )g rights( ) = f 2 j is of the form (p; )g Then if we let left = exists( ) [ neg exists( ) [ lefts( ) right = neg exists( ) [ rights( ) It is easy to show that is satis able if and only if both left and right are, so we can test left and right separately. To summarize, the function for determining the satis ability of a set of formulae is as follows: 5 function consis check( ) Split into left and right Compute L = left proto model( left), R = right proto model( right) Determine whether L and R are well-behaved if yes in both cases then return \ has a model" else return \ has no model" endif end consis check 5 Complexity The following result establishes the tractability of determining satis ability in the logic. Theorem 1 (Tractability) The time-complexity of the procedure described in section 4 is cubic in the number of connectives in . Proof: Consider rst the construction of the left proto-model as described in left proto model( ). Each pass through the loops A, B, C and D `uses up' at least one connective in . Loop E has cubic complexity, since, for each pair of proto-models, the requiredand banned-lists have to be searched for a common entry. (We assume constant-time table look-up to determine whether t 2 required(Rr) and t 2 banned(Rs) for any r, s, t.) The total number of events in the proto-model generated from (including all its sub-proto-models) is bounded by the size of . Moreover, testing the well-behavedness of a proto-model involves only checking for the disjointness of required(L) and banned(L), and for the absence of cycles in order(L). This step is clearly quadratic in the size of events(L). 2 6 Correctness and completeness The following two theorems establish that the procedure of section 4 actually works. We state here only the left-hand versions. The right-hand analogues have corresponding proofs. Theorem 2 (Correctness) Let be a left-handed set of formulae. If has a model, then left proto model( ) is well-behaved. Proof: Suppose M j=I and L = left proto model( ). We show by induction on the size of L that L is well-behaved. If L is empty, then L is trivially well-behaved. First, we map the events in events(L) to intervals in M at which the corresponding proposition letters are made true. The only events in events(L) will be those created by lines A4 and B5, corresponding to the proposition letters in blob arrows( ) and in diamonds( ), respectively. For each p 2 diamonds( ), let Jp be any interval such that hp; Jpi 2M ; the semantic rule S1 guarantees the existence of Jp. Map the event Ep in L to the interval Jp. For each p 2 blob arrows( ), let Jp be a rst interval (ordered by starting point) such that hp; Jpi 2M ; the semantic rule S3 guarantees the existence of Jp. Map the event Ep in L to the interval Jp. Let Gp be the interval [start(I); start(Jp)]. Note that, by the semantic rule S3, M j=Gp p. To show that L is well-behaved, it su ces to show that 1. Rp is well-behaved for every Ep 2 events(L) 2. required(L) \ banned(L) = ; 3. order(L) contains no cycles. 6 These conditions are then established as follows 1. If Ep 2 events(L) then p 2 blob arrows( ) or p 2 diamonds( ). If p 2 blob arrows( ), then, as we have observed, M j=Gp p. But since Rp = right proto model( p), then, by the inductive hypothesis, Rp is well-behaved. If p 2= blob arrows( ) but p 2 diamonds( ), then Rp is an empty proto-model and so is trivially well-behaved. 2. Let t be a proposition letter. By lemma 1 below, if t 2 required(L), then, since M j=I , we have M j=I t. Moreover, the only way in which t could have been added to banned(L) is by line C2, and this line is executed only if : t 2 , contradicting M j=I and M j=I t. So required(P ) \ banned(P ) = ;. 3. We need only check that the order imposed in lines D3, E4 and E7 is respected in M under the mapping given above. The absence of cycles then follows immediately. We illustrate with the case of line E4. Line E4 will add start(Er) > start(Es) to order(L) only if required(Rr) \ banned(Rs) 6= ;. Suppose, then that t 2 required(Rr) \ banned(Rs). M j=Gr r, by construction of Gr, and Rr = right proto model( r), so, by lemma 1, M j=Gr t and so, by semantic rule S3, M j=I !(r; t). Moreover, t could only have been inserted in banned(Rs) by line A3 (in which case t = s) or by line C2, in a recursive call (in which case : t 2 s). In the former case, we have M j=I !(r; s) so that start(Jr) > start(Js). In the latter case, we have !(s;: t) 2 which, given that M j=I !(r; t), again forces start(Jr) > start(Js), by the semantic rules S1, S2 and S3. Hence the order start(Er) > start(Es) is respected by the order start(Jr) > start(Js) on the corresponding intervals. The orderings added by lines D3 and E7 can be similarly shown to correspond to orderings on the intervals in M . 2 Theorem 3 (Completeness) Let be a left-handed set of formulae. If left proto model( ) is wellbehaved, then has a model. Proof: Let L be a well-behaved left proto-model. We de ne the notion of an expansion, ML, of L into a closed interval I. Intuitively, ML is a structure which assigns propositions to intervals according to the events in L, in such a way that the ordering imposed by L is respected. The de nition proceeds recursively. If L is an empty proto-model, the only expansion of L into a closed interval I is the empty structure ;. Otherwise, let M be any set of pairs M = fhp; JpijEp 2 events(L); Jp I an intervalg (1) such that the orderings on the startand end-points of the Jp's respect the orderings on the corresponding Ep's imposed in order(L). If L is well-behaved, then order(L) contains no cycles, so that there exist M . Since order(L) can never impose the conditions start(Jp) = start(Jq) or start(Jp) = start(I) for p 6= q, we may assume that start(Jp) 6= start(I) and start(Jp) 6= start(Jq), for all p 6= q. As before, for each Ep 2 events(L),we de ne Gp = [start(I); start(Jp)]; in addition, let Hp be a small closed interval such that end(Hp) = start(Jp) and start(Hp) > start(Hq) for all q such that start(Jp) > start(Jq) : (2) The idea is that Hp is a small interval wedged in between the start of Jp and the start of anything that starts before Jp (see gure 1). For each Ep 2 events(L), let Rp be the right proto-model belonging to Ep, and let MRp be an expansion of Rp into Hp. Then we say that ML = M [ [ Ep2events(L)MRp (3) 7 is an expansion of L into I. To prove theorem 3, we establish that if L is the well-behaved left proto-model generated from , and ML is an expansion of L into I, then, for all 2 , ML j=I : (4) The proof proceeds by induction on the structure of . Let be any formula in . We must show that ML j=I . There are four cases: Case 1: is p. The loop in B1{B11 will ensure that L contains an event Ep. The construction ofML in (1) and (3) will ensure that there exists a Jp I such that hp; Jpi 2ML. It follows that ML j=I . Case 2: is : p. The loop in C1{C3 will ensure that p 2 banned(L). Now, by lemma 2, if there exists a Jp I such that hp; Jpi 2ML, then p 2 required(L). Since, by the well-behavedness of L, required(L)\ banned(L) = ;, there is no Jp I such that hp; Jpi 2ML. It follows that ML j=I . Case 3: is !(p1; !(p2; : : : !(pn; ) : : :)) where is not of the form !(q; ). We show a) For i = 1; : : : ; n, Jpi is the rst (ordered by starting point) sub-interval J I such that hpi; Ji 2ML b) For i = 1; : : : ; n 1, Jpi+1 Gpi c) ML j=Gpn Regarding a), the loop in A1{A9 ensures that, for each i, L contains an event Epi . The construction of ML in (1) and (3) will ensure that there exists a Jpi I such that hpi; Jpii 2ML. We need to show that, if hpi; Ji 2ML, then start(J) start(Jpi ). Suppose, then, that hpi; Ji 2ML. By lemma 2, pi 2 required(L). Since there is only one event E 2 events(L) with proposition(E) = pi, by the construction of ML in (1) and (3), hpi; Ji 2 MRq for some Eq 2 events(L). In that case, J Hq, and pi 2 required(Rq) by lemma 2. Now, line A3 ensures that pi 2 banned(Rpi), whence lines E3{E5 ensure that order(L) contains start(Eq) > start(Epi). Since M must respect this ordering, we have start(Jq) > start(Jpi). But then start(Hq) > start(Jpi ), by construction of Hq in (2), and so start(J) > start(Jpi ), as required. Regarding b), the loop in D1{D5 ensures that, for each i = 1; : : : ; n 1, start(Jpi) end(Jpi+1 ). Hence, Jpi+1 Gpi . Regarding c), if L is well-behaved, then so is Rpn . Since MRpn in equation (3) is an expansion of Rpn into Hpn , and 2 pn , the inductive hypothesis yields: MRpn j=Hpn : (5) In fact, we can strengthen (5) to ML j=Hpn ; (6) since, if hr;Ki 2 ML nMRpn then K = Hpn . (In other words, the additional facts in ML cannot a ect what is true in Hpn .) To see this latter fact, it is su cient to note that Hp \ Hq = ; and Jp = Hqfor all distinct Ep; Eq 2 events(L). Finally, we strengthen (6) to ML j=Gpn : (7) 8 If is of the form p or (p; ), then (6) immediately implies (7) by the semantic rules S1 andS4, respectively. If, on the other hand, is of the form : p, then, when Rpn is built from pn ,the loop in lines C1{C3 will force p 2 banned(Rpn). Suppose, then that, for some J Gpn ,hp; Ji 2 ML. By (3), and lemma 2, we have some Epi 2 events(L) with either (i) p = pi or(ii) p 2required(Rpi). If (i), then the condition in line E7 will be satis ed, and we will haveend(J) > start(Jpn ) (by setting r = p and s = pn). Hence J = Gpn . If (ii), then the condition inline E3 will be satis ed, and we will have start(Ji) >start(Jpn) (by setting r = pi and s = pn).Hence,start(Hpi) >start(Jpn) and so start(J) > start(Jpn ), whence J = Gpn . Thus, there is noJ Gpn with hp; Ji 2ML, and (7) is established.Thus we have established a), b) and c) and hence that ML j=I .This completes the examination of all the possible cases for , so that we have established (4). Hencehas a model. 2Corollary 1 (Finite models) If a set of formulae has a model, then it has a nite model.Proof: If is left-handed, the result follows from theorems 2 and 3, noting that the model ML in theproof of theorem 3 is nite. The extension to the general case is straightforward. 2Finally, we present the proof of the lemmas used aboveLemma 1 Let L = left proto model( ), and let t be a proposition letter. If t 2 required(L) andM j=I , then M j=I t.Let R = right proto model( ), and let t be a proposition letter. If t 2 required(R) and M j=I , thenM j=I t.We give the proof for the left-hand case: the right-hand case is similar.Proof: The proof proceeds by induction on the size of L. The case L = ; is trivial. A propositionletter t can be inserted into required(L) only by lines A8 and B9. Therefore, either (i) must containthe formula t or (ii) contains a formula of the form !(r1; !(r2; : : : !(t; ) : : :)) or (iii) containsa formula of the form !(r1; !(r2; : : : !(q; ) : : :)) with p 2 required( q). In cases (i) and (ii), we haveimmediately that, if M j=I , then M j=I t. In case (iii), if M j=I , then M j=Gq q. But Rq issmaller than L, so that, by the inductive hypothesis, M j=Gq t. Since Gq I, M j=I t by semanticrule S1. 2Lemma 2 Let L be a proto-model constructed by the above algorithm, and let ML be an expansion of Linto a closed interval I. If there exists an interval J I and a proposition letter p such that hp; Ji 2ML,then p 2 required(L).(Similarly for right proto-models.)We give the proof for the left-hand case: the right-hand case is similar.Proof: The proof proceeds by induction on the size of L. The case L = ; is trivial. By the constructionof ML in (1) and (3), hp; Ji 2ML implies either (i) there is an Ep 2 events(L) or (ii) hp; Ji 2 MRq forsomeEq 2 events(L). If (i), then, since events(L) is modi ed only by lines A7 and B8, the lines A8 and A9immediately followingwill ensure that p 2 required(L). If (ii), then, since Rq is a proto-model constructedby the above algorithm,MRq is an expansion of Rq into a closed interval, and Rq is smaller than L, bythe inductive hypothesis, p 2 required(Rq). But line A8 will ensure that required(Rq) required(L). 27 ConclusionsThis paper demonstrates the tractability of a highly restricted temporal logic, into which English sen-tences containing a variety of temporal prepositional constructions can be translated.9 References[1] Pratt, Ian and D.S. Br ee: \An approach to the Semantics of some English Temporal Construc-tions", Proceedings, Seventeenth Annual Conference of the Cognitive Science Society Mahwah, NJ:Lawrence Erlbaum, pp. 118{123.10