Representations as Processes: Situation-theoretic Objects encoded in the -calculus

ion over roles The mobility of the calculus enables us to abstract over roles as well as over objects, relations, and situations, which is not usually allowed to do in Situation Theory. Suppose we have a type [ !x](s j= hhr; x; b; 1ii), which we would like to abstract further over the role, . We will abstract it by replacing a parameter y for the role and binding it with another role, %, to get a type such that[ %!y]. [ y!x](s j= hhr; x; b; 1ii) Such a type can be encoded either as ( x; y)(%(y) j y(x) jshr; x; b; 1i) with parallel composition or as ( x; y)(%(y).y(x) j shr; x; b; 1i) with sequential composition. There could be many reasons for introducing the abstraction over roles. In applications, for example, we would like to regard a role to be a discourse marker and allow processes to exchange it through ports. Such an interaction can occur not only between agents participating in a dialogue but even between processes processing di erent parts of sentences (See the next section 4). Constraints Constraints are relations between propositions or types. A constraint between propositions, (s1 j= hhr1; a; b; 1ii)) (s2 j= hhr2; c; d; 1ii), for example, means that if s1 supports hhr1; a; b; 1ii, then s2 supports hhr2; c; d; 1ii. We encode the antecedent as a test such that it receives data at s1, compares them with particular data, say hr1; a; b; 1i, then allows the other or following process encoding the consequent to be active. Suppose the consequent is encoded as a process s2hr2; c; d; 1i. Then, the constraint is encoded either as12: s1(x; y; z; p). [x = r1; y = a; z = b; p = 1] s2hr2; c; d; 1i with sequential composition, or ( x; y; z; p)(s1(x; y; z; p) j [x = r1; y = a; z = b; p = 1] s2hr2; c; d; 1i) with parallel composition. In the above de nitions, the action s2hr2; c; d; 1i can be performed only if the condition bracketed with `[ ]' is met, otherwise it behaves like 0. These de nitions are di erent only in that whether or not the input process runs concurrently. The reason why the names, x; y; z; and p, are not bound with in the former is that the preceeding action in sequential process binds them. See in the presence of the process s1hr1; a; b; 1i encoding the situation, s1 j= hhr1; a; b; 1ii, the system composed sequentially would be degenerated as follows: s1hr1; a; b; 1i.0 j s1(x; y; z; p). [x = r1; y = a; z = b; p = 1] s2hr2; c; d; 1i.0 ! 0j [r1 = r1; a = a; b = b; 1 = 1] s2hr2; c; d; 1i.0 12These processes de ned to encode constraints should be actually de ned as iterative processes so that they can interact with all the process encoding propositions of the form, s1hx; y; z; pi, for testing though `!' is omitted for simplicity. 8 where the last formula is equal to s2hr2; c; d; 1i.0 because the condition is met. Not only the constraints between propositions, but also would we like to express them between types such that (s1 j= hhr1; x; y; 1ii) ) (s2 j= hhr2; x; y; 1ii), where the same parameters are to be identical. Since names can be shared between processes, such a constraint can be encoded in the same way as the constraint between propositions is encoded. For example, it may be encoded either as: s1(z; x; y; p).[z = r1; p = 1] s2hr2; x; y; 1i, or ( z; x; y; p)(s1(z; x; y; p) j [z = r1; p = 1] s2hr2; x; y; 1i) In the same context, the system composed sequentially would be degenerated as below. Note in the transitions, s1(z; x; y; p) behaves as an abstractor with the role s1, and s1hr1; a; b; 1i as an anchoring function, which suggests in our framework constraints are treated as a complex form of abstraction and anchoring. s1hr1; a; b; 1i.0 j s1(z; x; y; p). [z = r1; p = 1]s2hr2; x; y; 1i.0 ! 0j [r1 = r1; 1 = 1]s2hr2; a; b; 1i.0 4 Processing sentences In this section, we see how sentences may be analyzed within our framework. First, we model a sentence as output actions. For example, a sentence, \He walks", is modeled as the sequence of output actions, he.walks. Secondly, such actions can trigger the processes encoding corresponding lexical items. Suppose lexical item for walks be encoded as a process L. The process would be pre xed with an input action walks so that it will become active only when the word is uttered, i.e. walks.L. Thirdly, the processes encoding lexical informaion consists of two parts, one encoding syntactic information and another encoding semantic informaiton. Let us see how a phrase structure, np vp ! s, may be encoded, where vp is a head. We encode the phrase structure as two processes. One encoding vp is a process that accepts the symbol np and turns into another process encoding s, say S, which we express as np.S. Another encoding np is a process that emits np and terminates, written as np.0. Then, the phrase structure is encoded as the system of np.0jnp.S,13 which can evolve into 0jS upon the interaction through np. Notice this encoding makes it possible for the process encoding np to emit data to another encoding vp, therefore to the other encoding s. If it is willing to emit a discourse marker, d, it can be written as nphdi.0jnp(y).S. We will implement our grammar in this way. The syntactic information is combied with the semantic information. Suppose the meaning of \he" is [ x!v](s j= hhmale; v; 1ii) and is encoded as shmale; v; 1i j x(v). Suppose also the meanign of \walks" is [ y!w](s j= hhwalk, w, 1ii) and is encoded as shwalk; w; 1i j y(w). The roles x and y will be further abstracted over and will be assinged values during parsing the sentence, which is made possible thanks to the 13The encoding is similar to the representation of syntactic structures in categorial grammar. 9 mobility of the calculus. We assume a process picking up a discourse marker, l1(x), runs concurrently with the processes encoding the meaning of \he". Combined with syntactic information, these lexical items are expressed as follows: [[he]] = he.( x; v)(nphxi.0 j shmale; v; 1i j x(v) j l1(x)) [[walk]] = walks.( y; w)(np(y).S j shwalk; w; 1i j y(w)) In the presence of another process encoding the utterance \He walks" and the other providing the discourse marker, d, these processes would be reduced to the following state14: ( v)(shmale; v; 1i j d(v)) j ( w)(shwalk; w; 1i j d(w)) which corresponds to a situation-theoretic type such that [ d!v, d!w] (s j= hhmale; v; 1ii ^ hhwalk; w; 1ii) Given a process encoding an anchoring function such as [ d!m], i.e. dhmi, the system would further be degenerated to: shmale; m; 1i j shwalk; m; 1i which corresponds to a proposition, (s j= hhmale; m; 1ii ^ hhwalk; m; 1ii) Note that the processes encoding lexical items of \he" and \walks" should actually be de ned with replication, i.e. !L1 and !L2 where L1 and L2 mean [[he]] and [[walk]] , respectively. This ensures that these processes can be created repeatedly. The change the utterance causes to the system of !L1 j !L2 is the spin o of the processes encoding the meaning of the sentence, say P. The whole transition is then expressed as !L1 j !L2 ! ! P j !L1 j !L2 and the con gurations of the system before and after the transition di ers by P. This shows our approach takes meaning as context change potential. 5 Conclusion Let us revise how we capture representations in terms of actions. Our approach is based on Channel Theory [4, 15], especially its application to Hoare-style logic, where the transition, P a !Q, will be regarded to be a channel from P to Q, and 14For detail, see the appendix A. 10 these states are de ned as possible courses of actions. The theory claims thatthe states, P and Q, classify some particular states, say s1 and s2, respectively.Such relations may be depicted as below, where s1:P (shown vertically) means s1 isclassi ed as P, and s1 7 ! s2 means s1 is connected to s2, which we can understandas the change from s1 to s2 upon the action. The arrow, !, appeared in thetransition, P a!Q, is replaced for =).Pa=) Qs1 7 ! s2Our intention is to t representations into the picture. With our de nition, torepresent something by the action a, P and Q must be equivalent. Since they areequivalent, we need not distinguish these states, s1 and s2. Hence, they can be saidto be identical, say to s1. Let us express such an identical connection as id7 !. Thenthe above picture is redrawn as:Pa=) Ps1id7 ! s1Now it should be clear what is classi ed by the action a; it is the identity con-nection of s1. One way to re ne Situation Semantics towards dynamics may be toreformulate propositions such as (s1 j= a) this way.The meaning of the meta-linguistic expression Going back to the extractof Japanese maptask corpus presented earlier, let us see how the problem may beapproached within our framework. In short, we think the meta-linguistic expression,to iu no, means a failure in updating their mutual understanding required fortheir collaboration. Ideally, it should be clear to all dialogue participants how theirmutual understanding would be updated upon an utterance. But certainly it is notalways the case. One may fail to understand others for various reasons: lack ofvocabulary, knowledge, and referents. The expression enables dialogue participantsin such a case to repair explicitely their mutual understanding. Channel Theoryseems to be useful to investigate the meaning too because it allows us to analyzevarious failures in communication.From functions to processes Throughout the paper, we have investigated howour view on computation, i.e. computation as processes, may work for linguisticapplication both in semantic and syntactic aspects. Comparing with functionalview, which is often adopted for analysing meaning of sentences, what bene ts canwe expect of our process-oriented view? Although we cannot answer exactly how itwould bene ts us because it is yet to be investigated, it seems at least more naturalto model discourse as processes rather than as functions. For example, in analysingthe meaning of a discourse composed of two sentences s1 and s2 whose individualmeanings are de ned as S1 and S2, respectively, the meaning of s1 is usually -abstracted over the meaning of the succeeding sentence, e.g. P (S1 ^P ), where Pwill be substituted for S2.Once we choose to represent meaning functionally, we have to commit ourselves tofunctional view, too. We may continue modifying the -calculus so that it ts withour linguistic intuition, but there might be some limitation in doing so due to itsunderlying philosophy, i.e. capturing meaning in terms of input and output. Mean-while, the other approach taking computation as processes allows us to describe11 state changes with ease. We should, however, pay attention to the price we have topay. We do not know yet how much the theory may be complicated and how it couldbe applied for. Moreover, compositionality may be lost to some extent because it isunpredictable how concurrent processes interact with each other. These are all leftfor future research.Acknowledgement The author thanks Robin Cooper for his supervision, Yuki-nori Takubo for the discussion on the meaning of the particle, and the researcherson the Japanese Maptask Corpus Project at Chiba University for the data cited inthe introduction.References[1] Abramsky, S. (1988) Domain theory in logical form. Research Report DOC88/15, Department of Computing, Imperial College of Science and Technology.[2] Aczel, P. (1988) Non-Well-Founded Sets . Stanford, Ca.: Center for the Studyof Language and Informaiton.[3] Aono, M., K. Ichikawa, H. Koiso, S. Sato, M. Naka, S. Tutiya, K. Yagi, N.Watanabe, M. Ishizaki, M. Okada, H. Suzuki, Y. Nakano, and K. Nonaka(1994) The Japanese map task corpus: an interim report. In Spoken LanguageProcessing, SLP3-5, pp. 25-30. in Japanese. 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ITK Research ReportNo. 34, Instituut voor Taalen Kennistechnologie, Tilburg University, Tilburg,Netherlands.[15] Seligman, J. and J. Barwise (1993) Channel theory: toward a mathematics ofimperfect information ow. Unpublished ms., May 1993.[16] Takubo, Y. and S. Kinsui (1992) Discourse management in terms of mentaldomains. Unpublished ms., to appear in Travaux de Linguistique Japonaise.A Parsing \He walks"The following shows how the system of processes encoding lexical items, \he" and\walks", would be degenerated upon the utterance, \He walks". We assume thediscourse marker, d, is available from the processl1hdi. The transitions would beas follows:1) The initial state.he. ( x; v)( nphxi.0 j shmale; v; 1i j x(v) j l1(x).0) jwalks. ( y; w)(np(y).S j shwalk; w; 1i j y(w)) jhe. walks.0j l1hdi.02) Given the utterance of \he", the process encodingthe lexicon of \he" becomes active.!( x; v)( nphxi.0 j shmale; v; 1i j x(v) j l1(x).0) jwalks. ( y; w)(np(y).S j shwalk; w; 1i j y(w)) jwalks.0j l1hdi.03) Then, it picks up the discourse marker, d, through l1,and substitutes it for x.!( v)( nphdi.0 j shmale; v; 1i j d(v) j 0) jwalks. ( y; w)(np(y).S j shwalk; w; 1i j y(w)) jwalks.0j 04) Given the utterance of \walks", the other processbecomes active.!( v)( nphdi.0 j shmale; v; 1i j d(v) j 0) j( y; w)(np(y).S j shwalk; w; 1i j y(w)) j0j 05) It picks up the marker through np and substitutes it for y.!( v)( 0j shmale; v; 1i j d(v) j 0) j( w)(Sj shwalk; w; 1i j d(w)) j0j 013