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


Representations as Processes: Situation-theoretic Objects encoded in the

 -calculus

Tsutomu Fujinami Centre for Cognitive Science University of Edinburgh e-mail: tsutomu@cogsci.ed.ac.uk
Abstract
The paper presents a way for encoding representations as concurrent processes. Based on linguistic data, we rst argue representations constitute a part of the meaning of sentences. Then, we regard representations as actions rather than static picture of things. To implement such an idea, we turn to the  -calculus and encode situation-theoretic objects into the calculus. Through the encoding, computational aspects of Situation Theory are investigated. We also show how sentences can be analyzed in our approach. Finally, we discuss what representations are and what outcome our approach would bring about.

1 Introduction
The dynamic approach to the meaning of sentences has been successful in analyzing various linguistic phenomena, e.g. donkey anaphora, taking the meaning of sentences as their context change potential 8, 9, 10, 14 . Such an approach may be extended to capture the meaning of sentences in dialogue, where participants collaborate to establish their mutual understanding through conversations. Shared mental states between participants are then involved in context, and to describe them we must need a ne-grained semantics such as Situation Semantics 5, 6 or Discourse Representation Theory DRT 9, 10 . To see that we really need more than rst order models to capture meaning, let us examine linguistic data. The following is an extract from Japanese Maptask corpus 3 . In the task, the route giver G has to conduct the route follower F to draw the correct route on his map, where they are provided with slightly di erent maps. In the extract, for example, the follower does not have iwase stone creek while the giver does. Suppose now the giver conducts the follower to recognize iwase stone creek, which he suspects the follower may not have in his map: G1 : jah, iwase tte iu no aru?1 then stone creek meta you have then, do you have a stone creek?
 Address: 2 Buccleuch Place, Edinburgh EH8 9LW, Scotland, U.K., Phone: +44 31 650 4419; Fax: +44 31 650 4587 The paper was presented at International Workshop on Computational Semantics, December 19-21, 1994, Tilburg, The Netherlands.

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F1 : iwase, iwase : : : stone creek stone creek, stone creek. . .  G2 : iwase tte iu no stone creek meta stone creek is . . .  F2 : hayase rapids wa
topic I have

wa. . .

topic

aru.

iwase wa nai. stone creek topic I do not have rapids, I have. stone creek, I do not have. Here we are concerned with the expression, tte iu no, appeared in G1 and G2 , which is in the literature analyzed as a sort of meta-linguistic expression 16 . Such an expression is used typically when the speaker is unsure if dialogue participants share the same concept and referent as to the intended item. In G1 , for example, the giver refers to iwase stone creek meta-linguistically because he is unsure if the follower recognizes it in his map. His suspicion is actually con rmed by the follower's lack of referent in his map as observed in F1 . Knowing the follower does not recognize it, the giver initiates his explanation on the item again by referring it meta-linguistically in G2 . The problem is how to capture the meaning of the meta-linguistic expression. One way to do this is, as mentioned above, to de ne it in terms of concepts and referents. That is, the speaker refers to an item meta-linguistically when he is unsure if dialogue participants share the same concept and referent. This explanation, however, leads to a problem. In F2 , the follower does not refer to iwase stone creek metalinguistically despite the fact that he cannot identify it in his map, which violates the explanation. To solve the problem, we postulate the use of the meta-linguistic expression di erently: the speaker considers whether or not dialogue participants mutually understand their attitudes to the item, where each participant's attitude may be di erent. The speaker refers to an item meta-linguistically when he is unsure if they mutually understand their attitudes to the item. In the above extract, the giver refers to iwase stone creek meta-linguistically in G1 and G2 because he believes they do not mutually understand their attitudes to the item. On the other hand, the reason why the follower refers to it normally | not meta-linguistically | in F2 is that he thinks it becomes obvious to them through their conversations that he cannot identify it while the giver can, which is their mutual understanding of their attitudes to the item2 .The example thus suggests that the use of the meta-linguistic expression concerns not simply the relation between agents and the world but a higher order
The utterances, G1 and F1 , are slightly simpli ed from the original by omitting irrelevant parts referring to other items. 2 One may think wa in F is used to make a contrast between hayase and iwase, where he is 2 actually unsure of their mutual understanding. But the uses of the meta-linguistic expression and the topical marker are independent as you can see from G2 . The follower must have referred to the item meta-linguistically if he is unsure of their mutual understanding in addition to the topical marker.
1

2

relation between agents and their attitudes, which is convincing enough for us to take account of representation as a constituent of meaning. Given the importance of representations, we may ask how such representations are constructed3 . Although many ideas exist on the topic, the one we will adopt is ecological realism : the idea that meaning thus representation arises from the interaction between cognitive agents and their environments. The idea lies in the core of Situation Theory, but the notion of interactions has not been fully investigated. Then, how interactions could be modelled? There are many di erent ideas for this, too, but we think concurrency is indispensable in any case, e.g. an agent can interact with more than one agent and can extract information from maps at the same time. Information gained through di erent channels is examined, compared with, and may be discarded or absorbed. While they are yet under development, some models have been proposed to analyse concurrent processes. The -calculus 12, 13 is one of such models, which we will turn to for our analysis. The reason why we adopt the calculus is that its underlying ontology ts with our needs to describe changing mental states of agents. In the calculus, states are modelled as con gurations of processes that can interact with each other, where processes are de ned as sequences of actions, and con gurations evolve into other forms upon interactions. Based on this de nition of states, we model mental states as con gurations of concurrent processes. Consequently, the change of con gurations caused by utterances is regarded to be the meaning of utterances. Having decided to look into interactions more fully, we may wonder how the notion of representation would t with our more dynamic view, which is often conceived of as the static pictures of things. In classical DRT 9 , for instance, DRSs are static objects describing agents' mental states at an abstract level, but obviously our mental activities do not stay and change continously. We can certainly encode representations into processes, which is one of our main purposes. To demonstrate it is possible, we will encode situation-theoretic objects in the -calculus. The encoding helps us in turn to investigate computational aspects of Situation Theory. To encode situation-theoretic objects into the calculus may not be so surprising because Situation Theory has to do with the calculus. One of the motivations of non-well-founded set theory 2 , which forms a basis of Situation Theory, is to de ne a semantics for Synchronous Calculus of Communicating SystemsSCCS 11 , an ancestor of the -calculus. Therefore, it should not be so di cult to implement an earlier version of Situation Semantics 7 using SCCS. But we are interested in here exploring the power of the -calculus to implement a more recent version of Situation Theory 5, 6 . Another intention of the encoding is to look for a computational model for Situation Theory, where computational means that it must be computable and be implementable. The strategy we will take is to regard situation-theoretic objects to be the speci cation of programs and to de ne the model as the set of programs satisfying the speci cation See 1 for viewing such an approach. Of course, the -calculus, which we can think of as a model, may not t with our purpose, and we may be forced to extend it. Nevertheless, the encoding should help us to understand what model we will need for computational Situation Theory. Apart from theoretical concern, we are interested in what bene ts we can expect by introducing concurrency. Modelling dialogue and multimodal communication are interesting applications, for which we will treat communicational events uniformally
3 Although the problem, the meaning of the meta-linguistic expression in Japanese, is quite interesting, it goes beyond the scope of the paper. However, we will come back to the point in the last and will discuss how the problem may be approached with our framework.

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as processes. We are certainly interested in modelling dialogue in the Map task where maps are given to participants. We need, however, to do more research in all aspects, especially on controlling interactions between processes, to apply our ideas to such applications. What we can do at the moment is just to suggest how sentences may be analysed within our framework. Though the example shown in x4 is far from complete, we include it to give the reader an insight of possible applications. The paper is organized as follows: the section 2 is an informal introduction to the -calculus, which we will use in the section 3 to encode situation-theoretic objects. The section 4 demonstrates how sentences may be analyzed within our framework. We conclude the paper with the discussion on functional vs. process-oriented view to meaning.

2 The -calculus
The section introduces the reader to the calculus informally. We start by examining simple automata, then will enrich them till we obtain the polyadic -calculus. The section is intended to give the reader basic ideas on the calculus to understand our encoding presented in the following section, not to formalize its syntax and semantics4. In what follows, greek letters range over automata or agents, capital letters processes, and italics actions.

Simple automata Let be a automaton that accepts a sequence of symbols, ha; bi, and terminates. We de ne the initial state of , P0 , by its possible courses of actions, e.g. accepting a and b, then terminating. We express such a state as P0 =def a:b:0, where 0 means termination5 . After accepting the symbol a, the state of turns into another state, P1 , such as accepting b and terminating, written as P1 =def b:0. To express the change of the state caused in by accepting a, we write P0 ,!P1 .
a

but we can think of another sort of action, emitting symbols. A system of automata which can interact with each other by emitting and accepting symbols forms a communicating system. Let the initial state of be P0 as before and the initial state of another automaton , Q0 , such as emitting the sequence of symbols, ha; bi. We distinguish such an output action with overline and write the de nition as Q0 =def a:b.0. Suppose a communicating system is composed of and , whose initial state is de ned as P0 jQ0, meaning that P0 and Q0 are concurrently active. Because they are active, they can communicate, e.g. by 's emitting a symbol and 's accepting it. Suppose now emits a and picks it up. Upon this interaction, the state of the system turns into another state, which is expressed as P0 jQ0 ,!P1 jQ1 , where P1 is as before and Q1 =def b.0. The symbol, `aa', means these actions are synchronised6. Hereinafter, we we may write ,! for ,! for readability, omitting actions.
aa aa

Simple communicating systems Simple automata can only accept symbols,

Interested readers are referred to 12, 13 . Hereinafter, 0 may be omitted for readability. We assume a synchronised version of the calculus. See 11 on synchronous and asynchronous calculi.
4 5 6

4

garded as establishing a communication channel or port through which data can be exchanged. Suppose is willing to receive a number from through a port c and wants to emit `5' through it. Let P10 be 's initial state, de ned as cx.P11 , where x is a parameter to be substituted for upon the interaction, whereas the 's initial state, Q10 , is de ned as ch5i.Q11 , where `5' is the number to be emitted. Upon the interaction, the number `5' is emitted to , and x is substituted for it, which we write as P10 jQ10 ,!P11 f5 g jQ11 . The expression, P11 f5 g , means any x appeared in P11 to be substituted for `5'.
=x =x

Data communicating systems The interaction between and can be re-

The monadic -calculus We have so far distinguished ports from data, but we

can treat them uniformally as names , which bene ts us with mobility: an access to a port too can be exchanged. Suppose needs to emit `5' to and wants to take over the job. Supose also is accessible only through a port a, and and communicates with only through another port b. Assume rst would have an access to a. Then, what needs to do is to send `5' so that it can emit the number to . 's state P20 may be de ned as bx.ahxi.P21 , 's state Q20 as bh5i.Q21 , and 's state R20 as ay.R21 . Upon 's emitting `5' to and his emitting it to , the state of the system would change as follows:

bx.ahxi.P21 j bh5i.Q21 j ay.R21  emits 5 to via b. ,! ah5i.P21 f5 g j Q21 j ay.R21 Receiving it, substitutes it for x, and emits it to via a.
=x

,! P21 f5

=x

 receives 5 and substitutes it for y.

g

j Q21

j R21 f5

=y

g

Now, suppose that does not have the access to , the port a. How can pass the job to ? Since does not have the access, has to rst emit it to , which is made possible in the -calculus. rst sends the port name a via b, then the number. Let 's initial state P30 be bx.bz .xhz i.P31 , 's state Q30 =def bhai.bh5i.Q31 , and 's state R30 =def ay.R31 . The state of the system may change upon interactions as follows:

bx.bz .xhz i.P31 j bhai.bh5i.Q31 j ay.R31  emits a to via b. ,! bz .ahz i.P31 f g j bh5i.Q31 j ay.R31  substitutes it for x7 . emits 5 to via b. j ay.R31 ,! ah5i.P31 f 5 g j Q31  substitutes it for z and emits it to via a.
a=x a=x; =z

,! P31 f

 substitutes 5 for y.

a=x; =z

5

g

j Q31

j R31 f5

=y

g

The polyadic -calculus In the polyadic version, agents can exchange sequences of data, not only a single item, upon a single interaction. Suppose, for instance, wants to receive a sequence of three items, while wants to emit a sequence, h4; 5; 6i,
7

Then the name standing for a port name in  .
x b z

x z

h i.P31

is substituted for .
a

5

where they are linked through a port d. Let the 's state P40 be dx; y; z .P41 and the 's state Q40 =def dh4; 5; 6i.Q41. The state of the system would change upon a single interaction as follows:

,! P41 f4

dx; y; z .P41
5 6

=x; =y; =z

g

j dh4; 5; 6i.Q41 j Q41

3 Encoding situation-theoretic objects
Memory as processes Representations are internally realized as active memory.
We, therefore, rst see how memory can be encoded8 . In the calculus, it is encoded as a process returning a value upon request. Let M0 be a memory storing a value v at an address l. Such a memory can be de ned as lhvi.0. To read the value, other processes interact with it at l. Suppose a process, R0 , reads the value, de ned as lx.R1 . Upon their interaction, the state of the system composed of M0 and R0 would change as follows:

lhvi.0j lx.R1 ,! 0 j R1 f g
v=x

In the de nition, the value can be read only once. For the memory to return it repeatedly, we may program it as an iterative process with the replication symbol `!' as !lhvi, which means to repeat the action lhvi some nite times. Thus, since !lhvi = lhvij  jlhvi, the memory can provide others with the value some nite times:

lhvij   j lhvij lx.R1 ,! lhvij   j lhvij R1 f g
v=x

Representations as invariant processes Our notion of representations is a
l v

generalization of the memory. Namely, an output action a is a representation if the action does not change the succeeding behavior of the process. The behavior of the h i memory, for instance, does not change after it emits the value because M0,!M1 and M1 is equal to M0 , i.e. !lhvi. We can extend such a case of single action to other composite actions by parallel and sequential operations. That is, a parallel actions, l0 hv0 ijl1 hv1 i, may represent something in the context of two parallel transitions such 0h 0 1h 1 that P0 ,!iP1 and P2 ,!iP3 if P0 is equivalent to P1 , and P2 to P3 . A sequence of 0h 0 1h 1 actions, l0 hv0 i.l1 hv1 i, may represent something in the context of P0 ,!iP1 ,!iP2 if P2 is equivalent9 to P0 .
l v l v l v l v

Propositions Propositions are of the form that s j= , where s is a situation and
See 13 for basic ideas and detail of encoding more complex data structures. To de ne the equivalence relation between processes is, however, not so easy though in the paper we limit our interest only to the simplest case, replication. We may de ne the relation as bisimilar , but leave it for future work. 10 `!' may be omitted in what follows for readability.
8 9

an infon, meaning that s supports ". In our encoding, we regard situations as ports and infons as data available through them. Hence, the proposition is encoded as a single action such as sh i, or more precisely as !sh i with replication10. An

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infon hhr; a; b; 1ii, whose relation is r, objects standing in r are a and b, with the polarity 1, is encoded as the sequence of data, hr; a; b; 1i. Thus the proposition s j= hhr; a; b; 1ii is encoded as shr; a; b; 1i, while s j= hhr; a; b; 0ii as shr; a; b; 0i. Conjunctions such as s j= ^&  are made true if and only if both processes, sh i and sh& i, are concurrently active, i.e. sh ijsh& i. The translation owes to the de nition that s j= ^ & i s j= and s j= & . On the other hand, to translate disjunctions is more troublesome. One way to do this is to use choice. In the calculus, P +Q means to execute either P or Q non-deterministically. With the function, a disjunction such as s j= _ &  may be de ned as sh i+sh& i, but the problem is that the de nition does not cover a case where both and & are true because the function allows only one process to survive. We may extend the calculus in this respect rather than encoding disjunctions in an ad-hoc way.

Abstraction Parametric propositions can be abstracted over the parameters to form situation-theoretic types. A parametric proposition, for instance, s j= hhr; x; b; 1ii containing a parameter x, can be abstracted over x with a role to form a type such that  !x s j= hhr; x; b; 1ii. We encode an abstractor as a process sharing names with others encoding parametric propositions. The type, for example, would be encoded as:  x x j shr; x; b; 1i, where  x binds x in its scope as a private
name like a local variable in procedual languages. Simultaneous abstractions, where the order of abstractions over parameters does not matter, is obtained thanks to the nature of concurrency. For example, a situationtheoretic type,  1 !x, 2 !y s j= hhr; x; y; 1ii equivalent to  2 !y, 1 !x s j= hhr; x; y; 1ii, can be encoded either as  x; y 1 x j 2 y j shr; x; y; 1i or  x; y 2 y j 1 x j shr; x; y; 1i because the calculus ensures the structual equivalence, P j Q = Q j P . Meanwhile, if one wants to abstract parameters in some order, i.e. rst over x then y, then he may express it sequentially as  x; y 1 x. 2 y j shr; x; y; 1i.

Assignments Types are anchored to environments by being assigned objects to roles. For example, a type  1 !x, 2 !y s j= hhr; x; y; 1ii is anchored to an environment by an anchoring function, 1 !a, 2 !b , which assigns a to 1 and b to 2 . Applied to the function, the type would be reduced to the proposition, s j= hhr; a; b; 1ii. Such anchoring functions are encoded as output processes. The function is, for example, encoded as 1 hai j 2 hbi or conversely. In the presence of
these processes, the processes encoding the type would be degenerated as follows11:  x; y  1 x.0j 2 y.0j shr; x; y; 1i j 1 hai.0 j 2 hbi.0 x is substituted for a. ,!  y  0 j 2 y.0j shr; a; y; 1i j0 j 2 hbi.0 y is substituted for b. ,! 0 j0 j shr; a; b; 1i j0 j 0
11 Although we explain the transitions sequentially, these interactions can occur in any order, even in parallel.

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Abstraction 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; yy j yx jshr; x; b; 1i with parallel composition or as -  x; yy.yx 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.

Then, the constraint is encoded either as12:

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 s2 hr2 ; c; d; 1i.
- s1 x; y; z; p. x = r1 ; y = a; z = b; p = 1 s2 hr2 ; c; d; 1i with sequential composition, or -  x; y; z; ps1x; y; z; p j x = r1 ; y = a; z = b; p = 1 s2 hr2 ; c; d; 1i with parallel composition.

In the above de nitions, the action s2 hr2 ; 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 s1 hr1 ; a; b; 1i encoding the situation, s1 j= hhr1 ; a; b; 1ii, the system composed sequentially would be degenerated as follows:

,! 0 j r1 = r1 ; a = a; b = b; 1 = 1 s2 hr2 ; c; d; 1i.0
12 These 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, 1 h i, for testing though `!' is omitted for simplicity.
s x; y; z; p

s1 hr1 ; a; b; 1i.0 j s1 x; y; z; p. x = r1 ; y = a; z = b; p = 1 s2 hr2 ; c; d; 1i.0

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where the last formula is equal to s2 hr2 ; 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 s2 hr2 ; x; y; 1i, or -  z; x; y; ps1z; x; y; p j z = r1 ; p = 1 s2 hr2 ; 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 s1 hr1 ; a; b; 1i as an anchoring function, which suggests in our framework constraints are treated as a complex form of abstraction and anchoring.

,! 0 j r1 = r1 ; 1 = 1 s2 hr2 ; a; b; 1i.0

s1 hr1 ; a; b; 1i.0 j s1 z; x; y; p. z = r1 ; p = 1 s2 hr2 ; x; y; 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.0jnpy.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 xv. Suppose also the meanign of walks" is  y!w s j= hhwalk, w, 1ii and is encoded as shwalk; w; 1i j yw. 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
13

The 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; vnphxi.0 j shmale; v; 1i j xv j l1 x walk = walks. y; wnpy.S j shwalk; w; 1i j yw 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 :  vshmale; v; 1i j dv j  wshwalk; w; 1i j dw 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 ,!Q , will be regarded to be a channel from P to Q , and
a

14

For detail, see the appendix A.

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these states are de ned as possible courses of actions. The theory claims that the 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 is classi ed as P , and s1 7,! s2 means s1 is connected to s2 , which we can understand as the change from s1 to s2 upon the action. The arrow, ,!, appeared in the transition, P ,!Q , is replaced for =. P = Q
a a

s1 7,! s2
Our intention is to t representations into the picture. With our de nition, to represent something by the action a, P and Q must be equivalent. Since they are equivalent, we need not distinguish these states, s1 and s2 . Hence, they can be said to be identical, say to s1 . Let us express such an identical connection as 7,!. Then the above picture is redrawn as: P = P
id a





Now it should be clear what is classi ed by the action a; it is the identity connection of s1 . One way to re ne Situation Semantics towards dynamics may be to reformulate propositions such as s1 j= a this way.

s1 7,! s1
id





The meaning of the meta-linguistic expression Going back to the extract
of Japanese maptask corpus presented earlier, let us see how the problem may be approached within our framework. In short, we think the meta-linguistic expression, to iu no, means a failure in updating their mutual understanding required for their collaboration. Ideally, it should be clear to all dialogue participants how their mutual understanding would be updated upon an utterance. But certainly it is not always the case. One may fail to understand others for various reasons: lack of vocabulary, knowledge, and referents. The expression enables dialogue participants in such a case to repair explicitely their mutual understanding. Channel Theory seems to be useful to investigate the meaning too because it allows us to analyze various failures in communication.

our view on computation, i.e. computation as processes, may work for linguistic application both in semantic and syntactic aspects. Comparing with functional view, which is often adopted for analysing meaning of sentences, what bene ts can we expect of our process-oriented view? Although we cannot answer exactly how it would bene ts us because it is yet to be investigated, it seems at least more natural to model discourse as processes rather than as functions. For example, in analysing the meaning of a discourse composed of two sentences s1 and s2 whose individual meanings 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 P will be substituted for S2 . Once we choose to represent meaning functionally, we have to commit ourselves to functional view, too. We may continue modifying the -calculus so that it ts with our linguistic intuition, but there might be some limitation in doing so due to its underlying philosophy, i.e. capturing meaning in terms of input and output. Meanwhile, the other approach taking computation as processes allows us to describe 11

From functions to processes Throughout the paper, we have investigated how

state changes with ease. We should, however, pay attention to the price we have to pay. We do not know yet how much the theory may be complicated and how it could be applied for. Moreover, compositionality may be lost to some extent because it is unpredictable how concurrent processes interact with each other. These are all left for 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 researchers on the Japanese Maptask Corpus Project at Chiba University for the data cited in the introduction.

References
1 Abramsky, S. 1988 Domain theory in logical form. Research Report DOC 88 15, Department of Computing, Imperial College of Science and Technology. 2 Aczel, P. 1988 Non-Well-Founded Sets . Stanford, Ca.: Center for the Study of 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 Language Processing, SLP3-5, pp. 25-30. in Japanese. Information Processing Society of Japan. 4 Barwise, J. 1993 Constraints, channels, and the ow of information. In P. Aczel, D. Israel, Y. Katagiri and S. Peters, eds., Situation Theory and its Applications, Vol.3, pp. 3-27. Stanford, Ca.: Center for the Study of Language and Informaiton. 5 Barwise, J. and R. Cooper 1991 Simple situation theory and its graphical representation. In J. Seligman, ed., Partial and Dynamic Semantics III, pp. 38-74. Centre for Cognitive Science, University of Edinburgh. DYANA Report R2.1.C. 6 Barwise, J. and R. Cooper 1993 Extended Kamp Notation: a graphical notation for situation theory. In Situation Theory and its Applications, Vol.3, pp. 29-53. 7 Barwise, J. and J. Etchemendy 1987 The Liar: An Essay on Truth and Circularity . Oxford University Press. 8 Groenendijk, J. and M. Stokhof 1991 Dynamic Montague Grammar. In M. Stokhof, J. Groenendijk and David Beaver, eds., Quanti cation and Anaphora I. Centre for Cognitive Science, University of Edinburgh. DYANA Report R2.2.A. 9 Kamp, H. 1984 A theory of truth and semantic representation. In J. A. G. Groenendijk, T. M. V. Janseen and M. Stokhof, eds., Truth, Interpretation and Information: Selected Papers from the Third Amsterdam Colloquium, pp. 1-41. Dordrecht: Foris Publications. 10 Kamp, H. and U. Reyle 1993 From Discourse to Logic: Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representatin Theory. Dordrecht: Kluwer. 12

11 Milner, R. 1983 Calculi for Synchrony and Asynchrony. Theoretical Computer Science, Vol. 25, pp. 267-310. 12 Milner, R. 1991 The polyadic -calculus: a tutorial, Research Report LFCS91-180, Laboratory for Foundations of Computer Science, Computer Science Department, Edinburgh University. 13 Milner, R., J. Parrow and D. Walker 1992 A calculus of mobile processes, Parts I and II, Journal of Information and Computation, Vol. 100, pp. 1-40 and pp. 41-77. 14 Muskens, R. 1992 Anaphora and the logic of change. ITK Research Report No. 34, Instituut voor Taal- en Kennistechnologie, Tilburg University, Tilburg, Netherlands. 15 Seligman, J. and J. Barwise 1993 Channel theory: toward a mathematics of imperfect information ow. Unpublished ms., May 1993. 16 Takubo, Y. and S. Kinsui 1992 Discourse management in terms of mental domains. 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 the discourse marker, d, is available from the process l1 hdi. The transitions would be as follows: 1 The initial state. he.  x; v nphxi.0 j shmale; v; 1i j xv j l1 x.0 j walks.  y; wnpy.S j shwalk; w; 1i j yw j he. walks.0 j l1 hdi.0 2 Given the utterance of he", the process encoding the lexicon of he" becomes active. ,!  x; v nphxi.0 j shmale; v; 1i j xv j l1 x.0 j walks.  y; wnpy.S j shwalk; w; 1i j yw j walks.0 j l1 hdi.0 3 Then, it picks up the discourse marker, d, through l1 , and substitutes it for x. ,!  v nphdi.0 j shmale; v; 1i j dv j 0 j walks.  y; wnpy.S j shwalk; w; 1i j yw j walks.0 j0 4 Given the utterance of walks", the other process becomes active. ,!  v nphdi.0 j shmale; v; 1i j dv j 0 j  y; wnpy.S j shwalk; w; 1i j yw j 5 It picks up the marker through np and substitutes it for y. ,!  v 0 j shmale; v; 1i j dv j 0 j  wS j shwalk; w; 1i j dw j

0 0

j0 j0

13


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