Compositionality Through Sentence Interfaces

In a compositional semantic theory it should be possible to determine the interpretation of each expression irrespective of the contexts in which it possibly occurs. However, in order to account for the inter-sentential anaphoric links a coherent text usually contains, it seems that each sentence in the text should be interpreted with respect to the context set up by the interpretation of the sentences which precede it. A problem thus arises concerning whether a compositional theory of text interpretation is possible at all and, in case, what the distinctive features of such a theory should be. According to [JvEB81] and [Jan97], a non compositionl semantic theory can be made compositional by refining the notion of meaning it is based on. By assuming the formal framework of Discourse Representation Theory (DRT), in the paper it will be considered how the notion of (sentence) meaning DRT is based on should be refined in order to support a compositional theory of text interpretation. By considering texts as systems in which, through inter-sentential anaphoric links, information concerning entities flows from sentence to sentence, it will be argued that a compositional theory of text interpretation should be based on the "anaphoric possibilities" allowed by each sentence in the text. In order to represent such possibilities, the concept of "sentence interfaces" will be introduced which allows a description of the composition of representation structures in terms of communication through anaphoric ports. What makes a sequence of sentences a genuine text is what text theorists usually define as text coherence. A precise definition of text coherence is out of the aims of this paper (note, however, that in this context coherence has a meaning which is different from the one common in logic). For our purposes, a text will be any sequence of sentences in which inter-sentential anaphoric links (both nominal and temporal) can be identified, which connect expressions occurring in different sentences. Examples of inter-sentential anaphoric links can be seen in the sequence of sentences below: 1. (a) Leo arrived at the top of the mountain at 5 p.m. (b) He had left home two hours before. In order to understand correctly this very simple text, the pronoun he has to be interpreted as referring to the same entity referred to by the proper name Leo which occurs in a different sentence. Similarly, the location in time of the event of Leo's leaving home, described in (1b), has to be determined as two hours in the past of the event of Leo's arriving at the top of the mountain, described in the sentence (1a). Inter-sentential links like those exemplified in (1) cannot be interpreted by considering the component sentences in isolation. Rather, this information can be determined only by interpreting (1b) in the context set up by the interpretation of (1a). Discourse Representation Theory ([KaR93]) is a formal theory of meaning which accounts sistematically for the kind of inter-sentential links exemplified in (1). Standard DRT associates with natural language sentences (and texts, as well) Discourse Representation Structures (DRS) which are pairs K = , where U is a set of discourse markers (formal entities which stand for the entities the discourse is about) and C is a set of conditions (formulas which represent information concerning such entities). With respect to a first order model M = and a set of discourse markers V, the semantic value of a DRS K = (in symbols ||K||M) can be determined by the pair , where F ⊆ DV is a set of assignments which satisfy in M all the conditions in C. Given a first order model M = , a DRS K = and a function s: V → D, s verifies K in M if and only if ||K||M = and there is s'∈DV such that s' is the same as s but possibly the values it assigns to the members of U (in symbols s[U]s'), and s'∈F. Finally, the DRS K = is true in M if and only if ||K||M = and F ≠ ∅. In the DRT approach, anaphoric links are accounted for in terms of relations between discourse markers; the representation of inter-sentential anaphoric links thus requires the sharing of discourse markers which occur in DRSs which have been associated with different sentences. Given a text T = S1,...,Sn, in order to make the sharing of discourse markers possible, in the standard DRT approach the model-theoretic interpretation of the text is deferred until the DRS KT, which represents the global content of the text (including the resolution of all the inter-sentential anaphoric links it contains), has been constructed. The DRS KT is built incrementally by processing each sentence Si with respect to the DRS Ki-1,which represents the content of the fragment of text S1,...,Si-1. Although, the DRS KT has been built compositionally, the model-theoretic interpretation of the text T is not compositional at all, since it cannot be determined as a function of the model-theoretic interpretation of the component sentences. This makes standard DRT a non compositional semantic theory. The problem thus arises whether DRT can be made compositional, while retaining its ability of representing the sharing of discourse markers, which allows the treatment of inter-sentential anaphoric links. Besides this theory specific problem, the general problem arises concerning whether a compositional theory of text interpretation is possible at all and, in case, what such a theory should amount to. According to [JvEB81] and [Jan97], there is a standard strategy which can be assumed in order to make compositional a non-compositional semantic theory; such a strategy consists in the refinement of the notion of meaning the theory is based on. Thus, by considering how DRT can be made a compositional theory of text interpretation, it is possible to determine how the meaning of sentences should be represented in order to account for the role sentences play in texts. In [vEK97] a compositional version of DRT has been introduced which accounts for the sharing of discourse markers across DRSs which have been associated with different sentences. Let A be a set of constants, V a set of discourse markers and P a set of n-ary predicates. The language of pDRS (proto-DRSs) can be defined as follows (; is a sequencing operator which comes from the dynamic logic for programming languages): terms: t::= v | a pDRS: K ::= v | T | Pt1...tn| t=v | ¬K | (K1;K2) Given a first order model M = , the value of the term t in M with respect to the assignment f (|t|M,f), and the value of the DRS K in M (||D||M) can be defined as follows: |t| M,f = I(t), if t∈A |t| M,f = f(t), if t∈V ||v||M = <{v}, DV> ||T||M = <∅, DV> ||Pt1...tn||M = <∅, {f∈DV:< |t1| M,f,..., |tn| M,f> ∈I(P)} ||t=v||M = <∅, {f∈DV: f(v) = |t| M,f}> ||¬K||M = -||K ||M ||K1;K2||M = ||K1||M ° ||K2||M What makes the theory of pDRSs able to account for the sharing of discourse markers is the semantics of the composition of pDRS, which is defined as follows: ° = Given a pDRS Ki, in order to characterize the set of the discourse markers which can be referred to also in a different pDRS Kj, in [vEK97] three different kinds of discourse markers occurrences have been distinguished which roughly correspond to different types of variables in imperative programming languages: (i) discourse markers whose interpretation is fixed in a larger context (variables in read memory); (ii) discourse markers which get introduced directly in K (variables in write memory) and (iii) discourse markers which are introduced in subordinate contexts (variable in scratch memory). Let var be a function such that var(Pt1...tn) = {ti : 1≤i≤n, ti∈V} var(v=t) = {v,t} if t∈V, else var(v=t) = {v}. Define three functions fix: pDRSs → 2V, intro: pDRSs → 2V and cbnd: pDRSs → 2V such that: fix(v) = ∅ intro(v) = {v} cbnd(v) = ∅ fix(T) = ∅ intro(T) = ∅ cbnd(T) = ∅ fix(Pt1...tn) = var(Pt1...tn) intro(Pt1...tn) = ∅ cbnd(Pt1...tn) = ∅ fix(t=v) = var(t=v) intro(t=v) = ∅ cbnd(t=v) = ∅ fix(¬K) = fix(K) intro(¬K) = ∅ cbnd(¬K) = intro(K) ∪ cbnd(K) fix(K1;K2) = fix(K1) ∪ (fix(K2) intro(K1)) intro(K1;K2) = intro(K1) ∪ intro(K2) cbnd(K1;K2) = cbnd(K1) ∪ cbnd(K2) Given a pDRS K = (K1;K2) the discourse markers which K1 can share with K2 are those determined by fix(K2), that is the discourse markers which occur in the conditions of K2 without having been introduced in the universe of K2. However, due to the way they get their values by means of assignments in DV, such an implementation of the sharing of discourse markers could be problematic. Consider, for instance, the pDRS K = (K1;K2), where K1 = [x; y; man(x); woman(y)] K2 = [y; loves(x,y)]). Given the definition of ||K1;K2||M, the occurrences of the marker y in K1 and K2 get different values, which makes impossible to interpret the occurrence of y in K1 as anaphorically related to the occurrence of y in K2. In order to account for anaphoric links across the composition of pDRSs, it must be ensured that the composition will not modify the anaphoric possibilities allowed by the component pDRSs. In the theory of [vEK97] this has been achieved by requiring that the composition K1;K2 satisfies the condition (fix(K1) ∪ intro(K1)) ∩ intro(K2) = ∅. From This condition it follows that, for each pDRS K satisfying it, the discourse markers introduced in K are new, that is intro(K) ∩ fix(K) = ∅ (pDRSs which satisfy this condition are called proper pDRSs, or simply DRSs). In the standard DRS construction algorithm (as described in [KaR93]) the condition is always satisfied, since each time a discourse marker is introduced in the universe of a DRS it is required to be a new marker. In order to satisfy this condition in the compositional DRT as well, in [vEK97] a merge operator • has been introduced which operates on unreduced DRSs (RDRSs), that is pDRSs in which the introduction of discourse markers is completely unconstrained. Given two RDRSs R1 and R2 the result of the composition R1 • R2 is the merging of R2 in R1, given a non deterministic renaming of the markers introduced in R2 such that intro(R2) ∩ fix(R2) = ∅. If the semantic value of the RDRS R = is given by the triple , where X = fix(R), Y = intro(R) and F is a set of assignments which satisfy all the conditions in C, the semantics of non deterministic merge is the following: 2. ||R1 • R2|| = where θ is an appropriate renaming of (some of) the markers in intro(R2). By means of a set of equations which determine the appropriate renaming of the discourse markers, reducible RDRSs can then be reduced to proper DRSs, that is DRSs in which only the sequencing operator ; occurs. The following example illustrates how the non deterministic merge-reduction works: [x; man(x); enter(x)] • [x; woman(x); smile(x)] ⇓ reduction [x; man(x); enter(x)] ; (x/y) [y; woman(y); smile(y)] ⇓ [x; y; man(x); enter(x); woman(y); smile(y)] The theory of non deterministic merge-reduction of RDRS sets up the basis for a compositional approach to the representation and the model theoretic interpretation of texts. Given a text T = S1, ...,Sn and a set of RDRS {R1, ..., Rn}, such that Ri represents the content of Si, the compositional interpretation of T can be determined in terms of the model-theoretic interpretation of the sequence of merging R1•R2•...•Rn1•Rn. In the frame of DRT, a compositional theory of text interpretation should (i) allow for the model theoretic interpretation of the sharing of discourse markers across different DRSs and (ii) avoid the "clash of discourse markers" which would make some anaphoric links impossible to represent. The compositional theory described in [vEK97] satisfies both these requirements. Each DRS Ki can be considered as providing a set of "anaphoric possibilities" determined by the sets of markers fix(Ki) and fix(Ki)∪intro(Ki); more specifically, through fix(Ki) the DRS Ki can be anchored to the representation of the preceding context, whereas fix(Ki)∪intro(Ki) represents the set of the anaphoric possibilities Ki allows for the interpretation of the subsequent discourse. The basic view of representation structures which the theory of non deterministic merge-reduction is based on can thus be summarized diagrammatically as follows: anaphoric plugs with fixed names fix(R ) 1 anaphoric sockets R 1 anaphoric plugs with fixed names fix(K ) 1 anaphoric sockets with fixed names