Generalising Discontinuity

This paper makes two generalisations of categorial calculus of discontinuity. In the rst we introduce unary modalities which mediate between continuous and discontinuous strings. In the second each of the modes of adjunction of the proposal to date, concatenation, juxtaposition and interpolation, are augmented with variants. Linguistic illustration and motivation is provided, and we show how adherence to a discipline of sorting renders the generalisations tractable within a particularly e cient logic programming paradigm. 1 Generalising Discontinuity The present work continues in the line of others seeking to develop categorial type calculus of discontinuity and associated automated theorem proving/parsing (Moortgat 1988, 1990, 1991; Solias 1992; Morrill and Solias 1993; Morrill 1994, ch. 4, 1995a, 1995b, 1995c; Llor e and Morrill 1995, Calcagno 1995). In particular, it generalises the sorted discontinuity calculus outlined in the appendix of Morrill (1995b) and implemented in a linear clausal fragment by compilation as described in Morrill 1995c; familiarity with these two works is assumed in what follows. We begin by summarising the point of departure for the present proposals. We then introduce two generalisations: unary \split" and \bridge" operators mediating between strings and split strings, and binary operators for staggered concatenation, and juxtaposition and interpolation adjunctions which inherit split points from their operands. We go on to show how these proposals t into the linear logic programming paradigm for categorial parsing as deduction. 1 Sorted Discontinuity Calculus The associative Lambek calculus (Lambek 1958) provides a logic of concatenation. Its types are speci cations of concatenative comportment and by classifying words with respect to types, properties of strings are de ned which are the deductive consequences. The non-associative Lambek calculus (Lambek 1961) is similarly a logic of juxtaposition, by which we mean putting side-byside in a way which imposes grouping (concatenation, being associative, forgets grouping). But the existence of discontinuous phenomena in natural grammar guarantees that such logic of itself cannot be adequate. In discontinuity calculus, as presented for example in Morrill (1994, ch. 4, 1995b), it is sought to combine and extend logic of concatenation and juxtaposition with logic of interpolation. In one, unsorted, approach concatenation, juxtaposition and interpolation are each assumed to be total functions in a single abstract total algebra and the categorial types are formed from unsorted type-constructors without restriction. The sorted discontinuity calculus is brie y introduced in the appendix of (Morrill 1995b). It is distinguished from the unsorted version in that instead of assuming all adjunctions to be total functions in an unsorted algebra, two sorts of object (string and split string) are assumed so that the adjunctions are sorted operations in a sorted algebra, and the categorial types come in a restricted form according to the sorted type-constructors. This formulation has particularly good computational properties; while the unsorted version has a logic programming implementation depending on matching under associativity and partial commutativity (Morrill 1995a), the sorted version has one depending on just uni cation of unstructured terms (i.e. constants and variables; Morrill 1995c). The sorted discontinuity calculus is as follows. Let us assume a monoid hL;+; i comprising the set of strings over some vocabulary, with + the associative operation of concatenation (so that s1+(s2+s3) = (s1+s2)+s3), and with the empty string (so that s+ = +s = s). The concatenation adjunction + has functionality L;L ! L. We de ne a juxtaposition adjunction (:; :) which is Cartesian product formation over L, of functionality L;L! L2; (s1; s2) =df hs1; s2i. And we further de ne an interpolation adjunction W of functionality L2; L! L; hs1; s2iWs =df s1+s+s2. Because these operations are sorted, the categorial types and type-constructors de ned with respect to them are correspondingly sorted. We refer to sort L as sort string, and sort L2 as sort split string. The family of concatenation connectives f=; n; g are de ned by \residuation" with respect to the concatenation adjunction +, which is of functionality L;L ! L. The existential conjunction (product) A B (A product B) is the setwise sum of the concatenation adjunction over A and B; AnB (A under B) and B=A (B over A) are the universal directional implications (divisions). D(AnB) = fsj 8s0 2 D(A); s0+s 2 D(B)g D(B=A) = fsj 8s0 2 D(A); s+s0 2 D(B)g D(A B) = fsj 9s1; s2; s = s1+s2 & s1 2 D(A) & s2 2 D(B)g (1) 2 Each of these type-constructors requires its operands to be of sort string and produces a composite type of sort string. The family of juxtaposition connectives f<;>; g are de ned by residuation with respect to the juxtaposition adjunction (:; :), which is of functionality L;L ! L2. The product A B is the setwise sum of the juxtaposition adjunction over A and B; A>B (B to A) and BB) = fsj 8s0 2 D(A); hs0; si 2 D(B)g D(BG j G#F j G F G ::= F F j F"F (4) Each formula A of sort string has an interpretation D(A) L and each formula A of sort split string has an interpretation D(A) L2. The system mixing the three families of connectives is sublinear in the space of logics arising from removing standard structural rules. Thus while linear logic (Girard 1987) results from removal of freely applying contraction and weakening, but not exchange, the present system lacks free commutativity also. This means that all theorems must be valid when reading divisions and products as the linear (multiplicative) implication and conjunction. Linear validity is a necessary condition for validity, though of course it is not su cient because further sublinear structural conditions must be respected. We present a tree-style natural deduction proof format in which linear logical resource-consciousness is re ected by closure of a unique assumption in conditionalisation. The sublinear conditions are expressed in labels (Gabbay 1991) re ecting the interpretation. We use boldface romans as constants naming elements of L, and we use ; ; ; : : : as variables over L term labels. A labelled formula of sort string has the form : A and a labelled formula of sort split string is of the form ( ; ): A. The labelled natural deduction rules can be seen as a restatement left-to-right and right-to-left of the bidirectional interpretation clauses rotated ninety degrees clockwise for the elimination (E) rules and anticlockwise for the introduction (I) rules,1 with metavariables or Skolem constants according to quanti ers and the polarity of their context. We give only introduction rules for existentials since the elimination rules are both problematic, and apparently unmotivated linguistically. (5) ... ... : A : AnBnE + : B a: An ... a+ : BnIn : AnB 1Cf. Ranta (1994), who attributes the observation to Martin-Lof (1987). 3 (6) ... ... : B=A : A/E + : B a: An ... +a: B=In : B=A (7) ... ... : A : B I + : A B (8) ... ... : A : A>B>E ( ; ): B a: An ... (a; ): B>In : A>B (9) ... ... : BG j G#F j G F j ^G G ::= F F j F"F j $F (22) Interpretation is now made with respect to a connection set J which is a subset of L that contains . Intuitively, connections allow for hypothetical reasoning over discontinuous strings as if they were continuous, by supposing that they are connected; the connection set includes because the empty string always connects adjacent strings. The interpretations of the earlier binary operators are independent of the connection set and are as given before. However, ^ and $ behave as an existential and a universal respectively with respect to the connection set. Signs in $A are split strings which, joined by any connection, give a string in A; signs in ^A are the results of joining 6 by some connection some split string in A.2 D($A) = fhs1; s2ij 8s 2 J; s1+s+s2 2 D(A)g D(^A) = fsj 9s1; s2; s0; s = s1+s0+s2 & s0 2 J & hs1; s2i 2 D(A)g (23) Labelled natural deduction rules are read o the interpretation clauses as before, and again the problematic existential elimination is excluded. Connections are represented j( ); because is a connection in all models, j( ) may be assumed freely, but other connections assumed will have to be conditionalised in logical deductions. (24) ... ... ( 1; 2): $A j( )$E 1+ + 2: A j(b)n ... 1+b+ 2: A$In ( 1; 2): $A (25) ... ... ( 1; 2): A j( )^I 1+ + 2: ^A The bridge and split type-constructors form a conjugate pair of respectively existential and universal modal operators (see Moortgat 1995) and thus satisfy laws such as A ) $^A: (26) ( 1; 2): A 1 j(b)^I 1+b+ 2: ^A$I1 ( 1; 2): $^A By way of linguistic illustration, we observe that assignment of a type R/^(S"N) to a relative pronoun allows medial relativisation such as `that Mary sent to John' to be generated as a continuous string: (27) that: R/^(S"N) Mary: N sent: ((NnS)/PP)/N to+John: PP j( ) 1 b: N/E sent+b: (NnS)/PP /E sent+b+to+John: NnSnE Mary+sent+b+to+John: S "I1 (Mary+sent, to+John): S"N ^I Mary+sent+to+John: ^(S"N)/E that+Mary+sent+to+John: R The subsequent section introduces the second generalisation, relating to binary operators. 2The operators are identity mappings with respect to the semantic dimension of signs. Note that as a logical basic type J would obey A ) A J (and A ) J A), but not A J ) A (or J A ) A), whereas a true product unit I with D(I) = f g would obey all four such laws. With an I-type connection set the interpretations become: D($A) = fhs1; s2ij s1+s2 2 D(A)g D(^A) = fsj 9s1 ; s2; s = s1+s2 & hs1; s2i 2 D(A)g We emphasise the J-version because the commutativity of the neutral element (s+ = +s) would undermine certain intended applications. Observe that so far as prosodic dimensions of signs are concerned we could de ne $A as A"J and ^A as A J , but semantically it is unclear how to give meanings to unit types, or a meaning in A"J which, applied to that of J returns itself as value. For these reasons we propose unary modalities and not unit types. 7 3 Generalised Sorted Discontinuity Calculus The concatenation, juxtaposition, and interpolation adjunctions of the discontinuity calculus can be illustrated as follows: + = ( , ) = W = (28) These are natural operations in the realm of strings and split strings, but others are imaginable, and linguistically motivated. In particular we consider here a generalisation of the discontinuity calculus in which each initial adjunction is augmented with variants (though these do not exhaust the conceivable options). Juxtaposition and interpolation have left and right variants according to positions of split points; concatenation has a single staggered variant because of the absence of a split point. Discont. Calculus Generalised Discont. Calculus +: concatenation +2: staggered concatenation (., .): juxtaposition (.,l .): juxtaposition inheriting split on the left (.,r .): juxtaposition inheriting split on the right W : interpolation Wl: interpolation at the left interior Wr: interpolation at the right interior (29) The staggered concatenation adjunction is of functionality L2; L2 ! L; left and right inheriting juxtaposition are of functionality L2; L ! L2 and L;L2 ! L2 respectively; the left and right interior interpolations are of functionality L2; L! L2. The variants perform the same basic role as their mother operations: concatenation outputs strings, juxtaposition outputs split strings, preserving operand order; interpolation performs a wrapping of the rst operand around the second. The de nitions of the new adjunctions are as follows. h ; i+2h ; i =df + + + (h ; i;l ) =df h ; + i ( ;r h ; i) =df h + ; ) h ; iWl =df h + ; ) h ; iWr =df h ; + ) (30) More graphically, we have (31). +2 = ( ,l ) = ( ,r ) = ( Wl ) = ( Wr ) = (31) The community of discontinuity connectives becomes generalised to (32) where m 2 fl; r; 0g, n 2 f2; 0g, and the zero variants, those we already had, will continue to be written without explicit subscript. +n nn under (staggered) /n over (staggered) n (.,m .) >m to (left, right) lX)/^X where X is S"TV and TV is (NnS)/N, with semantics x y z[(y z) ^ (x z)]. (35)(John, logic): S"TV and Charles: N phn: N j( ) 1 a: TV Charles+a+phn: S "I1 (Charles;phn): S"TV ^I Charles+phn: ^(S"TV)/E and+Charles+phn: X>lX>rE (John; logic+and+Charles+phn): S"TV Starting at the top right hand corner, `Charles a phonetics' is derived straightforwardly as a sentence from the hypothetical transitive verb a. The hypothetical can be withdrawn to yield a split form which wants to wrap around a transitive verb to form a sentence. This is mapped by ^I which fuses the right hand conjunct to a string of the right type for the coordinator to consume by over elimination, which pre xes the coordinator. The left hand conjunct `John logic' is also derivable as S"TV, in just the same way as `Charles phonetics'; when the coordinator combines with this conjunct, by to left elimination, the split marking of this conjunct is inherited by the result, again in type S"TV. So this will wrap around the transitive verb interpolating it in the rst conjunct, and distributing its semantics over the conjuncts. The semantics is spelled out in (36). 3The problem is that the new interaction principle [MA], p.113 requires 2 to be of split string sort qua the split operand of wrap (her notation is swapped relative to ours); but then since the new g-mode gets a string sort left operand in the top line, it cannot take rst operand 2, of split string sort, in the second line. 9 (36) x((x logic) j) x y z[(y z) ^ (x z)] c x phn 1 ((x phn) c) "I1 x((x phn) c) ^I x((x phn) c) /E y z[(y z) ^ ((z phn) c)]>rE z[((z logic) j) ^ ((z phn) c)] The last step illustrates inference with a inheriting juxtaposition. Our next example will illustrate staggered concatenation. 3.1.2 Comparative Subdeletion We make the second illustration of generalised discontinuity with reference to comparative subdeletion. Again the treatment is inspired by Hendriks (1995), but it uses the present sorted calculus, and the analysis assumes that `more : : : than' in examples such as the following has a unitary meaning. a. More sheep ran than sh swam. b. John ate more bagels than Mary ate donuts. (37) Our analytical perspective is that `more : : : than' combines with two sentences each lacking one quanti er; `more' occupies the determiner gap in the rst, and the two sentences are conjoined with `than'. Semantically there is a comparison, in the case of (37b) for example, between the cardinality of the set of bagels that John ate, and the cardinality of the set of donuts that Mary ate. The construction is triggered by the following lexical assignment, where Q abbreviates the quanti er type ((S"N)#S)/CN. (more, than) { x y[ z(x p q[(p z) ^ (q z)]) > z(y p q[(p z) ^ (q z)])] := (S"Q)n2(S/^(S"Q)) (38) Then there is the following derivation of (37b), where TV again abbreviates (NnS)/N. John: N ate: TV bagels: CN 1 a: Q John+ate+a+bagels: S "I1 (John+ate, bagels): S"Q more thann2E John+ate+more+bagels+than: S/^(S"Q) (Mary+ate, donuts): S"Q j( )^I Mary+ate+donuts: ^(S"Q)/E John+ate+more+bagels+than+Mary+ate+donuts: S (39) Observe in particular the staggered concatenation inference step n2E with combines (John+ate, bagels) with (more, than) to yield John+ate+more +bagels+than. The semantics of (39) is as follows. (40) j ate bagel 1 w ((w bagel) u((ate u) j)) "I1 w((w bagel) u((ate u) j)) more than n2 y[ z[(bagel z) ^ ((ate z) j)] > z(y p q[(p z) ^ (q z)])] w((w donut) u((ate u) m)) ^I w((w donut) u((ate u) m))/E [ z[(bagel z) ^ ((ate z) j)] > z[(donut z) ^ ((ate z) m)] The relevant comparison of cardinalities is indeed made. Rather than continue here with linguistic illustration of interior edge interpolation we pass on directly to consider computational aspects. 10 4 Computation The current section shows how the generalised discontinuity proposals t into the paradigm for logic programming of categorial deduction developed in Morrill (1995a, 1995c) and Llor e and Morrill (1995). The general proposal is to compile categorial assignments into clauses of linear logic. The compilation is performed systematically, according to the interpretations of categorial type-constructors; the target formalism of a linear logic programming fragment (Hodas and Miller 1994) is suitable because it is the most speci c level of propositional logic embracing all the sublinear categorial calculi with their discontinuity, partial commutativity, and so forth, and because in structuring resources as bags rather than lists, we eliminate the need to conjecture partition points of ordered sequents, a source of ine cient don't know non-determinism indigenous to Lambek sequent deduction. It is possible to work just with algebraic interpretation, as in Morrill (1995a), but in Morrill (1995c) and Llor e and Morrill (1995) it is observed that by exploiting the binary relational models (van Benthem 1991) of associative Lambek calculus one can avoid computation of matching under associativity, and instead propagate constraints under associativity by methods analogous to the use of string positions/di erence lists in the logic programming of DCGs. Both the original sorted discontinuity calculus and its generalisation here can be interpreted and implemented according to just binary relational models. However, because linguistically we wish to be able to represent not only precedence relations, but also dominance relations (e.g. using bracket operators; Morrill 1992, 1994 ch. 7) the algebraic dimension is needed to induce hierarchical structure that cannot be captured in binary relations. For this reason, we deal here with the more general problem of interpretation and computation according to combined algebraic and relational models. In this general setting matching under associativity is not altogether avoided. However we combine algebraic and relational style models into multidimensional hybrid models which allow us to exploit constraint propagation and adopt a lazy approach to computation of matching under associativity, by only attempting to check the algebraic conditions once satisfaction of the binary relational conditions have been con rmed. 4.1 Hybrid Models We begin by reviewing the hybrid models for the sorted version of the original discontinuity calculus. Interpretation takes place relative to a monoid hL;+; i and a set V . Each formula A of sort string has an interpretation D(A) V 2 L and each formula A of sort split string has an interpretation D(A) V 4 L2. The family of connectives f=; n; g are de ned by residuation with respect to a concatenation adjunction of functionality V 2 L; V 2 L ! V 2 L. The adjunction is a partial operation, de ned on hv1; v2; s1i and hv3; v4; s2i (respectively) just in case v2 = v3, in which case its value is hv1; v4; s1+s2i. D(AnB) = fhv2; v3; sij 8hv1; v2; s0i 2 D(A); hv1; v3; s0+si 2 D(B)g D(B=A) = fhv1; v2; sij 8hv2; v3; s0i 2 D(A); hv1; v3; s+s0i 2 D(B)g D(A B) = fhv1; v3; sij 9v2; s1; s2; s = s1+s2 & hv1; v2; s1i 2 D(A) & hv2; v3; s2i 2 D(B)g (41) The family of connectives f<;>; g are de ned by residuation with respect to a juxtaposition adjunction of functionality V 2 L; V 2 L! V 4 L2. It is de ned as Cartesian product formation: 11 applied to hv1; v2; s1i and hv3; v4; s2i (respectively) its value is hv1; v2; v3; v4; s1; s2i. D(A>B) = fhv3; v4; sij 8hv1; v2; s0i 2 D(A); hv1; v2; v3; v4; s0; si 2 D(B)g D(BB>E (I { J;K { L) { ( ; ): B I { J { a: An ... (I { J;K { L) { (a; ): B>In K { L { : A>B 12 (48) ... ... I { J { : BlX>rE (0 { 1; 2 { 6) { (John; logic+and+Charles+phn): S"TV (67)The next section shows how the hybrid models we have introduced and illustrated support a clausal compilation and automated proof search. 4.2 Linear clausal fragment Logic programming is a paradigm of computation as proof search. We represent a program as a list of program clauses and the task of deciding whether a query A follows is the task of determining whether the sequent ) A is derivable (output takes the form of computing values for the variables in A such that ) A). We assume, following the works referenced above, the linear clausal fragment (68). PCLS ::= AT OM AGENDA AGENDA ::= 1 j GOAL AGENDA GOAL ::= AT OM j (AGENDA PCLS) (68) Thus a program clause PCLS comprises an atomic head which is the postcondition of a linear implication, and a body which is a (possibly empty) list of goals which is the precondition of the implication. Goals may be atoms or may themselves be implications from a program clause to a list of goals. Such implicational goals are the only extension to ordinary Horn clauses exploited here. The propositional linear logic programming rules are as follows. The termination condition requires all the program clauses to have been consumed so that the empty agenda only follows from the empty program database: 15 (69) ) 1 The resolution rule uses up and therefore removes from the program database the program clause against which resolution is performed: (70) ) B1 : : : Bn C RES A B1 : : : Bn 1; ) A C The deduction theorem rule distributes the program clauses between its two premises: (71) A; ) B ) CDT ; ) (B A) C The rule requires us to show that B A follows from some of the premises, , while the remainder of the premises yield the remainder C of the agenda. By the deduction theorem, yields A B if and only if A; ) B. In practice a lazy approach to the partitioning can be adopted whereby the whole bag of conclusion premises, plus A, are made available to try to prove B and, after checking that A has been consumed, those not used are supplied to try to prove C (Llor e and Morrill 1995). Compilation takes place according to an immediate correspondence with interpretation. In the case of the associative implications, for example, we have the following, which is just a restatement of the hybrid interpretation clauses. (72) a. 8I; (I { K { + : B I { J { : A) J { K { : AnB b. 8K; (I { K { + : B J { K { : A) I { J { : B=A Categorial type assignments are translated into quanti er-free linear clauses by polar translation functions; the polarity is used to indicate whether new symbols introduced for quanti ed variables in the interpretation clauses are metavariables or Skolem constants. They are identity functions on atomic assignments; on complex category predicates they are de ned mutually as follows (for related unfolding see Roorda 1991, Moortgat 1992, Hendriks 1993 and Oehrle 1994); p indicates the polarity complementary to p: (73) a. I { K { + : Bp I { J { : ApI; new variable/constant as p += J { K { : AnBp b. I { K { + : Bp J { K { : ApI; new variable/constant as p += I { J { : B=Ap The program clauses and agenda are read directly o the unfoldings, with the only manipulation being a attening of positive implications into uncurried form: ((X+ Y 1 ) : : :) Y n > X+ Y 1 : : : Y n (74) We shall allow an agenda X1 : : : Xn 1 to be written X1 : : : Xn and we shall also allow unit program clauses X 1 to be abbreviated X. The unfolding for the remaining connectives of the basic discontinuity calculus is as follows. Unfolding is provided for negative (succedent occurrences) of products, but not for positive (antecedent occurrences), the compilations of which would fall outside of our linear logic programming 16 fragment. I { J { : A J { K { : B J; ; new variables; = + I { K { : A B (75) (I { J; K { L) { ( ; ): Bp I { J { : ApI; J; new variables/constants as p += K { L { : A>Bp (76) (I { J;K { L) { ( ; ): Bp K { L { : ApK;L; new variables/constants as p += I { J { : BThe top goal on the subagenda is resolved against the prepositional ditransitive clause. The stringposition uni cation is performed; the monoid term uni cation postponed.fg j(Q;Q; )(1) 0 { I { that+a1+f+a2: R (1 { I { a1+b+a2: S J { K { b: N) j(J;K; f)() 1 { 2 { Mary: N(3) P { N { e+sent+c+d: S P { 2 { e: N M { N { d: PP 3 { M { c: N() 3 { 4 { to+John: PP1. 0 { 4 { g: R4=I; g = that+a1+f+a22. (1 { 4 { a1+b+a2: S J { K { b: N) j(J;K; f)()J { K { b: N3.1 { 4 { a1+b+a2: S1=P; 4=N; a1+b+a2 = e+sent+c+d4.1 { 2 { e: N M { 4 { d: PP 3 { M { c: N(86)The current goal is resolved against the subject unit program clause.fg j(Q;Q; )(1) 0 { I { that+a1+f+a2: R (1 { I { a1+b+a2: S J { K { b: N) j(J;K; f)(4) 1 { 2 { Mary: N(3) P { N { e+sent+c+d: S P { 2 { e: N M { N { d: PP 3 { M { c: N() 3 { 4 { to+John: PP1. 0 { 4 { g: R4=I; g = that+a1+f+a22. (1 { 4 { a1+b+a2: S J { K { b: N) j(J;K; f)()J { K { b: N3.1 { 4 { a1+b+a2: S1=P; 4=N; a1+b+a2 = e+sent+c+d4.1 { 2 { e: N M { 4 { d: PP 3 { M { c: Ne =Mary5.M { 4 { d: PP 3 { M { c: N(87)The next goal on the agenda is resolved against the prepositional phrase unit program clause.fg j(Q;Q; )(1) 0 { I { that+a1+f+a2: R (1 { I { a1+b+a2: S J { K { b: N) j(J;K; f)(4) 1 { 2 { Mary: N(3) P { N { e+sent+c+d: S P { 2 { e: N M { N { d: PP 3 { M { c: N(5) 3 { 4 { to+John: PP1. 0 { 4 { g: R4=I; g = that+a1+f+a22. (1 { 4 { a1+b+a2: S J { K { b: N) j(J;K; f)()J { K { b: N3.1 { 4 { a1+b+a2: S1=P; 4=N; a1+b+a2 = e+sent+c+d4.1 { 2 { e: N M { 4 { d: PP 3 { M { c: Ne =Mary5.M { 4 { d: PP 3 { M { c: N3=M; d = to+John6.3 { 3 { c: N(88)The current goal is now resolved against the subproof hypothesis. The subproof postcondition hasbeen shown, and the subproof precondition has been used, so we exit and pursue the remainder 19of the agenda after the implicational top goal.fg j(Q;Q; )(1) 0 { I { that+a1+f+a2: R (1 { I { a1+b+a2: S J { K { b: N) j(J;K; f)(4) 1 { 2 { Mary: N(3) P { N { e+sent+c+d: S P { 2 { e: N M { N { d: PP 3 { M { c: N(5) 3 { 4 { to+John: PP1. 0 { 4 { g: R4=I; g = that+a1+f+a22. (1 { 4 { a1+b+a2: S J { K { b: N) j(J;K; f)(6) J { K { b: N3.1 { 4 { a1+b+a2: S1=P; 4=N; a1+b+a2 = e+sent+c+d4.1 { 2 { e: N M { 4 { d: PP 3 { M { c: Ne =Mary5.M { 4 { d: PP 3 { M { c: N3=M; d = to+John6.3 { 3 { c: N3=J; 3=K; c= b7. j(3; 3; f)(89)This last goal is satis ed by resolution against the universal connection axiom.f7g j(Q;Q; )(1) 0 { I { that+a1+f+a2: R (1 { I { a1+b+a2: S J { K { b: N) j(J;K; f)(4) 1 { 2 { Mary: N(3) P { N { e+sent+c+d: S P { 2 { e: N M { N { d: PP 3 { M { c: N(5) 3 { 4 { to+John: PP1. 0 { 4 { g: R4=I; g = that+a1+f+a22. (1 { 4 { a1+b+a2: S J { K { b: N) j(J;K; f)(6)J { K { b: N3.1 { 4 { a1+b+a2: S j(J;K; f)1=P; 4=N; a1+b+a2 = e+sent+c+d4.1 { 2 { e: N M { 4 { d: PP 3 { M { c: Ne =Mary5.M { 4 { d: PP 3 { M { c: N3=M; d = to+John6.3 { 3 { c: N3=J; 3=K; c = b7. j(3; 3; f)f =(90)Only at this point are the monoid matchings chased back:7. f =6. c = b5. d = to+John4. e =Mary3. a1 =Mary+sent; a2 = to+John1. g = that+Mary+sent+to+John(91)That is, the parsing as deduction process requires just uni cation of unstructured terms (theconstants and variables of the binary relational labelling), and algebraic matching (one-way uni-cation) under associativity only after the conditions on word order have been veri ed. (In theabsence of hierarchical structure the algebraic matching succeeds if and only if the relational la-belling does, so algebraic labelling is only needed when some non-associativity is present.) Theseproperties are common to the basic discontinuity calculus, and the two generalisations of it madehere.Referencesvan Benthem, J.: 1991, Language in Action: Categories, Lambdas and Dynamic Logic, Studies inLogic and the Foundations of Mathematics Volume 130, North-Holland, Amsterdam. 20Calcagno, M.: 1995, Bulletin of the Interest Group in Propositional and Predicate Logic Vol. 3No. 4, pp. 555{578.Dalrymple, M., S.M. Sheiber and F.C.N. Pereira: 1991, `Ellipsis and higher-order uni cation',Linguistics and Philosophy 18: 399{452.Gabbay, D.: 1991, Labelled Deductive Systems, to appear, Oxford University Press, Oxford.Girard, J-Y.: 1987, `Linear Logic', Theoretical Computer Science 50, 1{102.Hendriks, H.: 1993, Studied Flexibility: Categories, and Types in Syntax and Semantics, Ph.D.dissertation, Institute for Logic, Language and Computation, Universiteit van Amsterdam.Hendriks, P.: 1995, `Ellipsis and multimodal categorial type logic' in G. Morrill and R.T. Oehrle(eds.) Formal Grammar, Proceedings of the Conference of the European Summer School inLogic, Language and Information, Barcelona, 107{122.Hodas, J. and D. Miller: 1994, `Logic Programming in a Fragment of Intuitionistic Linear Logic',Journal of Information and Computation 110(2), 327{365.Lambek, J.: 1958, `The mathematics of sentence structure', American Mathematical Monthly 65,154{170, also in Buszkowski, W., W. Marciszewski, and J. van Benthem (eds.): 1988, Catego-rial Grammar, Linguistic & Literary Studies in Eastern Europe Volume 25, John Benjamins,Amsterdam, 153{172.Lambek, J.: 1961, `On the calculus of syntactic types', in R. Jakobson (ed.) Structure of languageand its mathematical aspects, Proceedings of the Symposia in Applied Mathematics XII,American Mathematical Society, 166{178.Llor e, F.X. and G. Morrill: 1995, `Di erence Lists and Di erence Bags for Logic Programmingof Categorial Deduction', in Proceedings of SEPLN XI, Deusto. Also available as Reportde Recerca LSI-95-30-R, Departament de Llenguatges i Sistemes Inform atics, UniversitatPolitecnica de Catalunya.Martin-Lof, P.: 1987, `Truth of a proposition, evidence of a judgement, validity of a proof, Synthese73: 407{420.Moortgat, M.: 1988, Categorial Investigations: Logical and Linguistic Aspects of the LambekCalculus, Foris, Dordrecht.Moortgat, M.: 1990, `The Quanti cation Calculus: Questions of Axiomatisation', in DeliverableR1.2.A of DYANA Dynamic Interpretation of Natural Language, ESPRIT Basic ResearchAction BR3175.Moortgat, M.: 1991, `Generalised Quanti cation and Discontinuous type constructors', to appearin Sijtsma and Van Horck (eds.) Proceedings Tilburg Symposium on Discontinuous Con-stituency, Walter de Gruyter, Berlin.Moortgat, M.: 1992, `Labelled Deductive Systems for categorial theorem proving', OTS Work-ing Paper OTS{WP{CL{92{003, Rijksuniversiteit Utrecht, also in Proceedings of the EighthAmsterdam Colloquium, Institute for Language, Logic and Information, Universiteit van Am-sterdam.Moortgat, G.: 1995, `Multimodal Linguistic Inference', Bulletin of the Interest Group in Proposi-tional and Predicate Logic Vol. 3 No. 2, 3, pp. 371{401.Morrill, Glyn: 1992, `Categorial Formalisation of Relativisation: Pied Piping, Islands, and Ex-traction Sites', Report de Recerca LSI{92{23{R, Departament de Llenguatges i SistemesInform atics, Universitat Politecnica de Catalunya.Morrill, G.: 1994, Type Logical Grammar: Categorial Logic of Signs, Kluwer Academic Publishers,Dordrecht.Morrill, G.: 1995a, `Clausal Proofs and Discontinuity', Bulletin of the Interest Group in Proposi-tional and Predicate Logic Vol. 3 No. 2, 3, pp. 403{427.Morrill, G.: 1995b, `Discontinuity in Categorial Grammar', Linguistics and Philosophy 18: 175{219.Morrill, G.: 1995c, `Higher-order Linear Logic Programming of Categorial Deduction', Proceed-ings Meeting of the European Chapter of the Association for Computational Linguistics,Dublin.Morrill, G. and T. Solias: 1993, `Tuples, Discontinuity and Gapping', Proceedings Meeting of theEuropean Chapter of the Association for Computational Linguistics, Utrecht, 287{297. 21Oehrle, R.T.: 1994, `Term-Labeled Categorial Type Systems', Linguistics and Philosophy 17,633{678.Partee, Barbara and Mats Rooth: 1983, `Generalized conjunction and type ambiguity', inR. Bauerle, C. Schwarze and A. von Stechow (eds.) Meaning, Use, and Interpretation ofLanguage, Linguistic Analysis Volume 6, Walter de Gruyter, Berlin, 53{95.Ranta, R.: 1994, Type-Theoretical Grammar, Oxford University Press, Oxford.Rooth, Mats and Barbara H. Partee: 1982, `Conjunction, type ambiguity, and wide scope `or'',in Daniel Flickinger, Marlys Macken, and Nancy Wiegand (eds.) Proceedings of the FirstWest Coast Conference on Formal Linguistics, Stanford Linguistics Department, Stanford,353{362.Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Univer-siteit van Amsterdam.Solias, T.: 1992, Gramaticas Categoriales, Coordinacion Generalizada y Elisi on, Ph.D. disserta-tion, Universidad Autonoma de Madrid.