Clausal Proofs and Discontinuity

We consider the task of theorem proving in Lambek calculi and their generalisation to \multimodal residuation calculi". These form an integral part of categorial logic, a logic of signs stemming from categorial grammar, on the basis of which language processing is essentially theorem proving. The demand of this application is not just for e cient processing of some or other speci c calculus, but for methods that will be generally applicable to categorial logics. It is proposed that multimodal cases be treated by dealing with the highest common factor of all the connectives as linear (propositional) validity. The prosodic (sublinear) aspects are encoded in labels, in e ect the term-structure of quanti ed linear logic. The correctness condition on proof nets (\long trip condition") can be implemented by SLD resolution in linear logic with uni cation on labels/terms limited to one way matching. A suitable uni cation strategy is obtained for calculi of discontinuity by normalisation of the ground goal term followed by recursive decent and redex pattern matching on the head term. 1 Clausal Proofs and Discontinuity1 The associative Lambek calculus (Lambek 1958) and non-associative Lambek calculus (Lambek 1961) were originally proposed as \syntactic calculi" for characterisation of the well-formedness of (respectively) sequential (semigroup structure) and binary hierarchical (groupoid structure) expressions, and were provided with single-conclusioned Gentzen-style sequent presentations which lack the usual structural rules of weakening (or: thinning, or: monotonicity), contraction, and permutation (or: exchange), and which directly provide Cut-free backward-chaining decision procedures for theoremhood. More recently it has become possible to locate the Lambek calculi within a space of \substructural logics" (logics lacking structural rules; Do sen and Schroeder-Heister 1993) of which linear logic (Girard 1987) is a prominent instance. At the same time, Lambek calculi have been extended in their linguistic application to categorial logics (Morrill 1994d), versions of categorial grammar characterising prosodic and semantic dimensions, for which the task of parsing is essentially theorem proving. In particular we can identify as a generalisation \residuation calculi" in which the Lambek connectives (corresponding to linear logic multiplicatives) are de ned in a number of potentially interactive modes (Moortgat and Morrill 1991). In Morrill (1993, 1994d) an improvement of the logic of discontinuity of Moortgat (1988) is developed in this way. Given Cut-elimination, decidability is directly demonstrable from sequent formulations, but in applications to natural language processing our further objective is e ciency. There are two main approaches in existence: sequent proof normalisation, and proof nets. The former, which builds proofs backwards from the goal sequent, even if somehow broadly generalisable, necessarily faces non-determinism with information from subformulas only made available serially according to the construction of formulas. The latter provides a phase of unfolding in which all the parts of a formula are made available in parallel, and then a non-deterministic phase of linking which builds proofs from the axioms, but requires a certain correctness condition. Roorda (1991) expresses this condition by reference to labelling by lambda terms corresponding to proofs under the Curry-Howard correspondence. Roorda (1991) and Moortgat (1990, 1992) do so by reference to labelling by groupoid terms of the algebras in which we interpret by residuation. We aim to improve the latter method, which as it stands presents the task of correctness checking in terms of intractable problems such as semigroup uni cation, i.e. it leaves some more speci c structuring of the task, indicating an e cient strategy, to be desired. Moortgat (1990) presents a scheme for gathering groupoid-labelled unfoldings into de nite clauses directly executable in Prolog, and Moortgat (1992) proposes multimodal generalisation with uni cation under theory. In Morrill (1994a) such a compilation is achieved by a more direct structuring of unfolding relating to Horn clause resolution in linear logic, showing how one term in such uni cation can always be kept ground, and multimodality is exempli ed with the logic of discontinuity of Morrill (1993, 1994d). This re nement however shares with the Moortgat proposals transformation into rst order clauses, resulting in an in ation of the resolution database at compile time to deal with higher order type inferences. In Morrill (1994b) the situation is improved by compiling into higher order clauses such that hypotheticals are emitted dynamically only as they become germane. The present paper aims to explain and motivate these proposals. 1To appear in theBulletin of the Interest Group in Propositional and Predicate Logics. I thankMichaelMoortgat and Dick Oehrle for comments on this work. 21 Residuation Calculi 1.1 Lambek Calculi The types (or: formulas) of (product-free) Lambek calculus are freely generated from a set of primitives by binary in x connectives / (\over") and n (\under"). Models can be given in a variety of structures; we deal here with a simple and transparent interpretation in groupoids. With respect to a groupoid algebra hL;+i (i.e. a set L closed under a binary operation +) for the non-associative Lambek calculus NL, and with respect to a semigroup algebra hL;+i (i.e. a set L closed under an associative binary operation +) for the associative Lambek calculus L, each formula A is \prosodically" interpreted as a subset D(A) of L by residuation as follows (Lambek 1988).D(AnB) = fsj8s0 2 D(A); s0+s 2 D(B)g D(B=A) = fsj8s0 2 D(A); s+s0 2 D(B)g (1) A sequent, ` A, comprises a succedent formula A and one or more formula occurrences in the antecedent con guration which is organised as a binary bracketed sequence for NL, and as a sequence for L. A sequent is valid if and only if in all interpretations applying the prosodic construction indicated by the antecedent con guration to objects inhabiting its formulas always yields an object inhabiting the succedent formula. The Gentzen-style sequent presentations for NL in (2) and for L in (3) are sound and complete for this interpretation (Buszkowski 1986, Do sen 1992); furthermore they enjoy Cut-elimination: every theorem can be generated without the use of Cut. In the following the parenthetical notation ( ) represents a con guration containing a distinguished subcon guration . (2) a. A ` A id ` A (A) ` BCut ( ) ` B b. ` A (B) ` CnL ([ ; AnB]) ` C [A; ] ` BnR ` AnB c. ` A (B) ` C/L ([B=A; ]) ` C [ ; A] ` B/R ` B=A (3) a. A ` A id ` A (A) ` BCut ( ) ` B b. ` A (B) ` CnL ( ; AnB) ` C A; ` BnR ` AnB c. ` A (B) ` C/L (B=A; ) ` C ; A ` B/R ` B=A By way of example, \lifting" A ` B/(AnB) is generated as follows in NL; it is similarly derivable in L. (4) A ` A B ` BnL [A, AnB] ` B/R A ` B/(AnB) On the other hand \composition" AnB, BnC ` AnC, while derivable as follows in L, is NLunderivable in its non-associative form: [AnB, BnC] ` AnC. 3 (5) A ` A B ` B C ` CnL B, BnC ` CnL A, AnB, BnC ` CnR AnB, BnC ` AnC 1.2 Multimodal Lambek Calculi In a slightly di erent formulation of the sequent calculus for L we may con gure antecedents with binary bracketing, and then use the NL rules together with an explicit structural rule of associativity (the double bar indicates bidirectionality): (6) ([ 1; [ 2; 3]]) ` AA ([[ 1; 2]; 3]) ` A From here it is a small step to give sequent calculus for \multimodal" Lambek calculi in which we have several families of connectives f=i; nigi2f1;:::;ng, each de ned by residuation with respect to their adjunction in a \multigroupoid" hL; f+igi2f1;:::;ngi (Moortgat and Morrill 1991): D(AniB) = fsj8s0 2 D(A); s0+is 2 D(B)g D(B=iA) = fsj8s0 2 D(A); s+is0 2 D(B)g (7) Sequent calculus can be given by indexing the brackets of NL-presentations to indicate mode of adjunction (and adding structural rules, including interaction postulates between di erent modes, as appropriate): (8) id A ` A ` A (A) ` BCut ( ) ` B (9) a. ` A (B) ` Cni L ([i ; AniB]) ` C [iA; ] ` Bni R ` AniB b. ` A (B) ` C/iL ([iB=iA; ]) ` C [i ; A] ` B/iR ` B=iA In particular cases of course we may choose non-composite notations for the connectives and brackets. With two modes interpreted in a \bigroupoid" understood as distinguishing left-headed and right-headed adjunction we have a \headed" calculus (Moortgat and Morrill 1991). With families f=; ng and f<;>g for adjunctions + (associative) and (:; :) (not assumed to be associative) respectively in a bigroupoid hL;+; (:; :)i we have a partially associative calculus L+NL (Oehrle and Zhang 1989, Morrill 1990). This latter forms two-thirds of the discontinuity calculus of Morrill (1993, 1994d) which we shall be considering. 1.3 Labelled Sequent Presentations \Labelling" (Gabbay 1991) is a means of presenting proof theory which will enable us to factor out the antecedent formulas of a sequent, and its associated prosodic construction, which is made more explicit. No essential use of sequent labelling is made here, in that the labelled presentation of calculus is just notational variation of ordered presentation. However, labelling is a step on the path to implementing residuation calculi. We notate a sequent ` A as a1: A1; : : : ; an: An ` : A where the multiset fA1; : : : ; Angm comprises the formula occurrences in , a1; : : : ; an are distinct 4atomic labels, and is a term over these labels representing explicitly the prosodic construction that was represented implicitly by the structured con guration . The labelled sequent calculus for NL is as follows: (10) a. a: A ` a: A id b. ` : A a: A; ` (a): BCut ; ` ( ): B c. ` : A b: B; ` (b): CnL ; d: AnB; ` (( + d)): C d. ; a: A ` (a+ ): BnR ` : AnB e. ` : A b: B; ` (b): C/L ; d: B=A; ` ((d + )): C f. ; a: A ` ( + a): B/R ` : B=A To obtain L an associativity equation on terms may be added: (( 1+ 2)+ 3) = ( 1+( 2+ 3)) (11) Or equivalence classes of terms represented by attening terms into lists. 1.4 Labelled Natural Deduction For labelled Fitch-style categorial derivation (Morrill 1993), there are lexical assignment, subderivation hypothesis, and term label equation rules thus ( { : A represents assignment to type A of the paired prosodic term and semantic term ; we include Curry-Howard semantic annotation intermittently in what follows; full explications are available in the references): a: n: { : A for any lexical entry (12) b: n: a1 { x1: A1 H ... ... n+m: am { xm: An H c: n: { : A 0 { 0: A = n; if = 0 & = 0 The logical rules are: a: n: { : A m: { : AnB ( + ) { ( ): B En n;m (13) b: n: a { x: A H m: (a+ ) { : B unique a as indicated { x : AnB In n;m a: n: { : A m: { : B=A ( + ) { ( ): B E/ n;m (14) 5 b: n: a { x: A H m: ( +a) { : B unique a as indicated { x : B=A I/ n;m A Fitch-style labelled calculus for the associative Lambek calculus L can be obtained from that for the non-associative calculus by adding a prosodic label equation. Alternatively, the associative Lambek calculus can be given by dropping parentheses in prosodic labels. By way of example, a simple instance of relativisation can be derived by hypothetical deduction as follows: 1. which { x y z[(y z) ^ (x z)]: (CNnCN)/(S/N) 2. John { j: N 3. talked { talk: (NnS)/PP 4. about { about: PP/N 5. a { x: N H 6. about+a { (about x): PP 4, 5 E/ 7. talked+about+a { (talk (about x)): NnS 3, 6 E/ 8. John+talked+about+a { ((talk (about x)) j): S 2, 7 En 9. John+talked+about { x((talk (about x)) j): S/N 5, 8 I/ 10. which+John+talked+about { ( x y z[(y z) ^ (x z)] x((talk (about x)) j)): CNnCN 1, 9 E/ 11. which+John+talked+about { y z[(y z) ^ ((talk (about z)) j)]: CNnCN = 10 (15) 1.4.1 Multimodal Fitch natural deduction Multimodal calculi can be presented Fitch-style by giving the same rules for each family of connectives with their associated adjunctions: a: n: { : A m: { : AniB ( +i ) { ( ): B Eni n;m (16) b: n: a { x: A H m: (a+i ) { : B unique a as indicated { x : AniB Ini n;m a: n: { : A m: { : B=iA ( +i ) { ( ): B E/i n;m (17) b: n: a { x: A H m: ( +ia) { : B unique a as indicated { x : B=iA I/i n;m Label equations are to be added according to the algebras of interpretation. 1.4.2 Multimodal labelled Prawitz natural deduction Labelled deduction can also be presented Prawitz-style; for the multimodal case (without semantics) there is the following. (18) ... ... : B=iA : A/iE +i : B ... ... : A : AniBniE +i : B 6(19) ... n a: A +ia: B/iIn : B=iA ... n a: A a+i : BniIn : AniB Each of the two styles of natural deduction have their merits so far as presentation of proofs is concerned: Fitch-style reasons serially while Prawitz-style indicates parallel (unordered) branches of inference; both avoid the sequent calculus reiteration of context formulas and both are practical for linguistic derivations. However, reading from premises to conclusion natural deduction o ers only retrospective justi cation of hypothetical assumptions: looking at the premises only we do not know which hypotheses might turn out to be useful. Sequent calculus is superior so far as proof discovery, as oppposed to presentation, is concerned because it shows which hypotheses are worth trying. 2 Automated Deduction Given Cut-elimination (the property that every theorem has a Cut-free proof; Lambek 1958, 1961 showed this for his calculi) the sequent calculi give decision procedures for determining whether a given sequent is a theorem. Backward-chaining Cut-free labelled sequent proof search admits only a nite number of possible rule applications for a given sequent, eliminating the principle connective of one of the ( nite number of) formulas, and choosing (one of the nite number of) antecedent partitionings in the case of binary rules. This creates subgoals the complexity of which in terms of connective occurrences totals exactly one connective occurrence less. Thus for any sequent there is a nite space of proof search. There are two sources of non-determinism however in (Cut-free) backward-chaining sequent proof search: in choosing on which formula to key rule application, and in choosing how to partition sequents in binary rules. With respect to the former, di erent sequences of choice can converge on the same subproblems; with respect to the latter, considerable space may need to be searched before determining whether a partitioning terminated in initial identity axiom sequents or not. The former, but not the latter, problem is addressed by \proof normalisation" (for the case of Lambek calculus see Hepple 1990, which re nes K onig 1989, see also Hendriks 1993): xing priorities of rule ordering to determine distinguished representatives of equivalence classes of proofs. Both drawbacks are addressed by proof nets (in linear logic), and the matrix methods of Bibel (1981) and Wallen (1990) (Bibel and Eder 1993). In these, formulas are unfolded, and proofs built from the initial sequents. Through unfolding, the parts comprising a formula are made available for examination in parallel, rather than only in serial according to the particular nesting of connectives. By building from initial sequents we ensure e ectively that only rule applications are tried which are already known to terminate successfully in initial sequents. In the context of linear logic then proof nets have been developed as a method of eliminating redundancy in the sequent representation of proofs. For these however a correctness check (the \long trip condition") is required. Corresponding proposals have been made for Lambek calculus by Roorda (1991), in which correctness is checked by semantic labelling, and Roorda (1991) and Moortgat (1990, 1992), in which correctness is checked by prosodic labelling. As such however, the method reduces proof net correctness to checking, by uni cation, satis ability of equations in groupoids, semigroups, and so on. Yet such problems as semigroup uni cation are in general intractable, even though the sequent formulations of the calculi show decidability. Somewhere the method loses control of constraints, and improved management is required in order to achieve e ciency. We shall provide the necessary structure by organising the proof nets used as clauses, in fact (higher order) Horn clauses, of linear logic, for which a resolution strategy is available in which at each uni cation step one term is ground, i.e. variable-free. This prepares the way for computational theorem proving in residuation calculi generally and illustration includes regular 7 and head-oriented discontinuity calculi. 3 Sequent Calculus for Classical Linear Logic The multiplicative fragment of linear logic, with which we shall be concerned, contains binary in x connectives (a conjunction \times") and z (a disjunction \par") and a unary post x negation ? (\neg"). Sequents are of the form ` where con gurations and are sequences of zero or more formulas. There are the following sequent rules, which are sound for classical logic, (and which would also be complete for classical logic if the structural rules of contraction and weakening were included). The calculus enjoys Cut-elimination. (20) a. id A ` A b. ` A; A; 0 ` 0Cut ; 0 ` ; 0 (21) a. ; A;B; 0 ` PL ; B;A; 0 ` b. ` ; A;B; 0PR ` ; B;A; 0 (22) a. ; A;B ` L ; A B ` b. ` A; 0 ` B; 0 R ; 0 ` A B; ; 0 (23) a. A; ` B; 0 ` 0zL AzB; ; 0 ` ; 0 b. ` ; A;B zR ` ; AzB (24) a. ` A; ?L ; A? ` b. ; A ` ?R ` A?; The linear implication is de ned by A B = A? zB, so any f ; z ;?; g formula can be considered an abbreviation for a f ; z ;?g one. 3.1 Negation Normal Form A f ; z ;?g formula is in negation normal form if and only if no connective occurrence falls within the scope of a negation, i.e. negations may only be immediately attached to (unnegated) atoms. We have the following proofs of the involution of negation: (25) a. A ` A ?L A;A? ` ?R A ` A?? b. A ` A ?R ` A?; A?L A?? ` A And there are the following proofs of a de Morgan law: (26) a. A ` A B ` B ?R ` A;A? ?L ` B;B? R ` A B;A?; B? zR ` A B;A?zB? ?L (A B)? ` A?zB? b. A ` A B ` B ?L A?; A ` ?L B?; B ` zL A?zB?; A;B ` L A?zB?; A B ` ?R A?zB? ` (A B)? The other de Morgan law is obtained similarly. Hence using the equivalences (27) to reduce redexes 8of the form on the left to contractums of the form on the right, any formula is converted to a negation normal form. A?? = A (A B)? = A?zB? (AzB)? = A? B? (27) So ? need not really be considered a connective in formulas but as a means of abbreviating f ; z g formulas in which atoms come in two avours: positive and negative. 4 Proof Nets for Linear Logic To validate a sequent using proof nets all formulas are rst put on one side of the sequent turnstyle using the negation rules: ` if and only if ; ? ` if and only if ` ?; (28) They are converted to negation normal form, after which a phase of \unfolding" ensues: (29) a. A B A B b. A B AzB We refer to the result of unfolding until all leaves are atomic as a proof frame. Then an attempt must be made to connect (or: link) all the (positive and negative) atoms in such a way that each is connected with exactly one other, of complementary polarity. Such connections correspond to initial sequent axioms. A connection of all the atoms is called a proof structure. The existence of a proof structure is a necessary, but not su cient, condition for theoremhood (note for example that conjunction and disjunction are not distinguished!): we shall further require the \long trip condition" which e ectively checks partitioning, i.e. that we have connected as initial sequent axioms atomic formula occurrences which were meant to be in the same sequent subproofs, and not ones meant to be in di erent subproofs. By way of example of unfolding and linking, there is the following: A ` (A B) B if and only if A ` (A?zB)?zB if and only if ` A?; (A B?)zB (30) (31) jjjjj A? j A j B? A B? jjj B (A B?)zB The proof structure (31) is also a proof net. The proof structure (32) is a proof net for @= (A, B ` A B is a theorem) but not for @=z (A, B ` AzB is not a theorem). (32) j A? j A j B A@B j B? A restriction to planar proof nets, ones with nested (i.e. non-crossing) connections, characterises (together with the long trip condition) the theorems of cyclic linear logic, i.e. linear logic in which exchange is limited to circular permutations. 9 5 Proof Nets for Lambek Calculus In the previous section proof nets were given by moving all formulas to the right of the sequent turnstyle. Here we shall do the converse: move all formulas to the left, that is we shall perform refutation proofs. From considerations of symmetry we see that the choice is not important, but it will enable us to present our proposal in the familiar context of resolution refutation. We consider the presentation of proof nets for Lambek calculus of Roorda (1991), but our polarities are reversed. Formulas composed from the implicational connectives / and n are signed positive for antecedent occurrences and negative for succedent occurrences, and unfolded as follows. (33) B+ A B=A+ A B+ AnB+ A+ B B=A B A+ AnB The transmission of polarities can be understood when we see an implication as the disjunction of its consequent with the negation of its antecedent. The steps given are compilations of decomposition accordingly, with unfolding, involution of negation, and de Morgan laws for multiplicative conjunction or disjunction compiled in. The ordering given, which swaps the components of negative (i.e. succedent) occurrences of implications allows restriction to planar connections. The following, for example, are proof nets for lifting A ` B/(AnB) and composition AnB, BnC ` AnC in L. (34) jjjjj A+ j A j B+ AnB+ jjj B B/(AnB) (35) jj j j j j jj A B+ B C+ C A+ AnB+ BnC+ AnC Just by considerations of symmetry however we can see that there will also be a proof structure for the invalid \lowering": B/(AnB) ` A. A long trip condition can express the required constraint. Roorda (1991) expresses such a condition in terms of the lambda terms that are notational variants of intuitionistic natural deductions corresponding to Lambek proofs, and which provide the semantic dimension of categorial logic. But which condition on axiom linking corresponds to NL? How would we express the L+NL hybrid? And what about multimodal calculi with interaction postulates? Since these varieties relate more directly to the groupoid prosodic dimension we shall follow Roorda/Moortgat in using prosodic terms to express the correctness conditions. 5.1 Labelled Proof Nets Roorda (1991) and Moortgat (1990, 1992) present unfolding with prosodic labelling as follows (see also Hendriks 1993 and Oehrle 1994). (36) a. +a: B+ a: A : B=A+ a: A a+ : B+ a new variable : AnB+ b. k: A+ +k: B : B=A k+ : B k: A+ k new constant : AnB 10 The succedent (negative) unfoldings introduce (Skolem-like) constants. The antecedent (positive) unfoldings introduce variables. Linking identi es the labels of linked atoms and the correctness condition is that a proof structure is a proof net if and only if the set of equations induced by linking is satis able. This can be checked by uni cation. Consider lifting, for which we assume labelling thus by a constant 1: 1: A ` 1: B/(AnB) (37) Then there is the proof net (38). (38) jjjjj 1: A+ j a: A j a+2: B+ 2: AnB+ jjj 1+2: B 1: B/(AnB) The linking yields the equations 1=a and a+2= 1+2 which are clearly satis ed. For composition as in (39) we obtain (40). 1: AnB, 2: BnC ` 1+2: AnC (39) (40) jj j j j j jj a: A a+1: B+ b: B b+2: C+ 3+(1+2): C 3: A+ 1: AnB+ 2: BnC+ 1+2: AnC This yields the equations a=3, a+1=b and b+2 = 3+(1+2) which are satis ed with b=3+1 in the associative case, but not in the non-associative one. For the invalid lowering however we have: (41) jjjjj 1: A j 2: A+ j 2+c: B c: AnB jjj 1+c: B+ 1: B/(AnB)+ The equations 1=2 and 1+c=2+c are clearly not satis able since they require the identi cation of two distinct constants. The method is attractive because we can see how it adapts to di erent residuation calculi, such as the partially associative calculus L+NL, just by unifying prosodic terms according to the laws of the groupoid algebras of interpretation. Such direct association of proof theory with interpretation is precisely the point of labelling: \bringing semantics back into syntax". Generality is obtained because we have identi ed the highest common factor of sublinear residuation calculi, linear validity. The degrees of variation are in e ect built in as non-logical properties of term structure of quanti er-free rst-order linear logic: unfolded signed labelled atoms (literals) : A+ and : B are atomic formulas of predicational linear logic A( ) and B( )? respectively. Linking corresponds to an application of the resolution principle, together with uni cation of terms. It is this relation which we shall exploit to resolve the computational shortcoming of the method as it stands: that testing satis ability of the linking equations appears to demand solution to such problems as semigroup uni cation, which are quite intractable. 5.2 Clausal Proofs In trying to shed light on how to compute the relevant labelled proof nets it is instructive to identify their rational: why they represent the relevant conditions. Recall the groupoid interpretation of 11 the Lambek connectives: D(AnB) = fsj8s0 2 D(A); s0+s 2 D(B)g D(B=A) = fsj8s0 2 D(A); s+s0 2 D(B)g (42) In labelled calculi categorial type assignment statements comprise a groupoid term and a type A; we write : A. Now we can formalise the metalanguage of the interpretation clauses in labels in order to translate type assignments to complex categorial types to statements of rst order logic: : AnB i 8a[a: A! a+ : B] : B=A i 8a[a: A! +a: B] (43) When we translate the type assignment statements of a labelled sequent in this way the problem of categorial theorem proving is compiled into that of theorem proving for a particular fragment of rst order logic. The quanti ers may turn out to be in positive or negative positions according to antecedent or succedent origen, and nesting within the negations implicit in implications. Hence the object variables correspond to either metavariables or Skolem constants in clausal refutation, and this is predictable from the transmission of polarities at the time of compilation. This explains the Roorda/Moortgat unfolding so far as it goes, but also suggests how to go further, preserving the relations between parts of formulas in such a way that a ow of information can be exploited. We can therefore adapt the compilation to include the implication (written in the logic programming direction), which is linear, dealing with occurrences. The polar translation functions are identity functions on atomic assignments; on complex category predicates they are de ned mutually as follows; p indicates the polarity complementary to p: (44) a. + : Bp : Ap new variable/constant as p += : AnBp b. + : Bp : Ap new variable/constant as p += : B=Ap The unfolding transformations have the same general form for the positive (con guration/database) and negative (succedent/agenda) occurrences; the polarity is used to indicate whether new symbols introduced for quanti ed variables in the interpretation clauses are metavariables or Skolem constants. The result of compilation is contained in a higher order logic programming fragment. 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 (45) This means that matching against the head and assembly of subgoals does not require recursion and restructuring as run time. We shall also allow unit program clauses X to be abbreviated X. We de ne this fragment and a suitable execution mechanism. Assume a set AT OM of atomic formulas, 0-ary, 1-ary, etc., formula constructors f : : : gn2f0;1;:::g and a binary (in x) formula constructor . A sequent comprises an agenda formula A and a bag (or: multiset) database = fB1; : : : ; Bngm; n 0 of program clauses; we write ` A. The set AGENDA of agendas is de ned by: AGENDA ::= GOAL : : : GOAL (46) The set PCLS of program clauses is de ned by: PCLS ::= AT OM AGENDA (47) For rst order programming the set GOAL of goals is de ned by: GOAL ::= AT OM (48) 12 For the higher order case the notion of GOAL is generalised to include implications: GOAL ::= AT OM j GOAL PCLS (49)In linear logic programming each rule is resource-conscious. The termination condition is: (50) id A ` A I.e. an atomic agenda is a consequence of its unit database; all program clauses must be \used up" by the resolution rule: (51) ` B1 : : : Bn C1 : : : Cm RES ; A B1 : : : Bn ` A C1 : : : Cm I.e. a program clause disappears from the database once it is resolved upon: each is used exactly once. The deduction theorem rule for higher-order clauses is also sensitised to the employment of antecedent contexts: (52) ; B ` A ` C1 : : : Cm DT ; ` (A B) C1 : : : Cm With respect to uni cation, the goal in resolution will always be ground so that potentially problematic uni cation problems reduce to simpler one way cases. The DT rule appears to need to hypothesise partitionings, however the form of compilation is such that A and B share a Skolem constant; so B can and must be used to prove A and a mechanism for the lazy splitting of contexts can be e ected. Starting from the initial database and agenda, a proof will be represented as a list of agendas, avoiding the context repetition of sequent proofs by indicating where the resolution rule retracts from the database (superscript coindexed overline), and where the deduction theorem rule adds to it (subscript coindexation): database ; A B1 : : : Bni agenda i: A C1 : : : Cn RES i+1: B1 : : : Bn C1 : : : Cm (53) database ; Bi agenda i: (A B) C1 : : : Cm DT i+1: A C1 : : : Cm (54)Consider the case of lifting: k: A ` k: B/(AnB) (55) The succedent unfolds as shown in (56). (56) k+l: B a+l: B a: A l: AnB+ k: B/(AnB) 13 Then the proof runs thus: database k: A3; a+l: B a: A12 agenda 1. k+l: B (a+l: B a: A) DT 2. k+l: B RES a = k 3. k: A (57) The uni cation is simply term uni cation: lifting is NLand L-valid. Composition depends on associativity and is only L-valid: k: AnB+, l: BnC+ ` k+l: AnC (58) (59) a+k: B a: A k: AnB+ b+l: C b: B l: BnC+ m+k+l: C m: A k+l: AnC database a+k: B a: A3, b+l: C b: B2, m: A14 agenda 1. m+k+l: C m: A DT 2. m+k+l: C RES b =m+k 3. m+k: B RES a =m 4. m: A (60) 6 Linguistic Examples Linguistic application cannot be explained in a few lines; for motivation see for example Moorgat (1988) and Morrill (1994d). Illustration should be instructive however given even a little familiarity. We consider linguistic examples starting with pure Lambek fragments. We shall then go on to multimodality and discontinuity. There is the following derivation of `John walks' as a sentence in L or NL. John: N+ walks: NnS+= a+walks: S+ a: N (61) database John: N2; a+walks: S a: N1 agenda 1. John+walks: S RES a=John 2. John: N (62)In the derivation of `John likes Bill' the transitive verb gives rise to an agenda of length two. Assuming an associative context, parentheses are omitted from the prosodic terms. John: N+ likes: (NnS)/N+= likes+a: NnS+ a: N = (b+likes+a: S+ b: N ) a: N (63) database John: N2; b+likes+a: S b: N a: N1; Bill: N3 agenda 1. John+likes+Bill: S RES b=John, a=Bill 2. John: N Bill: N RES 3. Bill: N (64) 14 The next example `John will walk' illustrates the e ect for an auxiliary verb treated as a functor over a verb phrase, which is itself a functor. The auxiliary creates an antecedent literal labelled by a Skolem constant, and this resolves with the subject literal of the embedded verb phrase. will: (NnS)/(NnS)+= will+a: NnS+ a: NnS = (b+will+a: S+ b: N ) (k+a: S k: N+) (65) database John: N2; b+will+a: S b: N (k+a: S k: N)1; c+walk: S c: N4; k: N35 agenda 1. John+will+walk: S RES b=John, a=walk 2. John: N (k+walk: S k: N) RES 3. k+walk: S k: N DT 4. k+walk: S RES c=k 5. k: N For the following minimal example of object relativisation `which John likes' associativity is essential. The relative pronoun is a higher order functor and the positive antecedent literal arising from its argument corresponds to an \empty category" or \trace" of the extraction. But note that such a literal arose in the non-extraction example (66) also. which: R/(S/N)+= which+a: R+ a: S/N = which+a: R+ (a+k: S k: N+) (67) database which+a: R (a+k: S k: N)1; John: N4; b+likes+c: S b: N c: N3; k: N25 agenda 1. which+John+likes: R RES a=John+likes 2. John+likes+k: S k: N DT 3. John+likes+k: S RES b=John, c=k 4. John: N k: N RES 5. k: N (68) 7 Multimodal Lambek calculi In multimodal calculi families of connectives f=i; nigi2f1;:::;ng are each de ned by residuation with respect to their adjunction in a \polygroupoid" hL; f+igi2f1;:::;ngi (Moortgat and Morrill 1991): D(AniB) = fsj8s0 2 D(A); s0+is 2 D(B)g D(B=iA) = fsj8s0 2 D(A); s+is0 2 D(B)g (69) Multimodal groupoid compilation is immediate: (70) a. +i : Bp : Ap new variable/constant as p += : AniBp b. +i : Bp : Ap new variable/constant as p += : B=iAp 15 This is entirely general. Any multimodal calculus can be implemented this way provided we have a (just one way) uni cation algorithm specialised according to the structural communication axioms. By way of example we shall deal with multimodality for discontinuity which involves varying internal structural properties (associativity vs. non-associativity) as well as \split/wrap" interaction between modes. We shall also consider unary operators projecting bracketed string structure and their use for head-oriented discontinuity. But before considering these we look at the simple multimodality of the partially associative calculus L+NL with operators / and n de ned with respect to +, and < and > de ned with respect to (:; :) in a bigroupoid hL;+; (:; :)i in which + (but not (:; :)) is associative. Assignment of a coordinator to (S>S)/S characterises a coordinate structure as a non-associative domain. For example in the following the complementised sentence `that it rains and it shines' is analysed as containing a domain [`it rains and it shines'] composed of subconstituents `it rains' and `and it shines' (the latter is itself unstructured). that: CP/S+= that+a: CP+ a: S and: (S>S)/S+= and+b: S>S+ b: S = ((c, and+b): S+ c: S ) b: S (71) database that+a: CP a: S1; it+rains: S3; (c, and+b): S c: S b: S2; it+shines: S4 agenda 1. that+(it+rains, and+it+shines): CP RES a=(it+rains, and+it+shines) 2. (it+rains, and+it+shines): S RES c=it+rains, b=it+shines 3. it+rains: S it+shines: S RES 4. it+shines: S (72) 8 Discontinuity We consider residuation calculi for two kinds of discontinuity: regular, for discontinuous functors, and for in x binders as in quanti er raising, re exivisation, pied piping and gapping; and headoriented, such as head in xation and head extraction in Germanic verb clusters and verb fronting. In each case the essential strategy is to specify discontinuous adjunction as a primitive (as opposed to derived) operation in the multigroupoid prosodic algebra of multimodal Lambek calculi, with respect to which discontinuity operators are de ned by residuation. 8.1 Regular Discontinuity In the discontinuity calculus of Morrill (1993, 1994d) connectives f=; ng, f<;>g and f"; #g are interpreted by residuation with respect to adjunctions +, (., .) andW respectively in a trigroupoid hL ;+; (:; :);W; i where + is associative and has (left and right) identity 2 L , and (., .) and W satisfy the \split-wrap" equation: (s1; s3)Ws2 = s1+s2+s3. We see that ( , ) is a left identity for W ; is required in the interest of linguistic generalisation: to include peripherality as a special case of discontinuity. Also for linguistic reasons however, formulas are interpreted as subsets of L = L nf g, preventing (but not ( ; )) from inhabiting types. The prosodic label equations then are as follows: a: s1+(s2+s3) = (s1+s2)+s3 b: s+ = +s = s c: (s1; s3)Ws2 = s1+s2+s3 (73) 16 A verb-particle construction is derived Fitch-style as in (74). 1. (rang ; up) { phone: (NnS)"N 2. John { j: N 3. Mary { m: N 4. ((rang ; up)WJohn) { (phone j): NnS 1, 2 E" 5. rang+John+up { (phone j): NnS = 4 6. Mary+rang+John+up { ((phone j) m): S 3, 5 En (74) Discussion of semantics would take us outside the direct concerns of the present article; the reader is referred to e.g. Moortgat (1988, 1991) and Morrill (1993, 1994d) for explication. The e ect of quanti er-raising, whereby quanti ers are to take sentential scope, is achieved by assignment of a quanti er phrase such as `everyone' to a \quantifying-in" type (S"N)#S. A simple instance of quanti er-raising is shown in (75). 1. John { j: N 2. likes { like: (NnS)/N 3. everything { x8y(x y): (S"N)#S 4. a { x: N H 5. likes+a { (like x): NnS 2, 4 E/ 6. John+likes+a { ((like x) j): S 1, 5 En 7. John+likes+a+ { ((like x) j): S = 6 8. ((John+likes; )Wa) { ((like x) j): S = 7 9. (John+likes; ) { x((like x) j): S"N 4, 8 I" 10. ((John+likes; )W everything) { 3, 9 E# ( x8y(x y) x((like x) j)): S 11. John+likes+everything { 8y((like y) j): S = 10 (75) The techniques given here show how proof net theorem-proving in implicational residuation calculi, and hence parsing in certain categorial logics, can be compiled into a form suitable for SLD-resolution in linear logic. This indicates a general strategy for checking the correctness of a proof net by uni cation in which one term is always ground, but leaves open the problem of computing uni ers in any particular case. This will be illustrated with respect to discontinuity. The unidirectionality of uni cation using the clausal implementationmakes it manageable through normalisation of and recursive decent through ground terms. For the regular discontinuity calculus the task concerns us with the equations (73). The uni cation procedure is to compute, for a given ground term , all the distinct assignments of ground terms to variables in another term 0 such that = 0[ ]. We treat the process in two stages; (75b) and (75c) are considered normalisation rules (with redex on the left and contractum on the right); associativity as in (75a) could be naturally treated by representing equivalence classes of associative bracketings as lists, though we shall not choose to do so. In the rst phase, the ground term is normalised by transforming redexes to their contractums. The second stage proceeds by recursion on the structure of 0. There are the cases that 0 is a variable, a constant, or has principle operator one of the three prosodic adjunctions. Finally there are the cases that 0 is, or can be instantiated to, a redex. If 0 is a variable v then simply put v = . If 0 is a constant k then if is k succeed, otherwise fail. If 0 is of the form ( 01; 02) then if is of the form ( 1; 2) unify 1 and 01 and 2 and 02. If 0 is of the form 01W 02 then if is of the form 1W 2 unify 1 and 01 and 2 and 02. If 0 is of the form 01+ 02 then nd representatives 1 and 2 satisfying = 1+ 2 (using associativity) and unify 1 and 01, and 2 and 02. It remains to consider the cases where 0 has the form of, or can be instantiated to the form of, a redex (the redexes in having been already removed in the rst phase). If 0 can be instantiated to + unify and . If 0 can be instantiated to + unify and . If 0 can be instantiated to ( 01; 03)W 02 then nd representatives 1; 2 and 3 satisfying = 1+ 2+ 3 and unify 1 with 01, 2 with 02, and 3 with 03. 17 As seen earlier a simple instance of discontinuity is obtained by assigning the compound particle verb prosodic form (rang;up) to a wrapping transitive verb type (NnS)"N. In the following derivation Mary+rang+John+up is uni ed with b+((rang;up)Wa) by the uni er fb =Mary; a = Johng. (rang, up): (NnS)"N+= ((rang, up)Wa): NnS+ a: N = (b+((rang, up)Wa): S+ b: N ) a: N (76) database Mary: N2; b+((rang, up)Wa): S b: N a: N1; John: N3 agenda 1. Mary+rang+John+up: S RES b=Mary, a=John 2. Mary: N John: N RES 3. John: N (77)In the next derivation, for `John likes everything', John+likes+everything is uni ed with cWeverything by c = (John+likes; ), and (John+likes; )Wk is subsequently uni ed with b+likes+a by fb = John; a = kg. everything: (S"N)#S+= (cWeverything): S+ c: S"N = (cWeverything): S+ ((cWk): S k: N+) (78) database John: N4; b+likes+a: S b: N a: N3; (cWeverything): S ((cWk): S k: N)1; k: N25 agenda 1. John+likes+everything: S RES c = (John+likes; ) 2. ((John+likes, )Wk): S k: N DT 3. John+likes+k: S RES b=John, a=k 4. John: N k: N RES 5. k: N (79) 8.2 Bracket Operators and Head-Oriented Discontinuity To treat head-oriented discontinuity we shall require bracket operators, interpreted in a prosodic algebra such as hL;+; [ . ]i where [ . ] is a unary operation (Morrill 1992); for nice symmetric proof theory we require that this is a 1-1 function (permutation) so that its inverse [ . ] 1 is total (Morrill 1994d, 1994c). [ [ s ] 1 ] = [ [ s ] ] 1 = s (80) D([ ]A) = f[ s ]js 2 D(A)g D([ ] 1A) = fsj[ s ] 2 D(A)g = f[ s ] 1js 2 D(A)g (81) Intuitively [ ]A is the result of appointing or crystalising prosodic objects as domains or constituents, and [ ] 1A the result of annulling or dissolving appointment as a domain. Labelled Prawitz-style natural deduction for bracket operators is as follows, including [ . ] and its inverse [ . ] 1 in prosodic labels, for which there is a prosodic label equation [ [ ] 1 ] = [ [ ] ] 1 = . (82) ... : A I[ ] [ ]: [ ]A ... : [ ]A E[ ] [ ] 1: A 18(83)...: A I[ ] 1[ ] 1: [ ] 1A...: [ ]1AE[ ] 1[ ]: AHead-oriented discontinuity is obtained by combining a bracketing operation and a primi-tive head adjunction in a headed calculus (Moortgat and Morrill 1991): the prosodic algebrais hL;+l;+r;+h; [ v: ]; i with implications fnl; /lg, fnr; /rg, and f"; #g interpreted by residuationw.r.t. +l, +r and +h respectively. There are the following interaction axioms:( +r )+h = +r( +h )( +l )+h = ( +h )+l(84)And [ v ] and [ v ] 1 are bracketing operators with respect to permutation [ v: ]; is a left andright identity for +l and +r; the bottoming-out interaction for head adjunction is (for our Dutchexample):[ v ]+h = [ v +l ](85)The following is a simple example of Dutch head-in xation:boeken+r [vkan+llezen]books can read\is able to read books"(86)It has the Prawitz-style derivation (87).boekenlezenkanlezen: [ v ]1(NnrIVi)boeken: NE[ v ] 1[vlezen]: NnrIViEnrboeken+r [vlezen]: IVikan: IVi#VPE#(boeken+r [vlezen])+hkan: VP=boeken+r([vlezen]+hkan): VP=boeken+r[vkan+llezen]: VPThe existing methods are applicable to head-oriented discontinuity, for which we also requirebracket unfolding:(88)[ ] 1: A+: [ ]A+[ ]: A+: [ ] 1A+(89)[ ] 1: A: [ ]A[ ]: A: [ ] 1AThen the verb-in xing example is derived as follows by clausal resolution:lezen: [ v ]1(NnrIVi)+= [ vlezen ]:NnrIVi+= a+r[ vlezen ]: IVi+ a: Nkan: IVi#VP+= b+hkan: VP+ b: IVi(90) database boeken: N3;a+r[ vlezen ]: IVi a: N2;b+hkan: VP b: IVi1agenda1.boeken+r [ vkan+llezen ]: VPRES b = boeken+r [ vlezen ]2.boeken+r [ vlezen ]: IViRES a=boeken3.boeken: N(91) 19The only non-trivial step is that resolving the initial unit goal, for which the uni cation is expli-cated by the following:boeken+r [ vkan+llezen ] = boeken+r([ vlezen ]+hkan) =(boeken+r[ vlezen ])+hkan(92)9 ConclusionWe hope that what we have shown here goes some way towards providing general and powerfulmethods for categorial parsing-as-deduction. There are some technical issues: to prove formallyproperties like completeness and the correctness of the recursive descent/redex pattern matchinguni cation algorithm used, especially in relation to classes of systems and not just speci c ones.It is also appropriate to look at how uni cation would work out for interpretation in orderedgroupoids, which allow more sensitive structural interaction, allowing inclusion rather than onlyequation relations between modes; there is good linguistic motivation for such interaction. Moregenerally, we can look at compilation with respect to other kinds of models. For example Morrill(1994b) uses the binary relational interpretation of L, and thereby avoids altogether the needto compute uni ers under associativity; that paper also makes some progress with respect toproducts, not considered here.Finally, we note that labelling and unfolding with respect to the semantic dimension, usingtyped lambda terms, renders the task of generation-as-deduction in the same light as parsing-as-deduction. That is, the problem of generation (computing prosodics given semantics), as opposedto parsing (computing semantics given prosodics), is rendered precisely as the task of one wayuni cation in the semantic algebra of typed lambda terms. This appears to o er the prospectof systematic generation in Lambek categorial grammar by simply one way uni cation in typedlambda calculus.BibliographyBibel, W.: 1981, `On matrices with connections', Journal of the Association for Computing Ma-chinery 28, 633{645.Bibel, W. and E. Eder: 1993, `Methods and calculi for deduction', in D.M. Gabbay, C.J. Hoggerand J.A. Robinson (eds.) Handbook of Logic in Arti cial Intelligence and Logic ProgrammingVolume 1: Foundations, Clarendon Press, Oxford, 67{102.Buszkowski, K.: 1986, `Competeness results for Lambek syntactic calculus', Zeitschrift fur Math-ematische Logik und Grundlagen der Mathematik 32, 13{28.Do sen, Kosta: 1992, `A Brief Survey of Frames for the Lambek Calculus', Zeitschrift fur Mathe-matische Logik und Grundlagen der Mathematik 38, 179{187.Do sen, Kosta and Peter Schroeder-Heister (eds.): 1993, Substructural Logics, Oxford UniversityPress, Oxford.Gabbay, D.: 1991, Labelled Deductive Systems, to appear, Oxford University Press, Oxford.Girard, Jean-Yves: 1987, `Linear Logic', Theoretical Computer Science 50, 1{102.Hendriks, Herman: 1993, Studied Flexibility: Categories and Types in Syntax and Semantics, Ph.Ddissertation, Institute for Logic, Language and Computation, Universiteit van Amsterdam.Hepple, Mark: 1990, `Normal form theorem proving for the Lambek calculus', in H. Karlgren (ed.),Proceedings of COLING 1990, Stockholm.Konig, E.: 1989, `Parsing as natural deduction', in Proceedings of the Annual Meeting of theAssociation for Computational Linguistics, Vancouver.Lambek, J.: 1958, `The mathematics of sentence structure', American Mathematical Monthly 65,154{170, also in W. Buszkowski, W. Marciszewski, and J. van Benthem (eds.): 1988, Catego-rial Grammar, Linguistic & Literary Studies in Eastern Europe Volume 25, John Benjamins,Amsterdam, 153{172. 20Lambek, J.: 1961, `On the calculus of syntactic types', in R. Jakobson (ed.) Structure of lan-guage and its mathematical aspects, Proceedings of the Symposia in Applied MathematicsXII, American Mathematical Society, 166{178.Lambek, J.: 1988, `Categorial and Categorical Grammars', in Richard T. Oehrle, Emmon Bach,and Deidre Wheeler (eds.) Categorial Grammars and Natural Language Structures, Studies inLinguistics and Philosophy Volume 32, D. Reidel, Dordrecht, 297{317.Moortgat, Michael: 1988, Categorial Investigations: Logical and Linguistic Aspects of the LambekCalculus, Foris, Dordrecht.Moortgat, Michael: 1990, `Categorial Logics: a computational perspective', Proceedings ComputerScience in the Netherlands.Moortgat, Michael: 1991, `Generalised Quanti cation and Discontinuous type constructors', toappear in Sijtsma and Van Horck (eds.) Proceedings Tilburg Symposium on DiscontinuousConstituency, Walter de Gruyter, Berlin.Moortgat, Michael: 1992, `Labelled Deductive Systems for categorial theorem proving', OTSWork-ing Paper OTS{WP{CL{92{003, Rijksuniversiteit Utrecht, also in Proceedings of the EighthAmsterdam Colloquium, Institute for Language, Logic and Computation, Universiteit van Am-sterdam.Moortgat, Michael and Glyn Morrill: 1991, `Heads and Phrases: Type Calculus for Dependencyand Constituent Structure', to appear in Journal of Language, Logic, and Information.Morrill, Glyn: 1990, `Rules and Derivations: Binding Phenomena and Coordination in Catego-rial Logic', in Deliverable R1.2.D of DYANA Dynamic Interpretation of Natural Language,ESPRIT Basic Research Action BR3175.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 Sistemes In-formatics, Universitat Polit ecnica de Catalunya.Morrill, Glyn: 1993, Discontinuity and Pied-Piping in Categorial Grammar, Report de RecercaLSI{93{18{R, Departament de Llenguatges i Sistemes Inform atics, Universitat Polit ecnica deCatalunya, to appear in Linguistics and Philosophy.Morrill, Glyn: 1994a, `Clausal Proof Nets and Discontinuity', Report de Recerca LSI{94{21{R,Departament de Llenguatges i Sistemes Informatics, Universitat Polit ecnica de Catalunya.Morrill, Glyn: 1994b, `Higher-Order Linear Logic Programming of Categorial Deduction, Reportde Recerca LSI{94{42{R, Departament de Llenguatges i Sistemes Informatics, UniversitatPolit ecnica de Catalunya, to appear in Proceedings of the Seventh Conference of the EuropeanChapter of the Association for Computational Linguistics, Dublin 1995.Morrill, Glyn: 1994c, `Structural Facilitation and Structural Inhibition', Report de Recerca LSI{94{26{R, Departament de Llenguatges i Sistemes Informatics, Universitat Polit ecnica deCatalunya, also in Michele Abrusci, Claudia Casadio and Michael Moortgat (eds.) LinearLogic and Lambek Calculus: Proceedings 1993 Rome Workshop, Dyana Occasional Publi-cations, DYANA-2, Dynamic Interpretation of Natural Language, ESPRIT Basic ResearchProject 6852, 183{210.Morrill, Glyn: 1994d, Type Logical Grammar: Categorial Logic of Signs, Kluwer Academic Pub-lishers, Dordrecht.Oehrle, Dick: 1994, `Term labelled categorial type systems', to appear in Linguistics and Philoso-phy.Oehrle, Richard T. and Shi Zhang: 1989, `Lambek Calculus and Preposing of Embedded Subjects',Chicago Linguistic Society 25, Chicago.Roorda, Dirk: 1991, Resource Logics: proof-theoretical investigations, Ph.D. dissertation, Univer-siteit van Amsterdam.Wallen, L.A.: 1990, Automated proof search in non-classical logics: e cient matrix proof methodsfor modal and intuitionistic logics, MIT Press.