Computational coverage of type logical grammar: The Montague test

It is nearly half a century since Montague made his contributions to the field of logical semantics. In this time, computational linguistics has taken an almost entirely statistical turn and mainstream linguistics has adopted an almost entirely non-formal methodology. But in a minority approach reaching back before the linguistic revolution, and to the origins of computing, type logical grammar (TLG) has continued championing the flags of symbolic computation and logical rigor in discrete grammar. In this paper, we aim to concretise a measure of progress for computational grammar in the form of the Montague Test. This is the challenge of providing a computational cover grammar of the Montague fragment. We formulate this Montague Test and show how the challenge is met by the type logical parser/theorem-prover CatLog2.Peer ReviewedPostprint (published version

A reply to Kubota and Levine on gapping

In a series of papers Kubota and Levine give an account of gapping and determiner gapping in terms of hybrid type logical grammar, including anomalous scopal interactions with auxiliaries and negative quantifiers. We make three observations: i) under the counterpart assumptions that Kubota and Levine make, the existent displacement type logical grammar account of gapping already accounts for the scopal interactions, ii) Kubota and Levine overgenerate determiner-verb order permutations in determiner gapping conjuncts whereas the immediate adaptation of their proposal to displacement type logical grammar does not do so, and iii) Kubota and Levine do not capture simplex gapping as a special case of complex gapping, but require distinct lexical entries for the two cases; we show how a generalisation of displacement type logical grammar allows both simplex and discontinuous gapping under a single type assignmentPostprint (author's final draft

Models for the displacement calculus

The displacement calculus D is a conservative extension of the Lambek calculus L* (with empty antecedent allowed in sequents). L* can be said to be the logic of concatenation, while D can be said to be the logic of concatenation and intercalation. In many senses, it can be claimed that D mimics L*, namely that the proof theory, generative capacity and complexity of the former calculus are natural extensions of the latter calculus. In this paper, we strengthen this claim. We present the appropriate classes of models for D and prove its completeness results, and strikingly, we see that these results and proofs are natural extensions of the corresponding ones for L*.Peer ReviewedPostprint (published version

Spurious ambiguity and focalization

Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on identifying the essential mathematical structure of derivations. This is trivial in the case of context free grammar, where the parse structures are ordered trees; in the case of type logical categorial grammar, the parse structures are proof nets. However, with respect to multiplicatives, intrinsic proof nets have not yet been given for displacement calculus, and proof nets for additives, which have applications to polymorphism, are not easy to characterize. In this context we approach here multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalization.Peer ReviewedPostprint (published version

Semantically inactive multiplicatives and words as types

The literature on categorial type logic includes proposals for semantically inactive additives, quantifiers, and modalities (Morrill 1994[17]; Hepple 1990[2]; Moortgat 1997[9]), but to our knowledge there has been no proposal for semantically inactive multiplicatives. In this paper we formulate such a proposal (thus filling a gap in the typology of categorial connectives) in the context of the displacement calculus Morrill et al. (2011[16]), and we give a formulation of words as types whereby for every expression w there is a corresponding type W(w). We show how this machinary can treat the syntax and semantics of collocations involving apparently contentless words such as expletives, particle verbs, and (discontinuous) idioms. In addition, we give an account in these terms of the only known examples treated by Hybrid Type Logical Grammar (HTLG henceforth; Kubota and Levine 2012[4]) beyond the scope of unaugmented displacement calculus: gapping of particle verbs and discontinuous idioms.Peer ReviewedPostprint (author’s final draft

Computational coverage of TLG: nonlinearity

We study nonlinear connectives (exponentials) in the context of Type Logical Grammar (TLG). We devise four conservative extensions of the displacement calculus with brackets, Db!, Db!?, Db!b and Db!b?r which contain the universal and existential exponential modalities of linear logic (LL). These modalities do not exhibit the same structural properties as in LL, which in TLG are especially adapted for linguistic purposes. The universal modality ! for TLG allows only the commutative and contraction rules, but not weakening, whereas the existential modality ? allows the so-called (intuitionistic) Mingle rule, which derives a restricted version of weakening. We provide a Curry-Howard labelling for both exponential connectives. As it turns out, controlled contraction by ! gives a way to account for the so-called parasitic gaps, and controlled Mingle ? iteration, in particular iterated coordination. Finally, the four calculi are proved to be Cut-Free, and decidability is proved for a linguistically suffcient special case of Db!b?r (and hence Db!b).Postprint (published version

Computational coverage of TLG : the Montague test

This paper reports on the empirical coverage of Type Logical Grammar (TLG) and on how it has been computer implemented. We analyse the Montague fragment computationally and we proffer this task as a challenge to computational grammar: the Montague TestPeer ReviewedPostprint (author's final draft

Computational coverage of TLG: displacement

This paper reports on the coverage of TLG of Morrill (1994) and Moortgat (1997), and on how it has been computer implemented. We computer-analyse examples of displacement: discontinuous idioms, quantification, (medial) relativisation, VP ellipsis, (medial) pied piping, appositive relativisation, parentheticals, gapping, comparative subdeletion, and reflexivisation, and, in the appendix, Dutch verb raising and crossserial dependency.Peer ReviewedPostprint (published version

Multiplicative-additive focusing for parsing as deduction

Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on identifying the essential mathematical structure of derivations. This is trivial in the case of context free grammar, where the parse structures are ordered trees; in the case of type logical categorial grammar, the parse structures are proof nets. However, with respect to multiplicatives intrinsic proof nets have not yet been given for displacement calculus, and proof nets for additives, which have applications to polymorphism, are not easy to characterise. Here we approach multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalisation.Peer ReviewedPostprint (published version

The hidden structural rules of the discontinuous Lambek calculus

Capítol de llibre d'homenatge "Categories and Types in Logic, Language, and Physics. Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday"The sequent calculus sL for the Lambek calculus L ([2]) has no structural rules. Interestingly, sL is equivalent to a multimodal calculus mL, which consists of the nonassociative Lambek calculus with the structural rule of associativity. This paper proves that the sequent calculus or hypersequent calculus hD of the discontinuous Lambek calculus ([7], [4] and [8]), which like sL has no structural rules, is also equivalent to an ¿-sorted multimodal calculus mD. More concretely, we present a faithful embedding translation (·)# between mD and hD in such a way that it can be said that hD absorbs the structural rules of mD.Peer ReviewedPostprint (published version