Oriol Valentín

The Displacement Calculus

If all dependent expressions were adjacent some variety of immediate constituent analysis would suffice for grammar, but syntactic and semantic mismatches are characteristic of natural language; indeed this is a, or the, central problem in grammar. Logical categorial grammar reduces grammar to logic: an expression is well-formed if and only if an associated sequent is a theorem of a categorial logic. The paradigmatic categorial logic is the Lambek calculus, but being a logic of concatenation the Lambek calculus can only capture discontinuous dependencies when they are peripheral. In this paper we present the displacement calculus, which is a logic of intercalation as well as concatenation and which subsumes the Lambek calculus. On the empirical side, we apply the new calculus to discontinuous idioms, quantification, VP ellipsis, medial extraction, pied-piping, appositive relativisation, parentheticals, gapping, comparative subdeletion, cross-serial dependencies, reflexivization, anaphora, dative alternation, and particle shift. On the technical side, we prove that the calculus enjoys Cut-elimination.

Dutch Grammar and Processing: A Case Study in TLG

The aim of this paper is to see type logical grammar (TLG) at work on an interesting linguistic case: the incremental processing of Dutch subordinate clause word order, namely the so-called cross-serial dependencies. With the help of proof net machinery adapted for the continuous and discontinuous Lambek calculus we are able to account for the increasing unacceptability of cross-serial dependencies with increasingly multiple embeddings.

A Count Invariant for Lambek Calculus with Additives and Bracket Modalities

The count invariance of van Benthem (1991[16]) is that for a sequent to be a theorem of the Lambek calculus, for each atom, the number of positive occurrences equals the number of negative occurrences. (The same is true for multiplicative linear logic.) The count invariance provides for extensive pruning of the sequent proof search space. In this paper we generalize count invariance to categorial grammar (or linear logic) with additives and bracket modalities. We define by mutual recursion two counts, minimum count and maximum count, and we prove that if a multiplicative-additive sequent is a theorem, then for every atom, the minimum count is less than or equal to zero and the maximum count is greater than or equal to zero; in the case of a purely multiplicative sequent, minimum count and maximum count coincide in such a way as to together reconstitute the van Benthem count criterion. We then define in the same way a bracket count providing a count check for bracket modalities. This allows for efficient pruning of the sequent proof search space in parsing categorial grammar with additives and bracket modalities.

On anaphora and the binding principles in categorial grammar

In type logical categorial grammar the analysis of an expression is a resource-conscious proof. Anaphora represents a particular challenge to this approach in that the antecedent resource is multiplied in the semantics. This duplication, which corresponds logically to the structural rule of contraction, may be treated lexically or syntactically. Furthermore, anaphora is subject to constraints, which Chomsky (1981)[1] formulated as Binding Principles A, B, and C. In this paper we consider English anaphora in categorial grammar including reference to the binding principles. We invoke displacement calculus, modal categorial calculus, categorial calculus with limited contraction, and entertain addition of negation as failure.

Displacement logic for anaphora

The displacement calculus of Morrill, Valentin and Fadda (2011) [25] aspires to replace the calculus of Lambek (1958) [13] as the foundation of categorial grammar by accommodating intercalation as well as concatenation while remaining free of structural rules and enjoying Cut-elimination and its good corollaries. Jager (2005) [11] proposes a type logical treatment of anaphora with syntactic duplication using limited contraction. Morrill and Valentin (2010) [24] apply (modal) displacement calculus to anaphora with lexical duplication and propose extension with a negation as failure in conjunction with additives to capture binding conditions. In this paper we present an account of anaphora developing characteristics and employing machinery from both of these proposals.

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.

Count-invariance including exponentials

We define infinitary count-invariance for categorial logic, extending countinvariance for multiplicatives (van Benthem, 1991) and additives and bracket modalities (Valentin et al., 2013) to include exponentials. This provides an effective tool for pruning proof search in categorial parsing/theorem-proving.

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.

The hidden structural rules of the discontinuous Lambek calculus

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 ω-sorted multimodal calculus mD. More concretely, we present a faithful embedding translation \((\cdot)^\sharp\) between mD and hD in such a way that it can be said that hD absorbs the structural rules of mD.

Generalized discontinuity

We define and study a calculus of discontinuity, a version of displacement calculus, which is a logic of segmented strings in exactly the same sense that the Lambek calculus is a logic of strings. Like the Lambek calculus, the displacement calculus is a sequence logic free of structural rules, and enjoys Cut-elimination and its corollaries: the subformula property, decidability, and the finite reading property. The foci of this paper are a formulation with a finite number of connectives, and consideration of how to extend the calculus with defined connectives while preserving its good properties.