A Frobenius Algebraic Analysis for Parasitic Gaps

The interpretation of parasitic gaps is an ostensible case of non-linearity in natural language composition. Existing categorial analyses, both in the typelogical and in the combinatory traditions, rely on explicit forms of syntactic copying. We identify two types of parasitic gapping where the duplication of semantic content can be confined to the lexicon. Parasitic gaps in adjuncts are analysed as forms of generalized coordination with a polymorphic type schema for the head of the adjunct phrase. For parasitic gaps affecting arguments of the same predicate, the polymorphism is associated with the lexical item that introduces the primary gap. Our analysis is formulated in terms of Lambek calculus extended with structural control modalities. A compositional translation relates syntactic types and derivations to the interpreting compact closed category of finite dimensional vector spaces and linear maps with Frobenius algebras over it. When interpreted over the necessary semantic spaces, the Frobenius algebras provide the tools to model the proposed instances of lexical polymorphism.Comment: SemSpace 2019, to appear in Journal of Applied Logic

Lexical and Derivational Meaning in Vector-Based Models of Relativisation

Sadrzadeh et al (2013) present a compositional distributional analysis of relative clauses in English in terms of the Frobenius algebraic structure of finite dimensional vector spaces. The analysis relies on distinct type assignments and lexical recipes for subject vs object relativisation. The situation for Dutch is different: because of the verb final nature of Dutch, relative clauses are ambiguous between a subject vs object relativisation reading. Using an extended version of Lambek calculus, we present a compositional distributional framework that accounts for this derivational ambiguity, and that allows us to give a single meaning recipe for the relative pronoun reconciling the Frobenius semantics with the demands of Dutch derivational syntax.Comment: 10 page version to appear in Proceedings Amsterdam Colloquium, updated with appendi

Non-associative, Non-commutative Multi-modal Linear Logic

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLΣ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLΣ

Positive Logic with Adjoint Modalities: Proof Theory, Semantics and Reasoning about Information

We consider a simple modal logic whose non-modal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4 and S5, such logics are useful, as shown in previous work by Baltag, Coecke and the first author, for encoding and reasoning about information and misinformation in multi-agent systems. For such a logic we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of "nested" or "tree-sequent" calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.Comment: This paper is the full version of the article that is to appear in the ENTCS proceedings of the 25th conference on the Mathematical Foundations of Programming Semantics (MFPS), April 2009, University of Oxfor

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

Comparing and evaluating extended Lambek calculi

Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expressive logical calculus was needed for many other cases. Therefore, many extensions and variants of the Lambek calculus have been proposed, since the eighties and up until the present day. As a result, there is now a large class of calculi, each with its own empirical successes and theoretical results, but also each with its own logical primitives. This raises the question: how do we compare and evaluate these different logical formalisms? To answer this question, I present two unifying frameworks for these extended Lambek calculi. Both are proof net calculi with graph contraction criteria. The first calculus is a very general system: you specify the structure of your sequents and it gives you the connectives and contractions which correspond to it. The calculus can be extended with structural rules, which translate directly into graph rewrite rules. The second calculus is first-order (multiplicative intuitionistic) linear logic, which turns out to have several other, independently proposed extensions of the Lambek calculus as fragments. I will illustrate the use of each calculus in building bridges between analyses proposed in different frameworks, in highlighting differences and in helping to identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona, Spain. 201

Logical Instrumentalism and Concatenation

Logical pluralism is the theory that there is more than one right logic. Logical instrumentalism is the view that a logic is a correct logic if it can be used to fruitfully pursue some deductive inquiry. Logical instrumentalism is a version of logical pluralism, since more than one logic can be used fruitfully. In this paper, I will show that a logical instrumentalist must accept linear logic as a correct logic, since linear logic is useful for studying natural language syntax. I further show that this means that the logical instrumentalist must accept a wide range of connectives, in particular concatenation. I end by explaining why this is a feature rather than a bug

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