Computational Coverage of TLG: The Montague Test Glyn Morrill and Oriol Valent´ın Universitat Polit` ecnica de Catalunya Barcelona Abstract This paper reports on the empirical coverage of Type Logical Grammar (TLG) and on how it has been com- puter implemented. We analyse the Montague fragment computationally and we proffer this task as a challenge to computational grammar: the Mon- tague Test. Keywords: logical syntax and semantics; pars- ing as deduction; Montague grammar; compu- tational grammar; Montague Test 1. Introduction The Type Logical Grammar of (Morrill, 1994) and (Moortgat, 1997) is a powerful formalism with a transparent syntax-semantics interface operating through the Curry-Howard isomor- phism. The version of the formalism used com- prises 50 connectives shown in Figure 1. The heart of the logic is the displace- ment calculus of (Morrill et al., 2011) which comprises twin continuous and discontinuous residuated families of connectives having a pure sequent calculus, the tree-based hyperse- quent calculus, and enjoying Cut-elimination (Valent´ın, 2012). Other primary connectives are additives, 1st order quantifiers, normal (i.e. distributive) modalities, bracket (i.e. nondis- tributive) modalities, and the non-linear expo- nentials, and contraction for anaphora. We can draw a clear distinction between these primary connectives and the semantically inactive connectives and synthetic connectives which are abbreviatory and there merely for convenience. There are semantically inactive variants of the continuous and discontinuous multiplicatives, including the words as types predicate W, and semantically inactive vari- ants of the additives, 1st order quantifiers, and normal modalities. Defined connectives di- vide into the continuous deterministic synthetic connectives of projection and injection, and the discontinuous, split and bridge, and the con- tinuous nondeterministic synthetic connectives of nondirectional division and unordered prod- uct, and the discontinuous, nondeterministic extract, infix, and discontinuous product. Finally there is the negation as failure of ‘except’ (formerly difference), a powerful de- vice for expressing linguistic exceptions (Mor- rill and Valent´ın, 2014). 2. Rules and linguistic applications for primary connectives In this section we present semantically labelled sequent rules for, and exemplify linguistic ap- plications of, the primary connectives. The continuous multiplicatives of Figure 2, the Lambek connectives, are the basic means of categorial categorization and subcategoriza- tion. The directional divisions over, /, and un- der, \, are exemplified by assignments such as the: N/CN for the man: N and sings: N\S for John sings: S , and loves:(N\S )/N for John loves Mary: S . The continuous product • is exemplified by a ‘small clause’ assignment such as considers:(N\S )/(N•(CN/CN)) for John considers Mary socialist: S . 1 The con- tinuous unit can be used together with ad- ditive disjunction to express the optionality of a complement as in eats:(N\S )/(N⊕I ) for 1 But this makes no different empirical predictions from the more standard type of analysis in CG and G/HPSG which simply treats verbs like consider as tak- ing a noun phrase and an infinitive.