The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the Curry-Howard derivational semantics, and structure-preservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural (...) language form and meaning. (shrink)

In this paper it is shown how simple texts that can be parsed in a Lambek Categorial Grammar can also automatically be provided with a semantics in the form of a Discourse Representation Structure in the sense of Kamp [1981]. The assignment of meanings to texts uses the Curry-Howard-Van Benthem correspondence.

We provide an algorithm for determining a categorial grammar from linguistic data that essentially uses unification of type-schemes assigned to atoms. The algorithm presented here extends an earlier one restricted to rigid categorial grammars, introduced in [4] and [5], by admitting non-rigid outputs. The key innovation is the notion of an optimal unifier, a natural generalization of that of a most general unifier.

This paper introduces λ-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, λ-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of linear combinators, closed pure λ-terms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable.

The paper proposes a semantics for contextual (i.e., Temporal and Locative) Prepositional Phrases (CPPs) like during every meeting, in the garden, when Harry met Sally and where I’m calling from. The semantics is embodied in a multi-modal extension of Combinatory Categoral Grammar (CCG). The grammar allows the strictly monotonic compositional derivation of multiple correct interpretations for “stacked” or multiple CPPs, including interpretations whose scope relations are not what would be expected on standard assumptions about surfacesyntactic command and monotonic derivation. A (...) type-hierarchy of functional modalities plays a crucial role in the specification of the fragment. (shrink)

For the most part, the papers collected in this volume stern from presentations given at a conference held in Tucson over the weekend of May 31 through June 2, 1985. We wish to record our gratitude to the participants in that conference, as well as to the National Science Foundation and the University of Arizona SBS Research Institute for their financial support. The advice we received from Susan Steele on organizational matters proved invaluable and had many felicitous consequences for the (...) success of the con­ ference. We also would like to thank the staff of the Departments of Linguistics of the University of Arizona and the University of Massachusetts at Amherst for their help, as weIl as a number of individuals, including Lin Hall, Kathy Todd, and Jiazhen Hu, Sandra Fulmer, Maria Sandoval, Natsuko Tsujimura, Stuart Davis, Mark Lewis, Robin Schafer, Shi Zhang, Olivia Oehrle-Steele, and Paul Saka. Finally, we would like to express our gratitude to Martin Scrivener, our editor, for his patience and his encouragement. Vll INTRODUCTION The term 'categorial grammar' was introduced by Bar-Rillel as a handy way of grouping together some of his own earlier work and the work of the Polish logicians and philosophers Lesniewski and Ajdukiewicz, in contrast to approaches to linguistic analysis based on phrase structure grammars. (shrink)

1 Recognition Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Classical Categorial Grammar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 (...) Classical Categorial Grammar. Examples . . . . . . . . . . . . . . . . . . . . . 5 4 Logic Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5 Lambek calculus. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Lambek calculus. Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7 Lambek calculus. Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 8 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 9 Residuated and Galois Connected Functions . . . . . . . . . . . . . . . . . . 14 10 Interpretation of the Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11 Nonveridical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 12 Dutch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 13 Classification of NPIs in Dutch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 14 Antilicensing Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.. (shrink)

We present a new axiomatization of the non-associative Lambek calculus. We prove that it takes polynomial time to reduce any non-associative Lambek categorial grammar to an equivalent context-free grammar. Since it is possible to recognize a sentence generated by a context-free grammar in polynomial time, this proves that a sentence generated by any non-associative Lambek categorial grammar can be recognized in polynomial time.

Even though residuation is at the core of Categorial Grammar (Lambek, 1958), it is not always immediate to realize how standard logical systems like Multi-modal Categorial Type Logics (MCTL) (Moortgat, 1997) actually embody this property. In this paper, we focus on the basic system NL (Lambek, 1961) and its extension with unary modalities NL() (Moortgat, 1996), and we spell things out by means of Display Calculi (DC) (Belnap, 1982; Goré, 1998). The use of structural operators in DC permits (...) a sharp distinction between the core properties we want to impose on the logical system and the way these properties are projected into the logical operators. We will show how we can obtain Lambek residuated triple \, / and of binary operators, and how the operators and introduced by Moortgat (1996) are indeed their unary counterpart.In the second part of the paper we turn to other important algebraic properties which are usually investigated in conjunction with residuation (Birkhoff, 1967): Galois and dual Galois connections. Again, DC let us readily define logical calculi capturing them. We also provide preliminary ideas on how to use these new operators when modeling linguistic phenomena. (shrink)

[1]. Recently renewed interest in non transformational approaches to syntax [2] suggests that it might be well to take another look at categorial grammars, since they seem to have been neglected largely because they had been shown to be equivalent to context free phrase structure grammars in weak generative capacity and it was believed that such grammars were incapable of describing natural languages in a natural way. It is my purpose here to sketch a theory of (...) grammar which represents a generalization of categorial grammars of the sort invented by Ajdukiewicz (1935) and further elaborated especially by Bar Hillel (1953) and Lambek (1961). (shrink)

Second-order abstract categorial grammars (de Groote in Association for computational linguistics, 39th annual meeting and 10th conference of the European chapter, proceedings of the conference, pp. 148–155, 2001) and hyperedge replacement grammars (Bauderon and Courcelle in Math Syst Theory 20:83–127, 1987; Habel and Kreowski in STACS 87: 4th Annual symposium on theoretical aspects of computer science. Lecture notes in computer science, vol 247, Springer, Berlin, pp 207–219, 1987) are two natural ways of generalizing “context-free” grammar formalisms for (...) string and tree languages. It is known that the string generating power of both formalisms is equivalent to (non-erasing) multiple context-free grammars (Seki et al. in Theor Comput Sci 88:191–229, 1991) or linear context-free rewriting systems (Weir in Characterizing mildly context-sensitive grammar formalisms, University of Pennsylvania, 1988). In this paper, we give a simple, direct proof of the fact that second-order ACGs are simulated by hyperedge replacement grammars, which implies that the string and tree generating power of the former is included in that of the latter. The normal form for tree-generating hyperedge replacement grammars given by Engelfriet and Maneth (Graph transformation. Lecture notes in computer science, vol 1764. Springer, Berlin, pp 15–29, 2000) can then be used to show that the tree generating power of second-order ACGs is exactly the same as that of hyperedge replacement grammars. (shrink)

Discontinuity refers to the character of many natural language constructions wherein signs differ markedly in their prosodic and semantic forms. As such it presents interesting demands on monostratal computational formalisms which aspire to descriptive adequacy. Pied piping, in particular, is argued by Pollard (1988) to motivate phrase structure-style feature percolation. In the context of categorial grammar, Bach (1981, 1984), Moortgat (1988, 1990, 1991) and others have sought to provide categorial operators suited to discontinuity. These attempts encounter certain difficulties (...) with respect to model theory and/or proof theory, difficulties which the current proposals are intended to resolve.Lambek calculus is complete for interpretation byresiduation with respect to the adjunction operation of groupoid algebras (Buszkowski 1986). In Moortgat and Morrill (1991) it is shown how to give calculi for families of categorial operators, each defined by residuation with respect to an operation of prosodic adjunction (associative, non-associative, or with interactive axioms). The present paper treats discontinuity in this way, by residuation with respect to three adjunctions: + (associative), (.,.) (split-point marking), andW (wrapping) related by the equations 1+s 2+s 3=(s 1,s 3)Ws 2. We show how the resulting methods apply to discontinuous functors, quantifier scope and quantifier scope ambiguity, pied piping, and subject and object antecedent reflexivisation. (shrink)

This paper studies the relation between some extensions of the non-associative Lambek Calculus NL and their interpretation in tree models (free groupoids). We give various examples of sequents that are valid in tree models, but not derivable in NL. We argue why tree models may not be axiomatizable if we add finitely many derivation rules to NL, and proceed to consider labeled calculi instead.We define two labeled categorial calculi, and prove soundness and completeness for interpretations that are almost the (...) intended one, namely for tree models where some branches of some trees may be resp. all branches of all trees must be infinitely extending. Extrapolating from the experiences in our quite simple systems, we briefly discuss some problems involved with the introduction of labels in categorial grammar, and argue that many of the basic questions are not yet understood. (shrink)

The equivalence of (classical) categorial grammars and context-free grammars, proved by Gaifman [4], is a very basic result of the theory of formal grammars (an essentially equivalent result is known as the Greibach normal form theorem [1], [14]). We analyse the contents of Gaifman's theorem within the framework of structure and type transformations. We give a new proof of this theorem which relies on the algebra of phrase structures and exhibit a possibility to justify the key (...) construction used in Gaifman's proof by means of the Lambek calculus of syntactic types [15]. (shrink)

Geach’s rich paper ‘A Program for Syntax’ introduced many ideas into the arena of categorial grammar, not all of which have been given the attention they warrant in the thirty years since its first publication. Rather surprisingly, one of our findings (Section 3 below) is that the paper not only does not contain a statement of what has widely come to be known as “Geach’s Rule”, but in fact presents considerations which are inimical to the adoption of the rule (...) in question. With regard to at least some amongst the numerous other points extracted here from Geach’s discussion, we shall not be able to reach so definitive a conclusion, and content ourselves with giving the issues an airing. (shrink)

The article presents the first results we have obtained studying natural reasoning from a proof-theoretic perspective. In particular we focus our attention on monotonic reasoning. Our system consists of two parts: (i) A Formal Grammar – a multimodal version of classical Categorial Grammar – which while syntactically analysing linguistic expressions given as input, computes semantic information (In particular information about the monotonicity properties of the components of the input string are displayed.); (ii) A simple Natural Logic which derives (monotonicity) (...) inferences using as vehicle the parsed output. The monotonicity markers assigned in the lexicon are propagated through the proofs via a combination of the structural and the logical rules for the unary operators of Multimodal Categorial Grammar (MMCG) [Moo97]. We have chosen to work with an expressive ‘grammar logic’, in order to avoid the use of extra-logical marking devices and extra-logical structural reasoning. Having MMCG as parser, our system is able to make the derivations simply within the logic. This new approach makes the implementation of the theory an easier task. We have implemented the theoretical results, so far obtained, using Grail, a theorem prover for Categorial Grammar Logics [Moo98]. (shrink)

The ‘syntax’ and ‘combinatorics’ of my title are what Curry (1961) referred to as phenogrammatics and tectogrammatics respectively. Tectogrammatics is concerned with the abstract combinatorial structure of the grammar and directly informs semantics, while phenogrammatics deals with concrete operations on syntactic data structures such as trees or strings. In a series of previous papers (Muskens, 2001a; Muskens, 2001b; Muskens, 2003) I have argued for an architecture of the grammar in which finite sequences of lambda terms are the basic data structures, (...) pairs of terms syntax, semantics for example. These sequences then combine with the help of simple generalizations of the usual abstraction and application operations. This theory, which I call Lambda Grammars and which is closely related to the independently formulated theory of Abstract Categorial Grammars (de Groote, 2001; de Groote, 2002), in fact is an implementation of Curry’s ideas: the level of tectogrammar is encoded by the sequences of lambda-terms and their ways of combination, while the syntactic terms in those sequences constitute the phenogrammatical level. In de Groote’s formulation of the theory, tectogrammar is the level of abstract terms, while phenogrammar is the level of object terms. (shrink)

It is proved that for any k, the class of classical categorial grammars that assign at most k types to each symbol in the alphabet is learnable, in the Gold (1967) sense of identification in the limit from positive data. The proof crucially relies on the fact that the concept known as finite elasticity in the inductive inference literature is preserved under the inverse image of a finite-valued relation. The learning algorithm presented here incorporates Buszkowski and Penn's (1990) (...) algorithm for determining categorial grammars from input consisting of functor-argument structures. (shrink)

This article summarizes the main ideas for formalizing categorial languages genrated by classical categorial grammar originated by K. Ajdukiewicz [1935,1960].This formalization is presented in detail in the author's monographs in Polish "Teorie Języków Syntaktycznie Kategorialnych" ("Theories of Syntactically Categorical Languages"), PWN, Warszawa-Wrocław 1985 and in English "Theory of Language Syntax, Categorial Approach", Kluwer Academic Publishers, Boston-London-Dordrecht 1991.

A version of strong negation is introduced into Categorial Grammar. The resulting syntactic calculi turn out to be systems of connexive logic.

This paper solves a natural but still open question: can abstract categorial grammars (ACGs) respresent usual categorial grammars? Despite their name and their claim to be a unifying framework, up to now there was no faithful representation of usual categorial grammars in ACGs. This paper shows that Non-Associative Lambek grammars as well as their derivations can be defined using ACGs of order two. To conclude, the outcome of such a representation are discussed.

In [2], Bar-Hillel, Gaifman, and Shamir prove that the simple phrase structure grammars (SPGs) defined by Chomsky are equivalent in a certain sense to Bar-Hillel's bidirectional categorial grammars (BCGs). On the other hand, Cohen [3] proves the equivalence of the latter ones to what the calls free categorial grammars (FCGs). They are closely related to Lambek's syntactic calculus which, in turn, is based on the idea due to Ajdukiewicz [1]. For the reasons which will be (...) discussed in the last section, Cohen's proof seems to be at least incomplete. This paper yields a direct proof of the equivalence ofFCGs andSPGs. (shrink)

Two axiomatizations of the nonassociative and commutative Lambek syntactic calculus are given and their equivalence is proved. The first axiomatization employs Permutation as the only structural rule, the second one, with no Permutation rule, employs only unidirectional types. It is also shown that in the case of the Ajdukiewicz calculus an analogous equivalence is valid only in the case of a restricted set of formulas. Unidirectional axiomatizations are employed in order to establish the generative power of categorial grammars (...) based on the nonassociative and commutative Lambek calculus with product. Those grammars produce CF-languages of finite degree generated by CF-grammars closed with respect to permutations. (shrink)

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci it is shown that the basic properties known as Residuation laws can be characterized in the framework of Cyclic Multiplicative Linear Logic, a purely non-commutative fragment of Linear Logic. We present a summary of this result and, pursuing this line of investigation, we analyze a well-known set of categorial grammar laws: Monotonicity, Application, Expansion, Type-raising, Composition, (...) Geach laws and Switching laws. (shrink)

Abstract -/- Quantifier domain restriction (QDR) and two versions of nominal restriction (NR) are implemented as restrictions that depend on a previously introduced interpreter and interpretation time in a two-dimensional semantic framework on the basis of simple type theory and categorial grammar. Against Stanley (2002) it is argued that a suitable version of QDR can deal with superlatives like tallest. However, it is shown that NR is needed to account for utterances when the speaker intends to convey different restrictions (...) for multiple uses of the same quantifying determiner. We argue that NR generally fares better with such examples but also observe that examples like Every sailor waves at every sailor might be pragmatically anomalous. An account of contextual domain restriction is proposed that (i) excludes these anomalous readings (but it is shown how they could be included), (ii) makes it possible to express different contextual domain restrictions as long-range dependencies on an interpreter and an interpretation time, and (iii) additionally models restrictions based on locative constructions as general mereological constraints introduced by shifting the index. (shrink)

Classical categorial grammars are the grammars introduced by Ajdukiewicz [1] and formalized by Bar-Hillel [2], Bar-Hillel et al. [3]. In [3] there is proved the weak equivalence of CCG’s and context-free grammars [6]. In this note we characterize the strong generative capacity of finite and rigid CCG’s, i.e. their capacity of structure generation. These results are more completely discussed in [4], [5].

We describe a categorial system (PPTS) based on partial proof trees(PPTs) as the building blocks of the system. The PPTs are obtained byunfolding the arguments of the type that would be associated with a lexicalitem in a simple categorial grammar. The PPTs are the basic types in thesystem and a derivation proceeds by combining PPTs together. We describe theconstruction of the finite set of basic PPTs and the operations forcombining them. PPTS can be viewed as a categorial (...) system incorporating someof the key insights of lexicalized tree adjoining grammar, namely the notionof an extended domain of locality and the consequent factoring of recursionfrom the domain of dependencies. PPTS therefore inherits the linguistic andcomputational properties of that system, and so can be viewed as a middleground between a categorial grammar and a phrase structure grammar. We alsodiscuss the relationship between PPTS, natural deduction, and linear logicproof-nets, and argue that natural deduction rather than a proof-net systemis more appropriate for the construction of the PPTs. We also discuss howthe use of PPTs allows us to localize the management of resources, therebyfreeing us from this management as the PPTs are combined. (shrink)

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case (...) of linear context-free rewriting systems. This suggest the possibility of defining an Abstract Categorial Hierarchy. (shrink)

In this paper we show that the membership problem for second order non-linear Abstract Categorial Grammars is decidable. A consequence of that result is that Montague-like semantics yield to a decidable text generation problem. Furthermore the proof we propose is based on a new tool, Higher Order Intersection Signatures, which grasps statically dynamic properties of λ-terms and presents an interest in its own.

We show how to encode context-free string grammars, linear context-free tree grammars, and linear context-free rewriting systems as Abstract Categorial Grammars. These three encodings share the same constructs, the only difference being the interpretation of the composition of the production rules. It is interpreted as a first-order operation in the case of context-free string grammars, as a second-order operation in the case of linear context-free tree grammars, and as a third-order operation in the case (...) of linear context-free rewriting systems. This suggest the possibility of defining an Abstract Categorial Hierarchy. (shrink)