Categorial Grammar Research Articles

A Geometrical Representation of the Basic Laws of Categorial Grammar

We present a geometrical analysis of the principles that lay at the basis of Categorial Grammar and of the Lambek Calculus. In Abrusci (On residuation, 2014) 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.

The Applicative Combinatory Categorial Analysis of Arabic

The Applicative Combinatory Categorial Grammar (ACCG), like the whole of the other categorial models conceptualizes the languages as organized systems of linguistic units (words, morphemes, lexemes, etc) of which some function as operators whereas others function as operands. The ACCG is a sufficiently flexible model to give an account of several languages (SVO, VSO, SOV, etc.) and several forms of organisation of the linguistic units of these languages (coordination, subordination, with backward modifier, etc). However, the research tasks were limited almost exclusively to European languages, in particular French, English, German, Dutch. A language like Arabic, in spite of the richness of the tradition of its schools of thought, which go back for several centuries, remains unexplored by this current, except some research works. In our article, we show how the model of Applicative Combinatory Categorial Grammar can give an account of certain forms of Arabic.

Parsing Linear Context-Free Rewriting Systems with Fast Matrix Multiplication

We describe a recognition algorithm for a subset of binary linear context-free rewriting systems (LCFRS) with running time O(nωd) where M(m) = O(mω) is the running time for m × m matrix multiplication and d is the “contact rank” of the LCFRS—the maximal number of combination and non-combination points that appear in the grammar rules. We also show that this algorithm can be used as a subroutine to obtain a recognition algorithm for general binary LCFRS with running time O(nωd+1). The currently best known ω is smaller than 2.38. Our result provides another proof for the best known result for parsing mildly context-sensitive formalisms such as combinatory categorial grammars, head grammars, linear indexed grammars, and tree-adjoining grammars, which can be parsed in time O(n4.76). It also shows that inversion transduction grammars can be parsed in time O(n5.76). In addition, binary LCFRS subsumes many other formalisms and types of grammars, for some of which we also improve the asymptotic complexity of parsing.

Command injection attacks, continuations, and the Lambek calculus

This paper shows connections between command injection attacks, continuations, and the Lambek calculus: certain command injections, such as the tautology attack on SQL, are shown to be a form of control effect that can be typed using the Lambek calculus, generalizing the double-negation typing of continuations. Lambek's syntactic calculus is a logic with two implicational connectives taking their arguments from the left and right, respectively. These connectives describe how strings interact with their left and right contexts when building up syntactic structures. The calculus is a form of propositional logic without structural rules, and so a forerunner of substructural logics like Linear Logic and Separation Logic.

Unified correspondence and proof theory for strict implication

The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.

FULL LAMBEK CALCULUS WITH CONTRACTION IS UNDECIDABLE

AbstractWe prove that the set of formulae provable in the full Lambek calculus with the structural rule of contraction is undecidable. In fact, we show that the positive fragment of this logic is undecidable.

The Lambek Calculus Extended with Intuitionistic Propositional Logic

We present sound and complete semantics and a sequent calculus for the Lambek calculus extended with intuitionistic propositional logic.

A Novel Approach by Injecting CCG Supertags into an Arabic–English Factored Translation Machine

This study addresses the integration and incorporation of rich additional information into the phrase-based approach, aptly called factored translation, which is an extension of phrase-based statistic machine translation (PBSMT). This approach was proven successful when translating English into a morphologically rich language. PBSMT represents the baseline of this work. We extend the phrase-based translation approach by integrating additional linguistic knowledge, namely part-of-speech (POS) tags, to create a factored model. The main contribution of this study is the creation of a new approach for Arabic–English translation via the injection of the factored model into Combinatory Categorial Grammar (CCG) supertags to form an integrated model (POS + CCG). The system was trained on a freely available multi-UN corpus on Arabic–English language pairs. Moses decoder, which is an open-source factored SMT system, was used to integrate these data into the target language model and the target side of the translation model. Results showed improvements to the BLEU automatic score via various high n-gram language models (LMs). The integration of the featured factors (POS + CCG) of the translation has been successfully tested. Overall, the 3-, 5-, 7-, and 9-g LM evaluation with BLEU scores proved that our integrated model performed better than PBSMT. Compared with three other models (PBSMT, POS, and CCG models), the integrated model improved the translation quality by 1.54, 1.29, and 0.21 %, respectively, over the 3-g LM.

Term satisfiability in FLew-algebras

FLew-algebras form the algebraic semantics of the full Lambek calculus with exchange and weakening. We investigate two relations, called satisfiability and positive satisfiability, between FLew-terms and FLew-algebras. For each FLew-algebra, the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory; we identify and investigate the complements as fragments of its universal theory. We offer characterizations of those algebras that (positively) satisfy just those terms that are satisfiable in the two-element Boolean algebra providing its semantics to classical propositional logic. In case of positive satisfiability, these algebras are just the nontrivial weakly contractive FLew-algebras. In case of satisfiability, we give a characterization by means of another property of the algebra, the existence of a two-element congruence. Further, we argue that (positive) satisfiability problems in FLew-algebras are computationally hard. Some previous results in the area of term satisfiability in MV-algebras or BL-algebras are thus brought to a common footing with known facts on satisfiability in Heyting algebras.

Dynamic epistemic logic in update logic

We show that dynamic epistemic logic (DEL) is a substructural logic and that it is an extension of the update logic introduced in the companion article [12]. We identify axioms and inference rules that completely characterize the DEL product update, and we provide a sequent calculus for DEL. Finally, we show that DEL with a finite number of atomic events is as expressive as epistemic logic. In parallel, we provide a sequent calculus for update logic which turns out to be a generalization of the non-associative Lambek calculus.

Displaying updates in logic

Abstract We generalize the language of substructural logics interpreted over the ternary relational semantics. This is motivated by our intention to capture within the logical framework of substructural logics various logic-based formalisms dealing with common sense reasoning and logical dynamics. This initiative is based on the key observation that an update can be represented abstractly by the ternary relation of the substructural framework: the first argument of the ternary relation represents an initial situation, the second an informative event and the third the resulting situation after the occurrence of the informative event. Refining the language of substructural logics may allow us to account for the fine-grained reasoning present in common sense reasoning. Thus, we introduce three triples of connectives that are interconnected by means of cyclic permutations. The usual fusion, implication and co-implication connectives form one of these triples. We define a cut-free display calculus for our language that generalizes the display calculus for modal logic. We do not resort to structural connectives for truth constants and we show how we can obtain our display rules using Gaggle Theory. We prove the soundness and strong completeness of our display calculus via a Henkin-style construction. Using correspondence results from substructural logics, we also obtain sound and complete display calculi for a wide variety of classical and substructural logics. Then, we define dual substructural connectives and we provide a display calculus for our language extended with these dual connectives. Finally, we focus on the specific case of bi-intuitionistic logic for which we provide a novel sound and strongly complete display calculus. In a companion article (Aucher, 2016, J. Logic Comput.), we provide a sequent calculus for update logic that generalizes the non-associative Lambek calculus and a sequent calculus for dynamic epistemic logic that extends this sequent calculus for update logic.

Grammatical function and Semantic Representation of ‘zai’ in Chinese by categorial grammar

Grammatical function and Semantic Representation of ‘zai’ in Chinese by categorial grammar

From linguistics to deontic logic via category theory

The present paper aims to bridge the gap between deontic logic, categorial grammar and category theory. We propose to analyze Forrester's (1984) paradox through the framework of Lambek's (1958) syntactic calculus. We first recall the definition of the syntactic calculus and then explain how Lambek (1988) defines it within the framework of category theory. Then, we briefly present Forrester's paradox in conjunction with standard deontic logic, showing that this paradox contains some features that reflect many problems within the literature. Finally, we analyze Forrester's paradox within the framework of the syntactic calculus and we show how a typed syntax can provide conceptual insight regarding some of the problems that deontic logic faces.

Against ellipsis: arguments for the direct licensing of ‘noncanonical’ coordinations

Categorial grammar is well-known for its elegant analysis of coordination enabled by the flexible notion of constituency it entertains. However, to date, no systematic study exists that examines whether this analysis has any obvious empirical advantage over alternative analyses of nonconstituent coordination available in phrase structure-based theories of syntax. This paper attempts precisely such a comparison. We compare the direct constituent coordination analysis of non-canonical coordinations (right-node raising, dependent cluster coordination and Gapping) in categorial grammar with an ellipsis-based analysis of the same phenomena in the recent HPSG literature. We provide a set of empirical evidence, consisting of cases in which non-canonical coordinations interact with scopal operators of various sorts, which systematically falsifies the predictions of the latter, ‘linearization-based’ ellipsis approach to coordination. We propose an alternative analysis in a variant of categorial grammar called Hybrid Type-Logical Categorial Grammar. The proposed framework builds on both the Lambek-inspired variants of categorial grammar and a more recent line of work modelling word order via a lambda calculus for the prosodic component. The flexible syntax–semantics interface of this framework straightforwardly captures the interactions between non-canonical coordinations and scopal expressions, demonstrating the broader empirical payoff of the direct constituent coordination analysis of non-canonical coordinations pioneered by Steedman (Language 61(3):523–568, 1985; Linguist Philos 13(2):207–263, 1990) and Dowty (Categorial grammars and natural language structures, 1988) hitherto not explicitly recognized in the literature.

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 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 involved. Here we approach multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalisation.

A note on the substructural hierarchy

We prove that all axiomatic extensions of the full Lambek calculus with exchange can be axiomatized by formulas on the level of the substructural hierarchy.

The syntax-semantics interface of ‘respective’ predication: a unified analysis in Hybrid Type-Logical Categorial Grammar

This paper proposes a unified analysis of the ‘respective’ readings of plural and conjoined expressions, the internal readings of symmetrical predicates such as same and different, and the summative readings of expressions such as a total of $10000. These expressions pose significant challenges to compositional semantics, and have been studied extensively in the literature. However, almost all previous studies focus exclusively on one of these phenomena, and the close parallels and interactions that they exhibit have been mostly overlooked to date. We point out two key properties common to these phenomena: (i) they target all types of coordination, including nonconstituent coordination such as Right-Node Raising and Dependent Cluster Coordination; (ii) the three phenomena all exhibit multiple dependency, both by themselves and with respect to each other. These two parallels suggest that one and the same mechanism is at the core of their semantics. Building on this intuition, we propose a unified analysis of these phenomena, in which the meanings of expressions involving coordination are formally modelled as multisets, that is, sets that allow for duplicate occurrences of identical elements. The analysis is couched in Hybrid Type-Logical Categorial Grammar. The flexible syntax-semantics interface of this framework enables an analysis of ‘respective’ readings and related phenomena which, for the first time in the literature, yields a simple and principled solution for both the interactions with nonconstituent coordination and the multiple dependency noted above.

Gapping as hypothetical reasoning

The scope anomaly observed in sentences like Mrs. J can’t live in Boston and Mr. J in LA (¬◊>∧) and No dog eats Whiskas or cat Alpo (¬∃>∨) is known to pose difficult challenges to many analyses of Gapping. We provide new arguments, based on both the basic syntactic patterns of Gapping and standard constituency tests, that the so-called ‘low VP coordination analysis’—the only extant analysis of Gapping in contemporary syntactic theories which accounts for this scope anomaly—is empirically untenable. We propose an explicit alternative analysis of Gapping in Hybrid Type-Logical Categorial Grammar, a variant of categorial grammar which builds on both the Lambek-inspired tradition and a more recent line of work modelling word order via a lambda calculus for the prosodic component. The flexible syntax-semantics interface of this framework enables us to characterize Gapping as an instance of like-category coordination, via a crucial use of the notion of hypothetical reasoning. This analysis of the basic syntax of Gapping is shown to interact with independently motivated analyses of scopal operators to immediately yield their apparently anomalous scopal properties in Gapping, offering, for the first time in the literature, a conceptually simple and empirically adequate solution for the notorious scope anomaly in Gapping.

On the Learnability of Classes of Minimal Grammars With Respect to Some Linear Preordering Relations

Abstract In this paper we prove the learnability of the classes of minimal grammars with respect to some linear preorderings on the set of categorial grammars. We give some examples of the linear preordering relations on the set of categorial grammars satisfying necessary conditions.

Discriminative Syntax-Based Word Ordering for Text Generation

Word ordering is a fundamental problem in text generation. In this article, we study word ordering using a syntax-based approach and a discriminative model. Two grammar formalisms are considered: Combinatory Categorial Grammar (CCG) and dependency grammar. Given the search for a likely string and syntactic analysis, the search space is massive, making discriminative training challenging. We develop a learning-guided search framework, based on best-first search, and investigate several alternative training algorithms.The framework we present is flexible in that it allows constraints to be imposed on output word orders. To demonstrate this flexibility, a variety of input conditions are considered. First, we investigate a “pure” word-ordering task in which the input is a multi-set of words, and the task is to order them into a grammatical and fluent sentence. This task has been tackled previously, and we report improved performance over existing systems on a standard Wall Street Journal test set. Second, we tackle the same reordering problem, but with a variety of input conditions, from the bare case with no dependencies or POS tags specified, to the extreme case where all POS tags and unordered, unlabeled dependencies are provided as input (and various conditions in between). When applied to the NLG 2011 shared task, our system gives competitive results compared with the best-performing systems, which provide a further demonstration of the practical utility of our system.