Categorial Grammar Research Articles

The Mereological Basis of Truthmaker Semantics

AbstractThis articles explores the mereological foundation of truthmaker semantics. Building upon Kit Fine’s abstract theory of part in Fine [J Philos 107(11):559–589, 2010], we engage in an exploration of the mereological assumptions that determine the construction of truthmaker semantics. Our approach yields semantics for a diverse range of logics, including substructural logics such as the associative Lambek calculus, as well as the logics of analytic containment. Furthermore, we elucidate the philosophical implications that arise from this pioneering approach.

Interpolation in Linear Logic and Related Systems

We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that the exhibited extensions of classical and intuitionistic linear logic all enjoy a guarded form of Craig interpolation. We also give continuum-many axiomatic extensions of classical linear logic without the deductive interpolation property.

Predykaty wyższego vs. pierwszego rzędu jako problem syntaktyczny

Predykaty wyższego vs. pierwszego rzędu jako problem syntaktyczny

Sequent Systems for Consequence Relations of Cyclic Linear Logics

Linear Logic is a versatile framework with diverse applications in computer science and mathematics. One intriguing fragment of Linear Logic is Multiplicative-Additive Linear Logic (MALL), which forms the exponential-free component of the larger framework. Modifying MALL, researchers have explored weaker logics such as Noncommutative MALL (Bilinear Logic, BL) and Cyclic MALL (CyMALL) to investigate variations in commutativity. In this paper, we focus on Cyclic Nonassociative Bilinear Logic (CyNBL), a variant that combines noncommutativity and nonassociativity. We introduce a sequent system for CyNBL, which includes an auxiliary system for incorporating nonlogical axioms. Notably, we establish the cut elimination property for CyNBL. Moreover, we establish the strong conservativeness of CyNBL over Full Nonassociative Lambek Calculus (FNL) without additive constants. The paper highlights that all proofs are constructed using syntactic methods, ensuring their constructive nature. We provide insights into constructing cut-free proofs and establishing a logical relationship between CyNBL and FNL.

On the unified representation of continuity and discontinuity and its neurocognitive grounding

Abstract Discontinuity in natural language is characterized by the linear disruption of a continuous string of linguistic expressions forming a constituent. While dependency relations in Dependency Grammar (DG) can capture discontinuity well, phrase-structure-based approaches such as Phrase Structure Grammar (PSG) face difficulty in accommodating discontinuity. Categorial Grammar (CG) has correspondences with PSG, although it can handle discontinuity, if equipped with wrapping operations. Given the existing literature on discontinuity in natural language, it appears that constituency relations of PSG, dependency relations of DG and functor-argument relations of CG are distinct and independent. Here, we argue for a unified representation achieved by taking into account fundamental representational principles of PSG, DG and CG. For simplicity, we show this by considering an embedded clause from Wan, spoken in Ivory Coast, as an illustrative case. The paper then attempts to explain, based on available empirical pieces of evidence, the plausible connections between the unified representation and the neurocognitive representation of continuity and discontinuity in natural language.

Natural Language Generation and Parsing as Heuristic Planning Problems

This paper formulates the problems of natural language generation and parsing as particular instances of the classical planning problem. It assumes the existence of a Categorial Grammar lexicon from which the preconditions and effects of available actions are obtained. A declarative formalization of heuristics for action selection is used to guide the search for solutions. Heuristics for mapping formulas in the description logic DL-Lite (R,and) into English sentences and backwards, and examples of application to Human Robot Interaction (HRI) are presented to illustrate the effectiveness of the approach.

Algebraic Proof Theory for LE-logics

In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [ 34 ] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\) , as well as for its extensions with analytic structural rules satisfying certain additional properties.

Relational Models for the Lambek Calculus with Intersection and Constants

We consider relational semantics (R-models) for the Lambek calculus extended with intersection and explicit constants for zero and unit. For its variant without constants and a restriction which disallows empty antecedents, Andreka and Mikulas (1994) prove strong completeness. We show that it fails without this restriction, but, on the other hand, prove weak completeness for non-standard interpretation of constants. For the standard interpretation, even weak completeness fails. The weak completeness result extends to an infinitary setting, for so-called iterative divisions (Kleene star under division). We also prove strong completeness results for product-free fragments.

Making first order linear logic a generating grammar

It is known that different categorial grammars have surface representation in a fragment of first order multiplicative linear logic (MLL1). We show that the fragment of interest is equivalent to the recently introduced extended tensor type calculus (ETTC). ETTC is a calculus of specific typed terms, which represent tuples of strings, more precisely bipartite graphs decorated with strings. Types are derived from linear logic formulas, and rules correspond to concrete operations on these string-labeled graphs, so that they can be conveniently visualized. This provides the above mentioned fragment of MLL1 that is relevant for language modeling not only with some alternative syntax and intuitive geometric representation, but also with an intrinsic deductive system, which has been absent. In this work we consider a non-trivial notationally enriched variation of the previously introduced ETTC, which allows more concise and transparent computations. We present both a cut-free sequent calculus and a natural deduction formalism.

Modal Information Logics: Axiomatizations and Decidability

The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021) on the Lambek Calculus.

Residuated Basic Logic

Residuated basic logic (RBL) is the logic of residuated basic algebras, which constitutes a conservative extension of basic propositional logic (BPL). The basic implication is a residual of a non-associative binary operator in RBL. The conservativity is shown by relational semantics. A Gentzen-style sequent calculus GRBL, which is an extension of the distributive full non-associative Lambek calculus, is established for residuated basic logic. The calculus GRBL admits the mix-elimination, subformula, and disjunction properties. Moreover, the class of all residuated basic algebras has the finite embeddability property. The consequence relation of GRBL is decidable.

Multimodality in the Hypergraph Lambek Calculus

Multimodality in the Hypergraph Lambek Calculus

Combinatory categorial grammars as generators of weighted forests

Combinatory Categorial Grammar (CCG) is an extension of categorial grammar that is well-established in computational linguistics. It is mildly context-sensitive, so it is efficiently parsable and reaches an expressiveness that is suitable for describing natural languages. Weighted CCG (wCCG) are introduced as a natural extension of CCG with weights taken from an arbitrary commutative semiring. Their expressive power is compared to other weighted formalisms with special emphasis on the weighted forests generated by wCCG since the ability to express the underlying syntactic structure of an input sentence is a vital feature of CCG in the area of natural language processing. Building on recent results for the expressivity in the unweighted setting, the corresponding results are derived for the weighted setting for any commutative semiring. More precisely, the weighted forests generatable by wCCG are also generatable by weighted simple monadic context-free tree grammar (wsCFTG). If the rule system is restricted to application rules and composition rules of first degree, then the generatable weighted forests are exactly the regular weighted forests. Finally, when only application rules are allowed, then a proper subset of the regular weighted forests is generatable.

A role of functional morphemes in Korean categorial grammars

Abstract This study discusses a role of functional morphemes in Korean categorial grammars, providing the reviews of various types of Korean categorial grammars that have never been conducted so far, notwithstanding many previous studies on them. Previous work has presented different morphological segmentation because of Korean’s agglutinative characteristics, implying that Korean words may contain a different segmentation sequence of morphemes. We focus on functional morphemes in Korean categorial grammars, which have been explored in different ways by previous work. We present detailed analyses for postpositions and verbal endings in categorial grammars, insisting that the functional morphemes in Korean should be treated as part of a word, with the result that their categories do not require to be assigned individually in a syntactic level, and also that it would be more efficient to assign the syntactic categories on the fully inflected lexical word derived by the lexical rule of the morphological processes in the lexicon.

Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality

We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L∗, a modality that allows for the use of limited editions of contraction and permutation in the logic. Lambek Calculus has been introduced to analyse syntax of natural language and the linguistic motivation behind this modality is to extend the domain of the applicability of the calculus to fragments which witness the discontinuity phenomena. The categorical part of the semantics is a monoidal biclosed category with a !-functor, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the !-functor. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L∗: the derivation of a phrase with a parasitic gap. The efficacy of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrase one, using BERT, Word2Vec, and FastText vectors and relational tensors.

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

We study how to relate well-known hypergraph grammars based on the double pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL (called HL-grammars). It turns out that DPO rules can be naturally encoded by types of HL using methods similar to those used by Kanazawa for multiplicative-exponential linear logic. In order to generalize his reasonings we extend the hypergraph Lambek calculus by adding the exponential modality, which results in a new calculus HMEL0; then we prove that any DPO grammar can be converted into an equivalent HMEL0-grammar. We also define the conjunctive Kleene star, which behaves similarly to this exponential modality, and establish a similar result. If we add neither the exponential modality nor the conjunctive Kleene star to HL, then we can still use the same encoding and show that any DPO grammar with a linear restriction on the length of derivations can be converted into an equivalent HL-grammar.

From Combinatory Categorial Grammar to the Calculus of Constructions: A Review of ccg2lambda.

The following is an overview of the concepts and systems necessary to build an understanding of the framework from ccg2lambda: A Compositional Semantics System, which produces logical semantic representations of natural language sentences. These representations are used in tandem with a theorem prover to conduct inferences. But, to the reader unfamiliar with the various tools within computer science, linguistics, and mathematics that were used to construct ccg2lambda and its dependencies, it can be difficult to understand the inner workings of the framework. Thus, an explication of these concepts could be useful for fostering further development. Concepts such as mathematical logic, combinatory categorial grammar, the calculus of constructions, and interactive theorem provers are covered, as well as the two systems C&C and ccg2lambda. A functional version of the ccg2lambda framework can be found at https://github.com/ajung202/ccg2lambda.

Linear Logic Properly Displayed

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e., closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic, and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic and applies the guidelines of the multi-type methodology in the design of display calculi.

Implementing Natural Language Inference for comparatives

This paper presents a computational framework for Natural Language Inference (NLI) using logic-based semantic representations and theorem-proving. We focus on logical inferences with comparatives and other related constructions in English, which are known for their structural complexity and difficulty in performing efficient reasoning. Using the so-called A-not-A analysis of comparatives, we implement a fully automated system to map various comparative constructions to semantic representations in typed first-order logic via Combinatory Categorial Grammar parsers and to prove entailment relations via a theorem prover. We evaluate the system on a variety of NLI benchmarks that contain challenging inferences, in comparison with other recent logic-based systems and neural NLI models.

Fragments of quasi-Nelson: residuation

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts.