Conjunctive categorial grammars and Lambek grammars with additives

TL;DR

This work introduces conjunctive categorial grammars by adding a conjunction operator to basic categorial grammars and proves their expressive equivalence with conjunctive grammars for languages without $ε$. It establishes a tight embedding of these grammars into the Lambek calculus with additives ($MALC$), and shows that $MALC$ can describe an $NP$-complete language, indicating a strict power beyond conjunctive grammars under standard complexity assumptions. The authors also address the empty-string case using $MALC^*$, derive a disjunction-only embedding (no conjunction) from conjunctive grammars, and discuss distributivity’s impact on these embeddings. Overall, the paper clarifies the relationships among conjunctive grammars, conjunctive categorial grammars, and $MALC$ (and its distributive/disjunctive variants), and raises open questions about separations, language bounds, and potential PSPACE-complete cases.

Abstract

A new family of categorial grammars is proposed, defined by enriching basic categorial grammars with a conjunction operation. It is proved that the formalism obtained in this way has the same expressive power as conjunctive grammars, that is, context-free grammars enhanced with conjunction. It is also shown that categorial grammars with conjunction can be naturally embedded into the Lambek calculus with conjunction and disjunction operations. This further implies that a certain NP-complete set can be defined in the Lambek calculus with conjunction. We also show how to handle some subtle issues connected with the empty string. Finally, we prove that a language generated by a conjunctive grammar can be described by a Lambek grammar with disjunction (but without conjunction).

Figures (3)

• Figure 4: Axioms and inference rules in the Lambek Calculus (for all $A, B \in \mathbf{Cat}$ and $\Gamma, \Pi, \Delta \in \mathbf{Cat}^*$). For $(\to\mathop{/})$ and $(\to\mathop{\backslash})$, in the case of $\mathbf{L}$, $\Pi$ is required to be non-empty
• Figure 5: Rules for conjunction and disjunction
• Figure 6: Axioms and Rules of $\mathbf{MACLL}$

Theorems & Definitions (42)

• Example 1
• Example 2
• Theorem 1: Bar-Hillel et al. BGS1960
• Example 3
• Example 4
• Lemma 1
• proof
• Theorem 1
• Lemma 2: Okhotin and Reitwießner OkhotinReitwiessner
• Theorem 3: Okhotin and Reitwießner OkhotinReitwiessner