Mati Pentus

Lambek grammars are context free

Basic categorial grammars are the context-free ones. Another kind of categorial grammars was introduced by J. Lambek (1958). These grammars are based on a syntactic calculus, known as the Lambek calculus. Chomsky (1963) conjectured that these grammars are also equivalent to context-free ones. Every basic categorial grammar (and thus every context-free grammar) is equivalent to a Lambek grammar. Conversely, some special kinds of Lambek grammars are context-free. These grammars use weakly unidirectional types, or types of order at most two. The main result of this paper says that Lambek grammars generate only context-free languages. Thus they are equivalent to context-free grammars and also to basic categorial grammars. The Chomsky conjecture, that all languages recognized by the Lambek calculus are context-free, is thus proved.<<ETX>>

Lambek calculus is NP-complete

We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L* the derivability problem is NP-complete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NP-complete.

Product-Free Lambek Calculus and Context-Free Grammars

In this paper we prove the Chomsky Conjecture (all languages recognized by the Lambek calculus are context-free) for both the full Lambek calculus and its product-free fragment.For the latter case we present a construction of context-free grammars involving only product-free types.

Models for the Lambek calculus

Abstract We prove that the Lambek calculus is complete w.r.t. L-models, i.e., free semigroup models. We also prove the completeness w.r.t. relativized relational models over the natural linear order of integers.

Language completeness of the Lambek calculus

Proves that the Lambek calculus (J. Lambek, American Math. Monthly, vol. 65, no. 3, pp. 154-170, 1958), which is essentially a subsystem of noncommutative linear logic, is complete with respect to L-models, i.e. free semigroup models.<<ETX>>

The conjoinability relation in Lambek calculus and linear logic

AbstractIn 1958 J. Lambek introduced a calculusL of syntactic types and defined an equivalence relation on types: “x≡ y means that there exists a sequence x=x1,...,xn=y (n ≥ 1), such thatxi →xi+1 or xi+ →xi (1 ≤i ≤ n)”. He pointed out thatx ≡y if and only if there is joinz such thatx →z andy →z. This paper gives an effective characterization of this equivalence for the Lambeck calculiL andLP, and for the multiplicative fragments of Girards and Yetters linear logics. Moreover, for the non-directed Lambek calculusLP and the multiplicative fragment of Girards linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are.

Provability, complexity, grammars

Classification of propositional provability logics by L. D. Beklemishev Lambek calculus and formal grammars by M. Pentus Relativizability in complexity theory by N. K. Vereshchagin.

Equivalence of Multiplicative Fragments of Cyclic Linear Logic and Noncommutative Linear Logic

In this paper we consider two associative noncommutative analogs of Girards linear logic. These are Abruscis noncommutative linear logic and Yetters cyclic linear logic.

The Monotone Lambek Calculus Is NP-Complete

We consider the Lambek calculus with the additional structural rule of monotonicity (weakening). We show that the derivability problem for this calculus is NP-complete (both for the full calculus and for the product-free fragment). The same holds for the variant that allows empty antecedents. To prove NP-hardness of the product-free fragment, we provide a mapping reduction from the classical satisfiability problem \( \textit{SAT} \). This reduction is similar to the one used by Yury Savateev in 2008 to prove NP-hardness (and hence NP-completeness) of the product-free Lambek calculus.