On the closure properties of linear conjunctive languages

AbstractLinear conjunctive grammars are conjunctive grammars in which the body of each conjunct contains no more than a single nonterminal symbol. They can at the same time be thought of as a special case of conjunctive grammars and as a generalization of linear context-free grammars that provides an explicit intersection operation.Although the set of languages generated by these grammars is known to include many important noncontext-free languages, linear conjunctive languages are still all square-time, and several practical algorithms have been devised to handle them, which makes this class of grammars quite suitable for use in applications.In this paper we investigate the closure properties of the language family generated by linear conjunctive grammars; the main result is its closure under complement, which implies that it is closed under all set-theoretic operations. We also consider several cases in which the concatenation of two linear conjunctive languages is certain to be linear conjunctive. In addition, it is demonstrated that linear conjunctive languages are closed under quotient with finite languages, not closed under quotient with regular languages, and not closed under ε-free homomorphism

Grammars with two-sided contexts

In a recent paper (M. Barash, A. Okhotin, "Defining contexts in context-free grammars", LATA 2012), the authors introduced an extension of the context-free grammars equipped with an operator for referring to the left context of the substring being defined. This paper proposes a more general model, in which context specifications may be two-sided, that is, both the left and the right contexts can be specified by the corresponding operators. The paper gives the definitions and establishes the basic theory of such grammars, leading to a normal form and a parsing algorithm working in time O(n^4), where n is the length of the input string.Comment: In Proceedings AFL 2014, arXiv:1405.527

Representing a P-complete problem by small trellis automata

A restricted case of the Circuit Value Problem known as the Sequential NOR Circuit Value Problem was recently used to obtain very succinct examples of conjunctive grammars, Boolean grammars and language equations representing P-complete languages (Okhotin, http://dx.doi.org/10.1007/978-3-540-74593-8_23 "A simple P-complete problem and its representations by language equations", MCU 2007). In this paper, a new encoding of the same problem is proposed, and a trellis automaton (one-way real-time cellular automaton) with 11 states solving this problem is constructed

On equations over sets of integers

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in T} and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.Comment: 12 apges, 0 figure

Conjunctive Grammars, Cellular Automata and Logic

The expressive power of the class Conj of conjunctive languages, i.e. languages generated by the conjunctive grammars of Okhotin, is largely unknown, while its restriction LinConj to linear conjunctive grammars equals the class of languages recognized by real-time one-dimensional one-way cellular automata. We prove two weakened versions of the open question Conj ?? RealTime1CA, where RealTime1CA is the class of languages recognized by real-time one-dimensional two-way cellular automata: 1) it is true for unary languages; 2) Conj ? RealTime2OCA, i.e. any conjunctive language is recognized by a real-time two-dimensional one-way cellular automaton. Interestingly, we express the rules of a conjunctive grammar in two Horn logics, which exactly characterize the complexity classes RealTime1CA and RealTime2OCA

Parsing Unary Boolean Grammars Using Online Convolution

In contrast to context-free grammars, the extension of these grammars by explicit conjunction, the so-called conjunctive grammars can generate (quite complicated) non-regular languages over a single-letter alphabet (DLT 2007). Given these expressibility results, we study the parsability of Boolean grammars, an extension of context-free grammars by conjunction and negation, over a unary alphabet and show that they can be parsed in time O(|G| log^2(n) M(n)) where M(n) is the time to multiply two n-bit integers. This multiplication algorithm is transformed into a convolution algorithm which in turn is converted to an online convolution algorithm which is used for the parsing

Cotransforming Grammars with Shared Packed Parse Forests

SPPF (shared packed parse forest) is the best known graph representation of a parse forest (family of related parse trees) used in parsing with ambiguous/conjunctive grammars. Systematic general purpose transformations of SPPFs have never been investigated and are considered to be an open problem in software language engineering. In this paper, we motivate the necessity of having a transformation operator suite for SPPFs and extend the state of the art grammar transformation operator suite to metamodel/model (grammar/graph) cotransformations