[Paper Review] Some Subclasses of Linear Languages based on Nondeterministic Linear Automata

Benjamín Bedregal|arXiv (Cornell University)|Nov 30, 2016

semigroups and automata theory26 references3 citations

TL;DR

This paper introduces deterministic linear automata (DLA) and even linear automata (ELA) as natural, intrinsic automata models for deterministic and even linear languages, respectively, within the framework of $λ$-nondeterministic linear automata. It proves that the class of languages accepted by DLA properly contains previously proposed classes of deterministic linear languages and establishes an infinite hierarchy from DLA to the full class of linear languages.

ABSTRACT

In this paper we consider the class of lambda-nondeterministic linear automata as a model of the class of linear languages. As usual in other automata models, lambda-moves do not increase the acceptance power. The main contribution of this paper is to introduce the deterministic linear automata and even linear automata, i.e. the natural restriction of nondeterministic linear automata for the deterministic and even linear language classes, respectively. In particular, there are different, but not equivalents, proposals for the class of "deterministic" linear languages. We proved here that the class of languages accepted by the deterministic linear automata are not contained in any of the these classes and in fact they properly contain these classes. Another, contribution is the generation of an infinite hierarchy of formal languages, going from the class of languages accepted by deterministic linear automata and achieved, in the limit, the class of linear languages.

Motivation & Objective

• To define a natural, intrinsic automata model for deterministic linear languages, addressing limitations of existing models that rely on external input preprocessing or complex mechanisms.
• To introduce even linear automata (ELA) as a counterpart model for even linear languages, ensuring a direct correspondence with even linear grammars.
• To demonstrate that the class of languages accepted by deterministic linear automata (DLin) properly contains previously proposed classes of deterministic linear languages.
• To establish an infinite hierarchy of formal language classes extending from deterministic linear languages to the full class of linear languages.
• To provide a characterization of $λ$-nondeterministic linear automata that are equivalent to deterministic linear automata, enabling conversion and analysis of such automata.

Proposed method

• Proposes $λ$-nondeterministic linear automata (nla) as a foundational model, where $λ$-moves do not increase acceptance power, ensuring simplicity and intrinsic operation.
• Defines deterministic linear automata (DLA) as a restriction of nla, where transitions are uniquely determined at each step, ensuring determinism.
• Introduces even linear automata (ELA) as a subclass of nla whose transition graph is bipartite between left and right states, mirroring the structure of even linear grammars.
• Establishes a normal form for linear grammars to simplify proofs and enable direct construction of equivalent automata.
• Applies a transformation algorithm to convert linear grammars in normal form into equivalent nla, and vice versa, proving equivalence between ELA and even linear grammars.
• Uses the hierarchy of degree of explicit nondeterminism to generate an infinite chain of language classes from DLA to the full class of linear languages.

Experimental results

Research questions

• RQ1 How can a deterministic automaton model for linear languages be defined that is both intrinsic and simpler than existing models?
• RQ2 Does the class of languages accepted by deterministic linear automata (DLin) properly contain previously proposed classes of deterministic linear languages?
• RQ3 Can an intrinsic automata model for even linear languages be constructed that directly corresponds to even linear grammars?
• RQ4 Is there an infinite hierarchy of formal language classes between the class of deterministic linear languages and the full class of linear languages?
• RQ5 What is the relationship between the class of languages accepted by deterministic linear automata and the class of linear languages that are also deterministic context-free?

Key findings

• The class of languages accepted by deterministic linear automata (DLin) properly contains all previously proposed classes of deterministic linear languages, including those defined by de la Higuera and Oncina.
• DLin is not a proper subset of DCFL ∩ Lin, and the paper conjectures that DCFL ∩ Lin ⊂ DLin, suggesting DLin is broader than the intersection of deterministic context-free and linear languages.
• Even linear automata (ELA) are shown to be equivalent in expressive power to even linear grammars, with a constructive proof via transformation algorithms.
• An infinite hierarchy of formal language classes is established, starting from DLin and approaching the full class of linear languages, based on the degree of explicit nondeterminism.
• A characterization is provided for $λ$-nondeterministic linear automata that are equivalent to deterministic linear automata, enabling conversion and analysis of such automata.
• The model of $λ$-nondeterministic linear automata is shown to be intrinsic and simpler than other existing models, avoiding external preprocessing or complex string operations.