One-stack Automata as Acceptors of Context-free Languages*

This paper presents one-stack automata as acceptors of context-free languages; these are equivalent to Pushdown Automata which are well known in automata theory. As equivalence relations such as equivalence of Turing Machines and two-stack Pushdown Automata help in learning general properties of formal modeling, the equivalence relation of Pushdown Automata and one-stack automata also helps in learning general properties of context-free language modeling. One-stack automata are helpful to students for several reasons including: (1) their contrast with two-stack Pushdown Automata and multi-stack automata is revealing for computability; (2) their computer animation is helpful for learning their salient features; (3) their graphical representation is more easily obtained by augmenting Non-Deterministic Finite Automata for regular languages which usually precede context-free language acceptors in the logical sequence of ideas. INTRODUCTION The most elegant models of computation are the mathematically defined automata including Turing Machines (TMs), two-stack Pushdown Automata (2PDA), Linear Bounded Automata (LBA), Pushdown Automata (PDA) and Finite Automata (FA). These models are usually studied in the fields of theory of computation, automata theory and computability. TMs define the most powerful automata class for processing the most complex sets, namely, recursively enumerable sets. TMs are equivalent to two-stack CCSC: Northwestern Conference 119 automata or 2PDA as proven by Minsky [7]. Classical forms of Pushdown Automata (PDA) are required to have exactly one stack and they are non-deterministic unless otherwise explicitly stated. PDA are acceptors of the class of Context-Free Languages (CFL's) or sets. They are less powerful than TMs; they cannot accept non-CFL's. Finite Automata (FA) define a proper subset of CFL's called regular languages denoted by regular expressions; they are equivalent to Non-deterministic FA (NFA). The above narrative information with some additional information is usually presented in a tabular form called the Chomsky hierarchy of grammars and languages as shown in Table 1 [1]. The Chomsky Hierarchy of Grammars and Languages Type Language/Grammar Acceptor 0 Recursively Enumerable Turing Machines = 2PDA = Post Machines 1 Context-Sensitive Linear Bounded Automata (LBA) = Turing Machines with bounded tape. 2 Context-Free Pushdown Automata (PDA) 3 Regular Finite Automata (FA) = NFA = Transition Graphs Table 1: The Chomsky Hierarchy Unlike other automata classes, PDA are not shown to be equivalent to any other acceptors [1]. This paper proves that one-stack automata are equivalent to PDA and suggests that this equivalence relation helps in learning general properties of CFL acceptors. In addition, computer animation of one-stack automata (www.asethome.org/onestack) demonstrates their revealing features [3]. Due to numerous recent research activities with multi-stack automata [2, 6] the name and the representation of one-stack automata are carefully chosen to avoid possible confusions. This paper presents essential features of one-stack automata and describes how these features and their animation promote learning CFL modeling.