Alice Ter Meulen

Basic Concepts of Algebra

An algebra A is a set A together with one or more operations f i . We may represent an algebra by writing

Pushdown Automata, Context Free Grammars and Languages

{\rm{ }}A = \left\langle {A,{\rm{ }}{f_1},{\rm{ }}{f_2}{\rm{ }}...{\rm{, }}{f_n}} \right\rangle

Relations and Functions

(9-1)

Languages Between Context Free and Context Sensitive

A set is an abstract collection of distinct objects which are called the members or elements of that set. Objects of quite different nature can be members of a set, e.g. the set of red objects may contain cars, blood-cells, or painted representations. Members of a set may be concrete, like cars, blood-cells or physical sounds, or they may be abstractions of some sort, like the number two, or the English phoneme /p/, or a sentence of Chinese. In fact, we may arbitrarily collect objects into a set even though they share no property other than being a member of that set. The subject matter of set theory and hence of Part A of this book is what can be said about such sets disregarding the actual nature of their members.

Formal Systems, Axiomatization, and Model Theory

We turn next to a class of automata which are more powerful than the finite automata in the sense that they accept a larger class of languages. These are the pushdown automata (pda’s).

Boolean and Heyting Algebras

We have seen that a pushdown automaton can carry out computations which are beyond the capability of a finite automaton, which is perhaps the simplest sort of machine able to accept an infinite set of strings. At the other end of the scale of computational power is the Turing machine (after the English mathematician A. M. Turing, who devised them), which can carry out any set of operations which could reasonably be called a computation.

Basic Concepts of Logic and Formal Systems

Recall that there is no order imposed on the members of a set. We can, however, use ordinary sets to define an ordered pair, written 〈a, b〉 for example, in which a is considered the first member and b is the second member of the pair. The definition is as follows: