# Introduction to Programming Languages/Grammars

A programming language is described by the combination of its semantics and its syntax. The semantics gives us the meaning of every construction that is possible in that programming language. The syntax gives us its structure. There are many different ways to describe the semantics of a programming language; however, after decades of study, there is mostly one technology to describe its syntax. We call this formalism the context free grammars.
Notice that context-free grammars are not the only kind of grammar that computers can use to recognize languages. In fact, there exist a whole family of formal grammars, which have been first studied by Noam Chomsky, and today form what we usually call the Chomsky's hierarchy. Some members of this hierarchy, such as the regular grammars are very simple, and recognize a relatively small number of languages. Nevertheless, these grammars are still very useful. Regular grammars are at the heart of a compiler's lexical analysis, for instance. Other types of grammars are very powerful. As an example, the unrestricted grammars are as computationally powerful as the Turing Machines. Nevertheless, in this book we will focus on context-free grammars, because they are the main tool that a compiler uses to convert a program into a format that it can easily process.

### Grammars

A grammar lets us transform a program, which is normally represented as a linear sequence of ASCII characters, into a syntax tree. Only programs that are syntactically valid can be transformed in this way. This tree will be the main data-structure that a compiler or interpreter uses to process the program. By traversing this tree the compiler can produce machine code, or can type check the program, for instance. And by traversing this very tree the interpreter can simulate the execution of the program.
The main notation used to represent grammars is the Backus-Naur Form, or BNF for short. This notation, invented by John Backus and further improved by Peter Naur, was first used to describe the syntax of the Algol programming language. A BNF grammar is defined by a four-elements tuple represented by (T, N, P, S). The meaning of these elements is as follows:
* T is a set of tokens. Tokens form the vocabulary of the language and are the smallest units of syntax. These elements are the symbols that programmers see when they are typing their code, e.g., the while's, for's, +'s, ('s, etc.
* N is a set of nonterminals. Nonterminals are not part of the language per se. Rather, they help to determine the structure of the derivation trees that can be derived from the grammar. Usually we enclose these symbols in angle brackets, to distinguish them from the terminals.
* P is a set of productions rules. Each production is composed of a left-hand side, a separator and a right-hand side, e.g., <non-terminal> := <expr1> ... <exprN>, where ':=' is the separator. For convenience, productions with the same left-hand side can be abbreviated using the symbol '|'. The pipe, in this case, is used to separate different alternatives.
* S is a start symbol. Any sequence of derivations that ultimately produces a grammatically valid program starts from this special non-terminal.
As an example, below we have a very simple grammar, that recognizes arithmetic expressions. In other words, any program in this simple language represents the product or the sum of names such as 'a', 'b' and 'c'.
This grammar could be also represented in a more convenient way using a sequence of bar symbols, e.g.:
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