# Compilers

Start Lecture #3

### 2.4.5: Left Recursion

Another complication. Consider
expr → expr + term
expr → term

For the first production the RHS begins with the LHS. This is called left recursion. If a recursive descent parser would pick this production, the result would be that the next node to consider is again expr and the lookahead has not changed. An infinite loop occurs. (Also note that the first sets are not disjoint.)

Note that this is NOT a problem with the grammar per se, but is a limitation of predictive parsing. For example if we had the additional production
term → x
Then it is easy to find the unique parse tree for
x x
but we won't find it with predictive parsing.

expr → term + expr
expr → term
it would be right recursive, which is not a problem. But the first sets are not disjoint and addition would become right associative.

Consider instead of the first (left-recursive) grammar
expr → term rest
rest → + term rest
rest → ε

Both sets of productions generate the same possible token strings, namely
term + term + ... + term
The second set is called right recursive since the RHS ends (has on the right) the LHS. If you draw the parse trees generated, you will see that, for left recursive productions, the tree grows to the left; whereas, for right recursive, it grows to the right.

Using this technique to eliminate left-recursion will (next month) make it harder for us to get left associativity for arithmetic, but we shall succeed!

In general, for any nonterminal A, and any strings α, and β, we can replace the pair
A → A α | β
with the triple
A → β R
R → α R | ε
where R is a nonterminal not equal to A and not appearing in α or β, i.e., R is a new nonterminal.

For the example above A is expr, R is rest, α is + term, and β is term.

Yes, this looks like magic.
Yes, there are more general possibilities.
We will have more to say in chapter 4.

## 2.5: A Translator for Simple Expressions

Objective: an infix to postfix translator for expressions. We start with just plus and minus, specifically the expressions generated by the following grammar. We include a set of semantic actions with the grammar. Note that finding a grammar for the desired language is one problem, constructing a translator for the language, given a grammar, is another problem. We are tackling the second problem.

```  expr → expr + term { print('+') }
expr → expr - term { print('-') }
expr → term
term → 0           { print('0') }
. . .
term → 9           { print('9') }
```

One problem we must solve is that this grammar is left recursive.

### 2.5.1: Abstract and Concrete Syntax

Often one prefers not to have superfluous nonterminals as they make the parsing less efficient. That is why we don't say that a term produces a digit and a digit produces each of 0,...,9. Ideally the syntax tree would just have the operators + and - and the 10 digits 0,1,...,9. That would be called the abstract syntax tree. A parse tree coming from a grammar is technically called a concrete syntax tree.

### 2.5.2: Adapting the Translation Scheme

We eliminate the left recursion as we did in 2.4. This time there are two operators + and - so we replace the triple
A → A α | A β | γ
A → γ R
R → α R | β R | ε

This time we have actions so, for example
α is + term { print('+') }
However, the formulas still hold and we get

```    expr → term rest
rest → + term { print('+') } rest
| - term { print('-') } rest
| ε
term → 0           { print('0') }
. . .
| 9           { print('9') }
```

### 2.5.3: Procedures for the nonterminals expr, term, and rest

The C code is in the book. Note the else ; in rest(). This corresponds to the epsilon production. As mentioned previously. The epsilon production is only used when all others fail (that is why it is the else arm and not the then or the else if arms).

### 2.5.4: Simplifying the translator

These are (useful) programming techniques.

### The complete program

The program in Java is in the book.

## 2.6: Lexical Analysis

The purpose of lexical analysis is to convert a sequence of characters (the source) into a sequence of tokens. A lexeme is the sequence of characters comprising a single token.

Note that (following the book) we are going out of order. In reality, the lexer operates on the input and the resulting token sequence is the input to the parser. The reason we were able to produce the translator in the previous section without a lexer is that all the tokens were just one character (that is why we had just single digits).

### 2.6.1: Removal of White space and comments

These do not become tokens so that the parser need not worry about them.

Consider distinguishing x<y from x<=y.

After reading the < we must read another character. If it is y, we have found our token (<). However, we must unread the y so that when asked for the next token we will start at y. If it is never more than one extra character that must be examined, a single char variable would suffice. A more general solution is discussed in the next chapter (Lexical Analysis).

### 2.6.3: Constants

This chapter considers only numerical integer constants. They are computed one digit at a time by value=10*value+digit. The parser will therefore receive the token num rather than a sequence of digits. Recall that our previous parsers considered only one digit numbers.

The value of the constant can be considered the attribute of the token named num. Alternatively, the attribute can be a pointer/index into the symbol table entry for the number (or into a numbers table).

### 2.6.4: Recognizing identifiers and keywords

The C statement
sum = sum + x;
contains 6 tokens. The scanner (aka lexer; aka lexical analyzer) will convert the input into
id = id + id ;
(id standing for identifier).
Although there are three id tokens, the first and second represent the lexeme sum; the third represents x. These two different lexemes must be distinguished.

A related distinction occurs with language keywords, for example then, which are syntactically the same as identifiers. The symbol table is used to accomplishes both distinctions. We assume (as do most modern languages) that the keywords are reserved, i.e., cannot be used as program variables. The we simply initialize the symbol table to contain all these reserved words and mark them as keywords. When the lexer encounters a would-be identifier and searches the symbol table, it finds out that the string is actually a keyword.

As mentioned previously care must be taken when one lexeme is a proper subset of another. Consider
x<y versus x<=y
When the < is read, the scanner needs to read another character to see if it is an =. But if that second character is y, the current token is < and the y must be pushed back onto the input stream so that the configuration is the same after scanning < as it is after scanning <=.

Also consider then versus thenewvalue, one is a keyword and the other an id.

### 2.6.5: A lexical analyzer

A Java program is given. The book, but not the course, seems to assume knowledge of Java.

Since the scanner converts digits into num's we can shorten the grammar above. Here is the shortened version before the elimination of left recursion. Note that the value attribute of a num is its numerical value.

```    expr   → expr + term    { print('+') }
expr   → expr - term    { print('-') }
expr   → term
term   → num            { print(num.value) }
```
In anticipation of other operators with higher precedence, we could introduce factor and, for good measure, include parentheses for overriding the precedence. Our grammar would then become.
```    expr   → expr + term    { print('+') }
expr   → expr - term    { print('-') }
expr   → term
term   → factor
factor → ( expr ) | num { print(num,value) }
```

The factor() procedure follows the familiar recursive descent pattern: Find a production with factor as LHS and lookahead in FIRST, then do what the RHS says. That is, call the procedures corresponding to the nonterminals, match the terminals, and execute the semantic actions.

Note that we are now able to consider constants of more than one digit.

## 2.7: Incorporating a symbol table

The symbol table is an important data structure for the entire compiler. One example of its use is that the semantic actions or rules associated with declarations set the type field of the symbol table entry. Subsequent semantic actions or rules associated with expression evaluation use this type information. For the simple infix to postfix translator (which is typeless), the table is primarily used to store and retrieve <lexeme,token> pairs.

### 2.7.1: Symbol Table per Scope

There is a serious issue here involving scope. We will learn soon that lexers are based on regular expressions; whereas parsers are based on the stronger but more expensive context-free grammars. Regular expressions are not powerful enough to handle nested scopes. So, if the language you are compiling supports nested scopes, the lexer can only construct the <lexeme,token> pairs. The parser converts these pairs into a true symbol table that reflects the nested scopes. If the language is flat, the scanner can produce the symbol table.

The idea for a block structured language (i.e., one with nested scopes) is that, when entering a block, a new symbol table is created. Each such table points to the one immediately outer. This structure supports the most-closely nested rule for symbols: a symbol is in the scope of most-closely nested declaration. This gives rise to a tree of tables.

#### Interface

• Create table: A new table is created and points to the immediately outer table, which is passed as a argument.
• Insert entry (in the current table).
• Retrieve entry (from the most-closely nested table in which it appears).

#### Reserved keywords

Simply insert them into the symbol table prior to examining any input. Then they can be found when used correctly and, since their corresponding token will not be id, any use of them where an identifier is required can be flagged. For example a lexer for a C-like language would have insert(int) performed for every table prior to scanning the input.

### 2.7.2: The Use of Symbol Tables

Below is the grammar for a stripped down example showing nested scopes. The language consists just of nested blocks, a weird mixture of C- and ada-style declarations (specifically, type colon identifier), and trivial statements consisting of just an identifier.

```    program → block
block   → { decls stmts }     -- { } are terminals not actions
decls   →  decls decl | ε     -- study this one
decl    → type : id ;
stmts   → stmts stmt | ε      -- same idea, a list
stmt    → block | factor ;    -- enables nested blocks
factor  → id
```

Semantic Actions
ProductionAction

Program{top = null}
block

block{{ saved = top;
top = new Env(top);
print ("} "); }
decls stmts }{ top = saved;
print ("} "); }

declsdecls decl
| ε

decltype id ;{ s = new Symbol;
s.type = type.lexeme;
top.put(id.lexeme,s); }

stmtsstmts stmt
| ε

stmtblock
| factor ; { print("; "); }

factorid{ s = top.get(id.lexeme);
print(s.type); }

One possible program in this language is

```    { int : x ;  float : y ;
x ; y ;
{ float : x ;
x ; y ;
}
{ int : y ;
x ; y;
}
x ; y ;
}
```

To show that we have correctly parsed the input and obtained its meaning (i.e., performed semantic analysis), we present a translation scheme that digests the declarations and translates the statements so that the above example becomes

```    { int; float; { float; float; } { int; int; } }
```

The translation scheme, slightly modified from the book page 90, is shown on the right. First a formatting comment.

This translation scheme looks weird, but is actually a good idea (of the authors): it reconciles the two goals of respecting the ordering and nonetheless having the actions all in one column.

Recall that the placement of the actions within the RHS of the production is significant. The parse tree with its actions is processed in a depth first manner so that the actions are performed in left to right order. Thus an action is executed after all the subtrees rooted by parts of the RHS to the left of the action and is executed before all the subtrees rooted by parts of the RHS to the right of the action.

Consider the first production. We want the action to be executed before processing block. Thus the action must precede block in the RHS. But we want the actions in the right column. So we split the RHS over several lines and place an action in the rightmost column of the line that puts in the right order.

The second production has some semantic actions to be performed at the start of the block, and others to be performed at the bottom.

To fully understand the details, you must read the book; but we can see how it works. A new Env initializes a new symbol table; top.put inserts into the symbol table in the environment top; top.get retrieves from that symbol table.

In some sense the star of the show is the simple production
factor → id
together with its semantic actions. These actions look up the identifier in the (correct!) symbol table and print out the type name.

Question: Why do we have the trivial-looking production
program → block
Answer: We need to initialize top only once, not each time we enter a block.

Hard Question: I prefer ada-style declarations, which are of the form
identifier : type
What problem would have occurred had I done so here and how does one solve that problem?
Answer: Both a stmt and a decl would start with id, which would mean we need more than one token lookahead to see when the decls end and the stmts begin. The fix is to left-factor the grammar as we will learn in chapter 4.

## 2.8: Intermediate Code Generation

### 2.8.1: Two kinds of Intermediate Representations

There are two important forms of intermediate representations.

• Trees, especially parse trees and syntax trees.
Since parse trees exhibit the syntax of the language being parsed, it may be surprising to see them compared with syntax trees. One would think instead that they are syntax trees. In fact there is a spectrum of syntax trees, with parse trees within the class.

Another (but less common) name for parse tree is concrete syntax tree. Similarly another (also less common) name for syntax tree is abstract syntax tree.

Very roughly speaking, (abstract) syntax trees are parse trees reduced to their essential components, and three address code looks like assembler without the concept of registers.

### 2.8.2: Construction of (Abstract) Syntax Trees

Remarks:

1. Despite the words below, your future lab assignments will likely not require producing abstract syntax trees. Instead, you will be producing concrete syntax trees (parse trees). I may include an extra-credit part of some labs that will ask for abstract syntax trees.
2. Note however that real compilers do not produce parse trees since such trees are larger and have no extra information that the compiler needs. If the compiler produces a tree (many do), it produces an abstract syntax tree.
3. The reason I will not require your labs to produce the smaller trees is that to do so it is helpful to understand semantic rules and semantic actions, which come later in the course. Of course, authors of real compilers have already completed the course before starting the design so this consideration does not apply to them. :-)

Consider the production
while-stmt → while ( expr ) stmt ;
The parse tree would have a node while-stmt with 6 children: while, (, expr, ), stmt, and ;. Many of these are simply syntactic constructs with no real meaning. The essence of the while statement is that the system repeatedly executes stmt until expr is false. Thus, the (abstract) syntax tree has a node (most likely labeled while) with two children, the syntax trees for expr and stmt.

To generate this while node, we execute
new While(x,y)
where x and y are the already constructed (synthesized attributes!) nodes for expr and stmt.

#### Syntax Trees for Statements

The book has an SDD (syntax directed definition) on page 94 for several statements. The part for while reads
stmt → while ( expr ) stmt1   { stmt.n = new While(expr.n, stmt1.n); }
The n attribute gives the syntax tree node.

#### Representing Blocks in Syntax Trees

Fairly easy

```    stmt  → block          { stmt.n = block.n }
block → { stmts }      { block.n = stmts.n }
```
Together these two just use the syntax tree for the statements constituting the block as the syntax tree for the block when it is used as a statement. So
```    while ( x == 5 ) {
blah
blah
more
}
```
would give the while node of the abstract syntax tree two children as always:
1. The tree for x==5.
2. The tree for blah blah more.

#### Syntax trees for Expressions

When parsing we need to distinguish between + and * to insure that 3+4*5 is parsed correctly, reflecting the higher precedence of *. However, once parsed, the precedence is reflected in the tree itself (the node for + has the node for * as a child). The rest of the compiler treats + and * largely the same so it is common to use the same node label, say OP, for both of them. So we see
term → term1 * factor     { term.n = new Op('*', term1.n, factor.n); }

### 2.8.3: Static Checking

Static checking refers to checks performed during compilation; whereas, dynamic checking refers to those performed at run time. Examples of static checks include

• Syntactic checks such as avoiding multiple declarations of the same identifier in the same scope.
• Type checks.

#### L-values and R-values

Consider Q = Z; or A[f(x)+B*D] = g(B+C*h(x,y));. I am using [] for array reference and () for function call).

From a macroscopic view, we have three tasks.

1. Evaluate the left hand side (LHS) to obtain an l-value.
2. Evaluate the RHS to obtain an r-value.
3. Perform the assignment.

Note the differences between L-values, quantities that can appear on the LHS of an assignment, and and R-values, quantities that can appear only on the RHS.

• An l-value corresponds to an address or a location.
• An r-value corresponds to a value.
• Neither 12 nor s+t can be used as an l-value, but both are legal r-values.

Static checking is used to insure that R-values do not appear on the LHS.

#### Type Checking

These checks assure that the type of the operands are expected by the operator. In addition to flagging errors, this activity includes

• Coercions. The automatic conversion of one type to another. Later in this course we will employ the function widen(a,t,w) in certain semantic rules/actions. Widen(x,int,double) would for example generate the intermediate code needed to convert x of type int into a quantity of type double.
• Overloading. In Java, Ada, and other languages, the same symbol can have different meanings depending on the types of the operands. Static checks are used to determine the correct operation, or signal an error if none exists

These are primitive instructions that have one operator and (up to) three operands, all of which are addresses. One address is the destination, which receives the result of the operation; the other two addresses are the sources of the values to be operated on.

Perhaps the clearest way to illustrate the (up to) three address nature of the instructions is to write them as quadruples or quads.

```    ADD        x y z
MULT       a b c
ARRAY_L    q r s
ARRAY_R    e f g
ifTrueGoto x L
COPY       r s
```
But we normally write them in a more familiar form.
```    x = y + z
a = b * c
q[r] = s
e = f[g]
ifTrue x goto L
r = s
```

#### Translating Statements

We do this and the next section much slower and in much more detail later in the course.

Here is the example from the book, somewhat Java intensive.

```    class If extends Stmt {
Expr E; Stmt S;
public If(Expr x, Stmt y) { E = x;  S = y;  after = newlabel(); }
public void gen() {
Expr n = E.rvalue();
emit ("ifFalse " + n.toString() + " goto " + after);
S.gen();
emit(after + ":");
}
}
```

#### Translating Expressions

I am just illustrating the simplest case

```    Expr rvalue(x : Expr) {
if (x is an Id or Constant node) return x;
else if (x is an Op(op, y, z) node) {
t = new temporary;
emit string for t = rvalue(y) op rvalue(z);
return a new node for t;
else read book for other cases
```

#### Better Code for Expressions

So called optimization (the result is far from optimal) is a huge subject that we barely touch. Here are a few very simple examples. We will cover these since they are local optimizations, that is they occur within a single basic block (a sequence of statements that execute without any jumps).

1. For a Java assignment statement x = x + 1; we would generate two three-address instructions
```	temp = x + 1
x    = temp
```
The can be combined into the three-address instruction x = x + 1, providing there are no further uses of the temporary.
2. Common subexpressions occurring in two different expressions, need be computed only once.

## Two Questions

1. How come this compiler was so easy?
2. Why isn't the final exam next week?

One reason is that much was deliberately simplified. Specifically note that

• No real machine code generated (no back end).
• No optimizations (improvement to generated code).
• FIRST sets disjoint.
• No semantic analysis.
• Input language very simple.
• Output language very simple and closely related to input.

Also, I presented the material way too fast to expect full understanding.

# Chapter 3: Lexical Analysis

Two methods to construct a scanner (lexical analyzer).

1. By hand, beginning with a diagram of what lexemes look like. Then write code to follow the diagram and return the corresponding token and possibly other information.
2. Feed the patterns describing the lexemes to a lexer-generator, which then produces the scanner. The historical lexer-generator is Lex; a more modern one is flex.

Note that the speed (of the lexer not of the code generated by the compiler) and error reporting/correction are typically much better for a handwritten lexer. As a result most production-level compiler projects write their own lexers.

## 3.1: The Role of the Lexical Analyzer

The lexer is called by the parser when the latter is ready to process another token.

The lexer also might do some housekeeping such as eliminating whitespace and comments. Some call these tasks scanning, but others user the term scanner for the entire lexical analyzer.

After the lexer, individual characters are no longer examined by the compiler; instead tokens (the output of the lexer) are used.

### 3.1.1: Lexical Analysis Versus Parsing

Why separate lexical analysis from parsing? The reasons are basically software engineering concerns.

1. Simplicity of design. When one detects a well defined subtask (produce the next token), it is often good to separate out the task (modularity).
2. Efficiency. With the task separated out, it is easier to apply specialized techniques.
3. Portability. Only the lexer need communicate with the outside.

### 3.1.2: Tokens, Patterns, and Lexemes

• A token is a <name,attribute> pair. These are what the parser processes. The attribute might actually be a tuple of several attributes or a pointer to a table entry, which itself might contain many components.
• A pattern describes the character strings for the lexemes of the token. For example a letter followed by a (possibly empty) sequence of letters and digits.
• A lexeme for a token is a sequence of characters that matches the pattern for the token.

Note the circularity of the definitions for lexeme and pattern.

Common token classes.

1. One for each keyword. The pattern is trivial.
2. One for each operator or class of operators. A typical class is the comparison operators. Note that these have the same precedence. We might have + and - as the same token, but not + and *.
3. One for all identifiers (e.g. variables, user defined type names, etc).
4. Constants (i.e., manifest constants) such as 6 or hello, but not a constant identifier such as quantum in the Java statement.
static final int quantum = 3;. There might be one token for integer constants, a different one for real, another for string, etc.
5. One for each punctuation symbol.

### 3.1.3: Attributes for Tokens

We saw an example of attributes in the last chapter.

For tokens corresponding to keywords, attributes are not needed since the name of the token tells everything. But consider the token corresponding to integer constants. Just knowing that the we have a constant is not enough, subsequent stages of the compiler need to know the value of the constant. Similarly for the token identifier we need to distinguish one identifier from another. The normal method is for the attribute to specify the symbol table entry for this identifier.

We really shouldn't say symbol table. As mentioned above if the language has scoping (nested blocks) the lexer can't construct the symbol table, but just makes a table of <lexeme,token> pairs.

Homework: 1.

### 3.1.4: Lexical Errors

We saw in this movie an example where parsing got stuck because we reduced the wrong part of the input string. We also learned about FIRST sets that enabled us to determine which production to apply when we are operating left to right on the input. For predictive parsers the FIRST sets for a given nonterminal are disjoint and so we know which production to apply. In general the FIRST sets might not be disjoint so we have to try all the productions whose FIRST set contains the lookahead symbol.

All the above assumed that the input was error free, i.e. that the source was a sentence in the language. What should we do when the input is erroneous and we get to a point where no production can be applied?

In many cases this is up to the parser to detect/repair. Sometimes the lexer is stuck because there are no patterns that match the input at this point.

The simplest solution is to abort the compilation stating that the program is wrong, perhaps giving the line number and location where the lexer and/or parser could not proceed.

We would like to do better and at least find other errors. We could perhaps skip input up to a point where we can begin anew (e.g. after a statement ending semicolon), or perhaps make a small change to the input around lookahead so that we can proceed.

## 3.2: Input Buffering

Determining the next lexeme often requires reading the input beyond the end of that lexeme. For example, to determine the end of an identifier normally requires reading the first whitespace character after it. Also just reading > does not determine the lexeme as it could also be >=. When you determine the current lexeme, the characters you read beyond it may need to be read again to determine the next lexeme.

### 3.2.1: Buffer Pairs

The book illustrates the standard programming technique of using two (sizable) buffers to solve this problem.

### 3.2.2: Sentinels

A useful programming improvement to combine testing for the end of a buffer with determining the character read.

## 3.3: Specification of Tokens

The chapter turns formal and, in some sense, the course begins. The book is fairly careful about finite vs infinite sets and also uses (without a definition!) the notion of a countable set. (A countable set is either a finite set or one whose elements can be put into one to one correspondence with the positive integers. That is, it is a set whose elements can be counted. The set of rational numbers, i.e., fractions in lowest terms, is countable; the set of real numbers is uncountable, because it is strictly bigger, i.e., it cannot be counted.) We should be careful to distinguish the empty set φ from the empty string ε. Formal language theory is a beautiful subject, but I shall suppress my urge to do it right and try to go easy on the formalism.

### 3.3.1: Strings and Languages

We will need a bunch of definitions.

Definition: An alphabet is a finite set of symbols.

Example: {0,1}, presumably φ (uninteresting), ascii, unicode, ebcdic, latin-1.

Definition: A string over an alphabet is a finite sequence of symbols from that alphabet. Strings are often called words or sentences.

Example: Strings over {0,1}: ε, 0, 1, 111010. Strings over ascii: ε, sysy, the string consisting of 3 blanks.

Definition: The length of a string is the number of symbols (counting duplicates) in the string.

Example: The length of allan, written |allan|, is 5.

Definition: A language over an alphabet is a countable set of strings over the alphabet.

Example: All grammatical English sentences with five, eight, or twelve words is a language over ascii. It is also a language over unicode.

Definition: The concatenation of strings s and t is the string formed by appending the string t to s. It is written st.

Example: εs = sε = s for any string s.

We view concatenation as a product (see Monoid in wikipedia http://en.wikipedia.org/wiki/Monoid). It is thus natural to define s0=ε and si+1=sis.

Example: s1=s, s4=ssss.

#### More string terminology

A prefix of a string is a portion starting from the beginning and a suffix is a portion ending at the end. More formally,

Definitions:

1. A prefix of s is any string obtained from s by removing (possibly zero) characters from the end of s.
2. A suffix is defined analogously.
3. A substring of s is obtained by deleting a prefix and a suffix.

Note: Any prefix or suffix is a substring.

Examples: If s is 123abc, then

1. s itself and ε are each a prefix, suffix, and a substring.
2. 12 are 123a are prefixes and substrings.
3. 3abc is a suffix and a substring.
4. 23a is a substring

Definitions: A proper prefix of s is a prefix of s other than ε and s itself. Similarly, proper suffixes and proper substrings of s do not include ε and s.

Definition: A subsequence of s is formed by deleting (possibly zero) positions from s. We say positions rather than characters since s may for example contain 5 occurrences of the character Q and we only want to delete a certain 3 of them.

Example: issssii is a subsequence of Mississippi.

Note: Any substring is a subsequence.

Homework: 3(a,b,c).

### 3.3.2: Operations on Languages

Definition: The union of L1 and L2, written L ∪ M is simply the set-theoretic union, i.e., it consists of all words (strings) in either L1 or L2.

Example: Let the alphabet be ascii. The union of {Grammatical English sentences with one, three, or five words} with {Grammatical English sentences with two or four words} is {Grammatical English sentences with five or fewer words}.

Definition: The concatenation of L1 and L2 is the set of all strings st, where s is a string of L1 and t is a string of L2.

We again view concatenation as a product and write LM for the concatenation of L and M.

Examples:: The concatenation of {a,b,c} and {1,2} is {a1,a2,b1,b2,c1,c2}. The concatenation of {aa,b,c} and {1,2,ε} is {aa1,aa2,aa,b1,b2,b,c1,c2,c}.

Definition: As with strings, it is natural to define powers of a language L.
L0={ε}, which is not φ.
Li+1=LiL.

Definition: The (Kleene) closure of L, denoted L* is
L0 ∪ L1 ∪ L2 ...

Definition: The positive closure of L, denoted L+ is
L1 ∪ L2 ...

Example: {0,1,2,3,4,5,6,7,8,9}+ gives all unsigned integers, but with some ugly versions. It has 3, 03, 000003.
{0} ∪ ( {1,2,3,4,5,6,7,8,9} ({0,1,2,3,4,5,6,7,8,9}* ) ) seems better.

In these notes I may write * for * and + for +, but that is strictly speaking wrong and I will not do it on the board or on exams or on lab assignments.

Example: {a,b}* is {ε,a,b,aa,ab,ba,bb,aaa,aab,aba,abb,baa,bab,bba,bbb,...}.
{a,b}+ is {a,b,aa,ab,ba,bb,aaa,aab,aba,abb,baa,bab,bba,bbb,...}.
{ε,a,b}* is {ε,a,b,aa,ab,ba,bb,...}.
{ε,a,b}+ is the same as {ε,a,b}*.

The book gives other examples based on L={letters} and D={digits}, which you should read.

### 3.3.3: Regular Expressions

The idea is that the regular expressions over an alphabet consist of
the alphabet, and expressions using union, concatenation, and *,
but it takes more words to say it right. For example, I didn't include (). Note that (A ∪ B)* is definitely not A* ∪ B* (* does not distribute over ∪) so we need the parentheses.

The book's definition includes many () and is more complicated than I think is necessary. However, it has the crucial advantages of being correct and precise.

The wikipedia entry doesn't seem to be as precise.

I will try a slightly different approach, but note again that there is nothing wrong with the book's approach (which appears in both first and second editions, essentially unchanged).

Definition: The regular expressions and associated languages over an alphabet consist of

1. ε, the empty string; the associated language L(ε) is {ε}, which is not φ.
2. Each symbol x in the alphabet; L(x) is {x}.
3. rs for all regular expressions (REs) r and s; L(rs) is L(r)L(s).
4. r|s for all REs r and s; L(r|s) is L(r) ∪ L(s).
5. r* for all REs r; L(r*) is (L(r))*.
6. (r) for all REs r; L((r)) is L(r).