CS 131: Programming Languages and Compilers

Introduction

Definition

A Compiler is "A program that takes a source-code program and translates it into an equivalent program in target language"

String

A String is a sequence of characters.

• Alphabet: A finite set of symbols （ASCII characters）

• String has words and sentences

• $st$ is the concatenation of $s$and $t$

  • $s \epsilon = \epsilon s = s$

• $\epsilon$is the empty string

  • $s^0 = \epsilon$

• $\mid s \mid$ is the length of s

• Suppose s is banana

• Prefix:前缀

  • ban banana

• Suffix:后缀

  • banana, ana

• Substring:子字符串

• Subsequence:子字符

  • bnan, nn

A Language is a set of Strings over a fixed Alphabet $\Sigma$, constructed using a specific Grammar.

• e.g. $\{\varepsilon, 0,01,011,0111, \ldots\}$

• Not all Strings of chars in the Alphabet is in the certain Language, only those who satisfy the Grammar rules.

• Alphabet $={0,1}$ and using Grammar rule $\mathrm{RE}=01^{*}$ , we can specify the above example Language, while $01$ isn't.

Below is Operations and Examples.

A Grammar $G$ is the description of method （rules） of how to construct a certain Language over a certain Alphabet

• Type 0: Turing Machine Recursive Enumerable Gramma

• Type 1: Context-sensitive Grammar （CSG）

• Type 2: Context-free Grammar (CFG,上下文无关文法）， mostly recursive

• Type 3: Right-linear Grammar Regular Expressions (RE,正则表达式）， non-recursive

Phases

Lexical Analyzer （词法分析器）

Reads the source program character by character and returns the tokens of the source program

• 分析并把 identifiers 放入符号表

• 利用正则表达式来分析 tokens

• 利用有限自动机来分析 Token 以完成词法分析

• 词法分析器 = 读入（scanning） + 词法分析（lexical analysis）

Token

Describes a pattern of characters having some meaning in the source program (such as identifiers, operators, keywords, numbers, delimiters and so on)

• e.g., identifiers, operators, numbers

A Lexeme （词素） is an instance of a Token, along with its unique attributes.

• e.g. 17

  • INT

  • Token.value = 17

Regular Expression （正则表达式）

我们利用正则表达式来 描述 Tokens 匹配 Tokens

• 每个正则表达式 $r$ 描述一个语言 $L(r)$

  • $r$被叫做 regular set

• 正则表达式是左结合的，从左向右匹配的

• 正则表达式是一个 Type-3 Grammar Rule

Properties

• e.g.

  • Integers:

    • Digit = [0-9]*

    • Integer = Digit Digit*

  • Identifier

    • letter = [a-zA-Z]

    • identifier = letter （letter + digit）*

  • Whitespace

    • WS = ('\n' + '\t' + ' ')+

• Examples

  • Even Binary number

    • 1[0+1]*0 | 0

Finite Automata （有限自动机）

Transition Diagram

当一个 Token 被识别：

• 如果是关键字，那么会返回 Token of the keyword

• 如果是符号表里的 ID，那么返回 entry of symbol table

• 如果符号表没有这个ID，那么加入ID并返回新的 entry of symbol table

switch (state){
case 0:
    c = nextchar();
    if (c == [something])
        state = [state], lexeme_beginning ++;
    else if (c == [something])
        [do something]
    else if (c == [final state]) // terminal
        retract(1); //forward
        lexical_value = install_num(); // install something 
        return (NUM);
    else
        state = fail();

Finite automata

A recognizer for a language is a program that takes a string x, and answers “yes” if x is a sentence of that language, and “no” otherwise.

We call the recognizer of the tokens as a finite automaton.

Example

• Start Arrow

• State

• Transition Edge

• Accepting State:同心圆，接受并结束，状态3

• Death State:错误状态，未定义的 transition 指向该状态

• Transition table:

NFA

Non-Deterministic Finite Automata （NFAs） easily represent regular expression, but are less precise.

Accept s: an Accepting State that spells out s.

An NFA is a mathematical model that consists of:

• S, a finite set of states

• $\Sigma$, the symbols of the input alphabet

• move, a transition function

  • move(state, symbol) $\to$ sets of states

• A state $s_0 \in S$, the start state

• $F \subseteq S$, a set of final or accepting states

• $\epsilon$move: ε- transitions are allowed in NFAs. In other words, we can move from one state to another one without consuming any symbol

DFA

Deterministic Finite Automata （DFAs） require more complexity to represent regular expressions but offer more precision.

Does not allow $\epsilon$move, for every $s \in S$, there is ONLY ONE decision for every input Symbol.

Accept s: an Accepting State that spells out s and ONLY and ONLY ONE path.

Implementation of Lexers

One Token, A Recognizer. There are 4 ways. (r stands for RE)

1. $r \to NFA \to Recognizer$

2. $r \to NFA \to DFA \to Recognizer$

3. $r \to DFA \to Recognizer$

4. $r \rightsquigarrow DFA \to Minimized ~ DFA \to Recognizer$

RE to NFA

Algorithm is called Thompson's Construction.

There are some requirements on such construction:

• $N(s)$and $N(t)$CANNOT have any intersections

• REMEMBER to assign unique names to all states

Properties of the resulting NFA:

• Exactly 1 Start State & 1 Accepting State

• of States in NFA $\leq 2 \times$(# of Symbols + # of Operators) in $r$

• States do not have multiple outgoing edges with the same input symbol

• States have at most 2 outgoing $\epsilon$ edges

RE to DFA

[Step 1]: We make Augmented RE: concatenate with symbol # (meaning "finish").

• e.g. (a+b)*a#

• Ensures at least one operator in the RE

[Step 2]: Build syntax tree for this Augmented RE:

• $\epsilon$, # and $a \in \Sigma$ all are at leaves

• All other operators are inner nodes

• Non-$\epsilon$ leaves get its position number, increasing from left $\to$ right

[Step 3]: Compute nullable(), firstpos() & lastpos() for ALL nodes.

1. firstpos(n): Function returning the set of positions where the first Symbol can be at, in the sub-RE rooted at n

2. lastpos(n): Function returning the set of Positions where the last Symbol can be at, in the sub-RE rooted at n

3. nullable(n): Function judging whether the sub-RE rooted at n can generate $\epsilon$

[Step 4]: Compute followpos() for Leaf positions

followpos(i): Function returning the set of positions which can follow position i in the generated String

Conduct a Post-order Depth First Traversal on the syntax tree, and do the following oprations when leaving $\cdot$ / * nodes:

• $c{1} \cdot c{2}:$ For all $i \in$ lastpos(c1) , followpos(i)= followpo(i) $\cup$ firstpos(c2)

• $c^{*}:$ For all $i \in$ lastpos(c), followpos(i)$=$ followpos(i) $\cup$ firstpos(c)

[Step 5]: Construct the DFA.

void construct() {
    S0=firstpos(root);
    DStates= {(S0, unmarked)};
    while (DStates has an unmarked State U) {
        Mark State U;
        for (each possible input char c) 
        {
            V= {};
            for (each position p in U whose symbol is c)
                V=UnionofVandfollowpos(p);
            if (V is not empty) {
                if (V is not in DStates)
                    Include V in DStates, unmarked;
                Add the Transition U--c->V;
            }
        }
    }
}

• A State $S$in resulting DFA is an Accepting State iff # node $\in S$

• Start State of the resulting DFA is $S_0$

Calculate \epsilon-Closure

set epsClosure(set S) 
{
        for (each State s in S)
            Push s onto stack;
        closure = S;
        while (stack is not empty) 
        {
        Pop State u;
        for (each State v that u->v is an epsilon Transition) 
        {
              if (v is not in closure) 
              {
                  Include v in closure;
                  Push v onto stack;            
              }
         }    
         }
         return closure;
 }

Implement NFA as Recognizer

bool  recognizer() {
    S=epsClosure(s0);
    while ((c=getchar()) !=EOF)
        S=epsClosure(move(S, c));
    if (S and F has intersections)
        return ACCEPT;
    return REJECT;
}

Implement DFA as Recognizer

bool recognizer() {
    s=s_0;
    while ((c=getchar()) !=EOF)
        s=move(s, c);
    if (s is in F)
        return ACCEPT;
    return REJECT;
}

Convert NFA to DFA

Algorithm is called Subset Construction(子集构造法), since we make subset of States in original NFA into a single State in resulting DFA

void subsetConstruction() {
    S0=epsClosure({s0});
    DStates= {(S0, unmarked)};
    while (DStates has any unmarked State U) {
        MarkState U;
        for (each possible inputchar c) {
            V=epsClosure(move(U, c));
            if (V is not empty) {
                if (V is not in DStates)
                    Include V in DStates, unmarked;
                Add the Transition U--c->V;
            }
        }
    }
}

• A State $S$ in resulting DFA is an Accepting State iff $\exists s \in S, s$ is an Accepting State in original NFA

• Start State of the resulting DFA is $S_0$

Minimize DFA

void minimize() {
    PI = {G_A, G_n};
    do {
        for (every group G in PI){
            for (every pair of States (s,t) in G){
                if (for every possible input char c, transition s--c -> and t--c-> go to states in the same group)
                    s,t are in the same subgroup;
                else
                    s,t should split into different subgroups;
            }
            Split G according to the above information;
        }
    }while (PI changed in this iteration)
    Every Group in PI is a state in the minimal DFA;
}  

• A State S in the minimal DFA is an Accepting State iff $\exists s \in S$, s is an Accepting State in original DFA

• Start State of the minimal DFA is the one containing original Starting State

Other Issues for Lexers

Look ahead

Comment Skip

Symbol Table

Syntax Analyzer （句法分析）

如果说词法分析这一步提供了可供计算机识别的词，那么句法分析是为了理解句子结构。

通常这一步会生成 parse tree, parse tree 用以描述句法结构。

Difference with Lexical Analyzer

• The syntax analyzer deals with recursive constructs of the language

• Both do similar things; But the lexical analyzer deals with simple non-recursive constructs of the language.

• The lexical analyzer recognizes the smallest meaningful units （tokens） in a source program.

• The syntax analyzer works on the smallest meaningful units （tokens） in a source program to recognize meaningful structures （sentences） in our programming language.

Parse Tree Abstraction

A Parse Tree / Syntax Tree (语法树) is a graphical representation of the structure of a program, where leaf nodes are Tokens.

CFG （上下文无关文法）

A Context-free Grammar (CFG) is a Type-2 Grammar rule, which serves the construction of a Parse Tree from a streamof Tokens. We use a set of Production Rules to characterize a CFG

A Terminal (终结符号) is a Token; A Non-terminal (非终结符号) is a syntactic variable.

• The Start Symbol is the first one of Non-terminals; Usually represents the whole program

• A Sentence is a string of Terminals such that Start Symbol $S \Rightarrow{ }^{+} s$

A Production Rule (生成规则) is a law of production, from a Non-terminal to a sequence of Terminals & Non-terminals.

• e.g. $A \rightarrow \alpha A \mid \beta$, where $A$ is a Non-terminal and $\alpha, \beta$ are Terminals

• May be recursive

• The procedure of applying these rules to get a sentence of Terminals is called Sentential Form / Derivation

Derivation Directions（派生文法）&Ambiguity（二义性）

Left-most Derivation (左递归)$\left(\Rightarrow_{l m}\right)$ means to replace the leftmost Non-terminal at each step.

• If $\beta A \gamma \Rightarrow \operatorname{lm} \beta \delta \gamma$, then NO Non-terminals in $\mathcal{\beta}$

• Corresponds to Top Down Parsing

Right-most Derivation $(\Rightarrow r m)$means Replace the rightmost Non-terminal at each step.

• If $\beta A \gamma \Rightarrow_{r m} \beta \delta \gamma$, then NO Non-terminals in $\gamma$

• Corresponds to Bottom Up Parsing, in reversed manner

A CFG is Ambiguous when it produces more than one Parse Tree for the same sentence. Must remove Ambiguity for apractical CFG, by:

• Enforce Precedence (优先级) and Associativity (结合律)

  • e.g. $* > +$ , then $+$ gets expanded first

• Grammar Rewritten

Top-Down Parsers

Construction of the parse tree starts at the root, and proceeds towards the leaves.

• Recursive Predictive Parsing

• Non-Recursive Predictive Parsing （LL Parsing）. （L-left to right; L-leftmost derivation）

• 语法构架能力弱

Implement

1. Eliminate Left Recursion $\to$ Recursive-descent Parsing

2. Eliminate Left Recursion $\to$ Left Factoring $\to$Recursive Predictive Parsing

3. Eliminate Left Recursion $\to$Left Factoring $\to$Construct Parsing Table $\to$Non-recursive Predictive Parsing

Left Recursion Elimination (消除左递归)

$A \Rightarrow^{+} A_{\alpha}$: Left Recursion

• Top Down Parsing CANNOT handle Left-recursive Grammars

• Can be eliminated by rewriting

For Immediate Left Recursions (Left Recursion that may appear in a single step), eliminate by:

/* Non-terminals arranged in order: A1, A2, ... An. */
void eliminate() 
{
    for (i from 1 to n) {
        for (j from 1 to i-1)
            Replace Aj with its products in every Prodcution Rule Ai->Aj ...;
        Eliminate Immediate Left Recursions Ai->Ai ...;    
    }
}

Implementing Recursive-descent Parsing

/*  Example:
*    E -> T | T + E*    
     T -> int | int * T | ( E )
*/
bool term(TOKENtok)  { return*ptr++==tok; }
bool E1()             { returnT(); }
bool E2()             { returnT() &&term(PLUS) &&E(); }
bool E() {
     TOKEN*save=ptr;
     return (ptr=save, E1()) || (ptr=save, E2());
}
bool T1()             { returnterm(INT); }
bool T2()             { returnterm(INT) &&term(TIMES) &&T(); }
bool T3()             { returnterm (OPEN) &&E() &&term(CLOSE); }
bool T() {
     TOKEN*save=ptr;
     return (ptr=save, T1()) || (ptr=save, T2()) || (ptr=save, T3());
}

Left Factoring: Produce LL(1) Grammar

LL(1) means Only 1 Token Look-ahead ensures which Pruduction Rule to expand now.

To convert LL(1) to a CFG, for each Non-terminal :

Implementing Recursive Predictive Parsing

Parsing Table Construction

Implementing LL(1) Parsing

Bottom-Up Parsers

Construction of the parse tree starts at the leaves, and proceeds towards the root.

• Bottom-up parsing is also known as shift-reduce parsing

• LR Parsing – much general form of shift-reduce parsing: LR, SLR, LALR (L-left to right; R-rightmost derivation)

IR

Thanks

• [Guanzhou HU's Notes](https://www.josehu.com/)

• TAs: 季杨彪, 杨易为, 尤存翰