FE-Logo
  • Home
  • Study Material
  • Non-Deterministic Finite Automation
    • Introduction to Compiler
    • The Structure of a Compiler
    • Intermediate Code Generation
    • Building a Compiler
    • Applications of Compiler
    • Optimizations for Computer Architectures
    • Design of New Computer Architectures
    • Program Translations
    • Software Productivity Tools
    • Programming Language Basics
    • Minimisation of DFAs
    • Explicit Access Control
    • Parameter Passing Mechanisms
    • Introduction to Lexical Analysis
    • Regular expressions
    • Short hands
    • Nondeterministic finite automata
    • Converting a regular expression to an NFA
    • Deterministic finite automata
    • Converting an NFA to a DFA
    • The subset construction
    • Dead states
    • Lexers and lexer generators
    • Splitting the input stream
    • Lexical errors
    • Properties of regular languages
    • Limits to expressive power
    • The Role of the Lexical Analyzer
    • Input Buffering
    • Specification of Tokens
    • Operations on Languages
    • Regular Definitions and Extensions
    • Recognition of Tokens
    • The Lexical-Analyzer Generator Lex
    • Finite Automata
    • Construction of an NFA from a Regular Expression
    • Efficiency of String-Processing Algorithms
    • The Structure of the Generated Analyzer
    • Optimization of DFA-Based Pattern Matchers

  • Basic Parsing Techniques
    • Introduction to Syntax analysis
    • Context-free grammars
    • Writing context free grammars
    • Derivation
    • Syntax trees and ambiguity
    • Operator precedence
    • Writing ambiguous expression grammars
    • Other sources of ambiguity
    • Syntax analysis and Predictive parsing
    • Nullable and FIRST
    • Predictive parsing revisited
    • FOLLOW
    • LL(1) parsing
    • Methods for rewriting grammars for LL(1) parsing
    • SLR parsing
    • Constructions of SLR parse tables
    • Conflicts in SLR parse-tables
    • Using precedence rules in LR parse tables
    • Using LR-parser generators
    • Properties of context-free languages
    • Introduction to Syntax-Directed Translator
    • Evaluating an SDD at the Nodes of a Parse Tree
    • Evaluation Orders for SDD\'s
    • Ordering the Evaluation of Attributes
    • A larger example of calculating FIRST and FOLLOW
    • Syntax Definition
    • Associativity of Operators
    • Parse Trees
    • Ambiguity
    • Syntax-Directed Translation
    • Synthesized Attributes
    • Tree Traversals
    • Parsing
    • Predictive Parsing
    • Use e-Productions
    • Translator for Simple Expressions
    • Semantic Rules with Controlled Side Effects
    • Applications of Syntax-Directed Translation
    • The Structure of a Type of syntax
    • Switch-Statements
    • Syntax-Directed Translation Schemes
    • Postfix Translation Schemes
    • SDT\'s With Actions Inside Productions
    • Eliminating Left Recursion from SDT\'s
    • SDT\'s for L-Attributed Definitions
    • Implementing L-Attributed SDD\'s
    • On-The-Fly Code Generation
    • L-Attributed SDD\'s and LL Parsing
    • Bottom-Up Parsing of L-Attributed SDD\'s

  • Syntax-directed Translation
    • Register Allocation and Assignment
    • Semantic Analysis
    • Introduction to Intermediate Code Generation
    • Variants of Syntax Trees
    • Variants of Syntax Trees
    • The Value-Number Method for Constructing DAG\'s
    • Three-Address Code
    • Quadruples
    • Triples
    • Static Single-Assignment Form
    • Types and Declarations
    • Type Equivalence
    • Sequences of Declarations
    • Translation of Expressions
    • Incremental Translation
    • Addressing Array Elements
    • Translation of Array References
    • Type Checking
    • Type Conversions
    • Overloading of Functions and Operators
    • Type Inference and Polymorphic Functions
    • Algorithm for Unification
    • Control Flow
    • Flow-of-Control Statements
    • Control-Flow Translation of Boolean Expressions
    • Boolean Values and Jumping Code
    • Back patching
    • Backpatching for Boolean Expressions
    • Flow-of-Control Statements
    • Break-, Continue-, and Goto-Statements
    • Introduction to Run-Time Environments
    • Stack Allocation of Space
    • Activation Records
    • Calling Sequences
    • Variable-Length Data on the Stack
    • Access to Nonlocal Data on the Stack
    • Displays
    • Heap Management
    • Locality in Programs
    • Reducing Fragmentation
    • Managing and Coalescing Free Space
    • Manual Deallocation Requests
    • Reachability
    • Introduction to Garbage Collection
    • Reference Counting Garbage Collectors
    • Introduction to Trace-Based Collection
    • Basic Abstraction
    • Optimizing Mark-and-Sweep
    • Mark-and-Compact Garbage Collectors
    • Copying collectors
    • Short-Pause Garbage Collection
    • Incremental Reachability Analysis
    • Partial-Collection Basics
    • The Train Algorithm
    • Parallel and Concurrent Garbage Collection
    • Partial Object Relocation
    • Introduction Code Generation
    • Issues in the Design of a Code Generator
    • Instruction Selection
    • Register Allocation
    • The Target Language
    • Addresses in the Target Code
    • Stack Allocation
    • Run-Time Addresses for Names
    • Basic Blocks and Flow Graphs
    • Basic Blocks
    • Next-Use Information
    • Representation of Flow Graphs
    • Optimization of Basic Blocks
    • Dead Code Elimination
    • Representation of Array References
    • Pointer Assignments and Procedure Calls
    • A Simple Code Generator
    • The Code-Generation Algorithm
    • Design of the Function getReg
    • Peephole Optimization
    • Algebraic Simplification and Reduction in Strength
    • Register Assignment for Outer Loops
    • Instruction Selection by Tree Rewriting
    • Code Generation by Tiling an Input Tree
    • Pattern Matching by Parsing
    • General Tree Matching
    • Optimal Code Generation for Expressions
    • Evaluating Expressions with an Insufficient Supply of Registers
    • Dynamic Programming Code-Generation

  • Data Flow Analysis
    • The Lazy-Code-Motion Algorithm
    • Introduction to Machine-Independent Optimizations
    • The Dynamic Programming Algorithm
    • The Principal Sources of Optimization
    • Semantics-Preserving Transformations
    • Copy Propagation
    • Induction Variables and Reduction in Strength
    • Introduction to Data-Flow Analysis
    • The Data-Flow Analysis Schema
    • Reaching Definitions
    • Live-Variable Analysis
    • Available Expressions
    • Foundations of Data-Flow Analysis
    • Transfer Functions
    • The Iterative Algorithm for General Frameworks
    • Meaning of a Data-Flow Solution
    • Constant Propagation
    • Transfer Functions for the Constant-Propagation Framework
    • Partial-Redundancy Elimination
    • The Lazy-Code-Motion Problem
    • Loops in Flow Graphs
    • Depth-First Ordering
    • Back Edges and Reducibility
    • Natural Loops
    • Speed of Convergence of Iterative Data-Flow Algorithms
    • Region-Based Analysis
    • Necessary Assumptions About Transfer Functions
    • An Algorithm for Region-Based Analysis
    • Handling Non-reducible Flow Graphs
    • Symbolic Analysis
    • Data-Flow Problem Formulation
    • Region-Based Symbolic Analysis

  • Code Generation
    • Introduction to Software Pipelining of Loops
    • Matrix Multiply: An In-Depth Example
    • Software Pipelining of Loops
    • Introduction Instruction-Level Parallelism
    • Multiple Instruction Issue
    • A Basic Machine Model
    • Code-Scheduling Constraints
    • Finding Dependences Among Memory Accesses
    • Phase Ordering Between Register Allocation and Code Scheduling
    • Speculative Execution Support
    • Basic-Block Scheduling
    • List Scheduling of Basic Blocks
    • Global Code Scheduling
    • Upward Code Motion
    • Updating Data Dependences
    • Advanced Code Motion Techniques
    • Software Pipelining
    • Register Allocation and Code Generation
    • A Software-Pipelining Algorithm
    • Scheduling Cyclic Dependence Graphs
    • Improvements to the Pipelining Algorithms
    • Conditional Statements and Hardware Support for Software Pipelining
    • Basic Concepts of Parallelism and Locality
    • Parallelism in Applications
    • Loop-Level Parallelism
    • Introduction to Affine Transform Theory
    • Optimizations
    • Iteration Spaces
    • Affine Array Indexes
    • Controlling the Order of Execution
    • Changing Axes
    • Intermediate Code for Procedures
    • Data Reuse
    • Self Reuse
    • Self-Spatial Reuse
    • Array Data-Dependence Analysis
    • Integer Linear Programming
    • Heuristics for Solving Integer Linear Programs
    • Solving General Integer Linear Programs
    • Finding Synchronization-Free Parallelism
    • Affine Space Partitions
    • Space-Partition Constraints
    • Solving Space-Partition Constraints
    • A Simple Code-Generation Algorithm
    • Eliminating Empty Iterations
    • Synchronization Between Parallel Loops
    • The Parallelization Algorithm and Hierarchical Time
    • Pipelining
    • Solving Time-Partition Constraints by Farkas' Lemma
    • Code Transformations
    • Parallelism With Minimum Synchronization
    • Locality Optimizations
    • Partition Interleaving
    • Putting it All Together
    • Uses of Affine Transforms
    • Interprocedural Analysis
    • Context Sensitivity
    • Cloning-Based Context-Sensitive Analysis
    • Importance of Interprocedural Analysis
    • SQL Injection
    • A Logical Representation of Data Flow
    • Execution of Datalog Programs
    • Problematic Datalog Rules
    • A Simple Pointer-Analysis Algorithm
    • Flow Insensitivity
    • Context-Insensitive Interprocedural Analysis
    • Context-Sensitive Pointer Analysis
    • Adding Context to Datalog Rules
    • Datalog Implementation by BDD's
    • Relational Operations as BDD Operations

Branch : Computer Science and Engineering
Subject : Compiler design
Unit : Basic Parsing Techniques

Syntax trees and ambiguity


Introduction: We can draw a derivation as a tree: The root of the tree is the start symbol of the grammar, and whenever we rewrite a non-terminal we add as its children the symbols on the right-hand side of the production that was used. The leaves of the tree are terminals which, when read from left to right, form the derived string. If a non-terminal is rewritten using an empty production, an e is shown as its child. This is also a leaf node, but is ignored when reading the string from the leaves of the tree.

When we write such a syntax tree, the order of derivation is irrelevant: We get the same tree for left derivation, right derivation or any other derivation order. Only the choice of production for rewriting each non-terminal matters.

As an example, the derivations in figures 3.5 and 3.6 yield the same syntax tree, which is shown in figure 3.7.

The syntax tree adds structure to the string that it derives. It is this structure that we exploit in the later phases of the compiler. For compilation, we do the derivation backwards: We start with a string and want to produce a syntax tree. This process is called syntax analysis or parsing.

Even though the order of derivation does not matter when constructing a syntax tree, the choice of production for that non-terminal does. Obviously, different choices can lead to different strings being derived, but it may also happen that several different syntax trees can be built for the same string. As an example, figure 3.8 shows an alternative syntax tree for the same string that was derived in figure 3.7.

When a grammar permits several different syntax trees for some strings we call the grammar ambiguous. If our only use of grammar is to describe sets of strings, ambiguity is not a problem. However, when we want to use the grammar to impose structure on strings, the structure had better be the same every time. Hence, it is a desirable feature for a grammar to be unambiguous. In most (but not all) cases, an ambiguous grammar can be rewritten to an unambiguous grammar that generates the same set of strings, or external rules can be applied to decide which of the many possible syntax trees is the “right one”. An unambiguous version of grammar 3.4 is shown in figure 3.9.

T -> R

T -> aTc

R ->

R -> bR

Grammar 3.9: Unambiguous version of grammar 3.4

How do we know if a grammar is ambiguous? If we can find a string and show two alternative syntax trees for it, this is a proof of ambiguity. It may, however, be hard to find such a string and, when the grammar is unambiguous, even harder to show that this is the case. In fact, the problem is formally un-decidable, i.e., there is no method that for all grammars can answer the question “Is this grammar ambiguous?”. But in many cases it is not difficult to detect and prove ambiguity. For example, any grammar that has a production of the form

N -> NαN , where a is any sequence of grammar symbols, is ambiguous.

We will, in sections 3.12 and 3.14, see methods for constructing parsers from grammars. These methods have the property that they only work on unambiguous grammars, so successful construction of a parser is a proof of un-ambiguity. However, the methods may also fail on certain unambiguous grammars, so they cannot be used to prove ambiguity.

 

In the next section, we will see ways of rewriting a grammar to get rid of some sources of ambiguity. These transformations preserve the language that the grammar generates. By using such transformations (and others, which we will see later), we can create a large set of equivalent grammars, i.e., grammars that generate the same language (though they may impose different structures on the strings of the language).

Given two grammars, it would be nice to be able to tell if they are equivalent. Unfortunately, no known method is able to decide this in all cases, but, unlike ambiguity, it is not (at the time of writing) known if such a method may or may not theoretically exist. Sometimes, equivalence can be proven e.g. by induction over the set of strings that the grammars produce. The converse can be proven by finding an example of a string that one grammar can generate but the other not. But in some cases, we just have to take claims of equivalence on faith or give up on deciding the issue.

Questions of this topic


Ask your question

<
>