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Equivalence Checker Using ROBDD

Introduction

This tool is designed to check the equivalence of two Boolean expressions using Reduced Ordered Binary Decision Diagrams (ROBDD). It includes a series of modules that extract variables, parse Boolean expressions, build ROBDD trees, visualize these trees as graphs and compare them using variable mapping and truth table generation.

Features

1- Boolean Expression Parser Module

This module is divided into several functions, each handling a specific part of the parsing and evaluation process:

Logical Operators

Implementations for each Boolean operator are provided:

  • NOT(a): Logical NOT operation.
  • XOR(a, b): Exclusive OR operation.
  • AND(a, b): Logical AND operation.
  • OR(a, b): Logical OR operation.
  • NOR(a, b): Logical NOR operation.
  • NAND(a, b): Logical NAND operation.

Variable Extraction

  • extract_variables(expr): Extracts and returns a sorted list of unique variable names from the given Boolean expression.

Expression Parsing

  • parse_boolean_expression(expr): Main function to initiate the parsing of the given Boolean expression. It breaks down the expression into tokens and processes them to construct and evaluate the Boolean logic.

Parsing Details

The parsing process involves several nested functions designed to evaluate expressions based on the precedence of operations:

  • parse_factor(): Handles individual factors in the expression, managing NOT operations and parentheses.
  • parse_term(): Manages XOR operations.
  • parse_expression(): Handles AND and OR operations, including combinations with NAND and NOR.

Usage

To use this script, simply import and call parse_boolean_expression with a Boolean expression as a string:

from boolean_parser import parse_boolean_expression

# Take care |~ represents NOR gate and &~ represents NAND gate
expression = "a & (b | !c)"
result = parse_boolean_expression(expression)
print(result)
#Output : AND(a, OR(b, NOT(c)))

expression = "a & (b | ~c)"
result = parse_boolean_expression(expression)
print(result)
#Output : AND(a, NOR(b, c))

2- Truth Table Module

This module generates a truth table for a given Boolean expression and compares two truth tables to determine if their corresponding expressions are equivalent.

Evaluate Parsed Expression

  • evaluate_parsed_expression(parsed_expr, var_values): Evaluates a parsed Boolean expression based on the given variable values. This function dynamically replaces variable placeholders in the parsed expression with actual Boolean values and evaluates the result.

Generate Truth Table

  • generate_truth_table(expression): Generates and prints a truth table for a specified Boolean expression. This function:
    • Extracts variables from the expression.
    • Parses the Boolean expression to a form suitable for evaluation.
    • Iterates over all possible combinations of truth values for the variables.
    • Evaluates the expression for each combination and prints the result.

Compare Truth Tables

  • compare_truth_tables(table1, table2): Compares two truth tables to check if they represent equivalent expressions. It returns True if the tables are equivalent, otherwise False.

Usage

To generate a truth table for a Boolean expression:

from truth_table import generate_truth_table

# Example Boolean expression
expression = "a & (b | !c)"
truth_table = generate_truth_table(expression)

# a b c | Result
# --------------
# 0 0 0 | 0
# 0 0 1 | 0
# 0 1 0 | 0
# 0 1 1 | 0
# 1 0 0 | 1
# 1 0 1 | 0
# 1 1 0 | 1
# 1 1 1 | 1

3- ROBDD Graph Construction and Manipulation Module

This module implements a Reduced Ordered Binary Decision Diagram (ROBDD) for representing and manipulating Boolean expressions. It includes utilities for constructing an ROBDD from a Boolean expression, comparing two ROBDDs and generating a visual representation of an ROBDD using DOT format.

BDDNode Class:

  • BDDNode(var, high, low, terminal) defines a node in the Binary Decision Diagram (BDD), which can either represent a variable or a terminal value (True/False).

Shannon Expansion:

  • shannon_expansion(expr, var) decomposes a Boolean expression to build the BDD by recursively applying Shannon expansion.

Reduction Function:

  • reduction(node) implements the reduction rules to simplify the BDD and cache nodes to avoid redundant subtrees.

Expression Parsing:

  • parse_boolean_expression function (from an boolean parser module) is used to parse Boolean expressions.

Comparison of ROBDDs:

  • compare_bdds_with_variable_mapping(bdd1, bdd2, var_map=None) allows comparing two ROBDDs while handling variable renaming, ensuring that functionally equivalent ROBDDs are identified.

DOT Representation:

  • robdd_to_dot(robdd) provides a function to convert an ROBDD into DOT format for visualization in graph tools like graphviz.

Usage

from graphviz import Source
from robdd_graph import generate_robdd, robdd_to_dot

expression_1 = "(A & B) | (!A & C)"
# The equivalent expression using De-Morgan 
expression_2 = "!((!X | !Y) & (X | !Z))"

robdd_1 = generate_robdd(expression_1)
robdd_2 = generate_robdd(expression_2)

dot_representation_1 = robdd_to_dot(robdd_1)
dot_representation_2 = robdd_to_dot(robdd_2)

src_1 = Source(dot_representation_1)
src_2 = Source(dot_representation_2)

src_1.render("robdd_1", format='png', cleanup=True)
src_2.render("robdd_2", format='png', cleanup=True)

Installation

Prerequisites

  • Python 3.8 or higher
  • Graphviz for graph visualization

Setup

Clone the repository and install the required Python packages:

git clone https://github.com/ahmd-kamel/ROBDD-Based-Equivalence-Checker.git
cd ROBDD-Based-Equivalence-Checker
pip install graphviz
python3 equivalence_checker.py 'A&B' 'A^C'