Question

Here’s a problem that occurs in automatic program analysis. For a set of variables${\mathit{x}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{..}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{x}}_{\mathbf{n}}$, you are given some equality constraints, of the form “ ${\mathit{x}}_{\mathbf{i}}\mathbf{=}{\mathit{x}}_{\mathbf{j}}$” and some disequality constraints, of the form “ ${\mathit{x}}_{\mathbf{i}}\mathbf{\ne }{\mathit{x}}_{\mathbf{j}}$.” Is it possible to satisfy all of them?

For instance, the constraints.

$\mathbf{}{\mathit{x}}_{\mathbf{1}}\mathbf{=}{\mathit{x}}_{\mathbf{2}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}\mathbf{=}{\mathit{x}}_{\mathbf{3}}\mathbf{,}{\mathit{x}}_{\mathbf{3}}\mathbf{=}{\mathit{x}}_{\mathbf{4}}\mathbf{,}{\mathit{x}}_{\mathbf{1}}\mathbf{\ne }{\mathit{x}}_{\mathbf{4}}$

cannot be satisfied. Give an efficient algorithm that takes as input m constraints over $\mathit{n}$ variables and decides whether the constraints can be satisfied.

Short answer

Yes, it is possible to satisfy all of them and its efficient algorithm that takes as input and constraints over variables is given below.

Step-by-step solution

Automatic program analysis.

The Automatic program analysis program analysis is the process of automatically analyzing the behavior of computer programs regarding a property such as correctness, robustness, safety and likeness. Automatic program analysis program analysis focuses on two major areas that are program optimization and program correctness.

Efficient algorithm for satisfy the condition.

A problem where you maximize or minimize a real function by systematically choosing input values from an allowed set and computing the value of the function and a set of variables $x_1,......,x_n$, with some equality constraints of the form “$x_i=x_j$” and some disequality constraints, of the form “ $x_i=x_j$.” For that instance, the constraints is given in the question is,

$x_1=x_2,x_2=x_3,x_3=x_4,x_1\ne x_4$

Cannot be satisfied. And that will takes as input m constraints over n variables.

Let’s take input: m and n, which contain variables as x and y also contains constraints over these all variable, the program analysis using constraints is divisible into constraint generation and constraint resolution. Constraint generation produces constraints from a program text that give a declarative speciation of the desired information about the program. Constraint resolution (i.e., solving the constraints) then computes this desired information. In the author’s view, the constraint-based analysis paradigm is appealing for three primary reasons constraints separate speciation from implementation, constraints yield natural speculations and constraints enable sophisticated implementations.

Here, set of variables$x_1,......,x_n$, with some equality constraints of the form “ $x_i=x_j$” and some disequality constraints, of the form is$x_i\ne x_j$. An inequality constraint can be either active, ε-active, violated, or inactive at a design point. On the other hand, an equality constraint is either active or violated at a design point.

For equality constraints of the form the condition must be follows: “$x_i=x_j$”.

And for disequality constraints, of the form the condition must be follows: “inequality constraints”.

And by the definition of the equality constraints and from the disequality constraints these above both the conditions satisfy the instances which are showing in the equation.

$x_1=x_2,x_2=x_3,x_3=x_4,x_1\ne x_4$, hence it is shown in the equation that in the prefix constraints it follows equality constraints and in the end equation it shows the inequality constraints with the variable here and treated it as input variable $x_1=x_2,x_2=x_3,x_3=x_4,$. follows equality constraints. And here$x_1\ne x_4$is inequality constraints with this condition .Hence, it satisfy the constraints.