This paper presents an efficient unification algorithm for first-order predicate logic. The unification problem is described as finding a substitution that makes two terms equal. The authors propose a new algorithm that incorporates an efficient acyclicity test and compare it with other well-known unification algorithms. The algorithm is implemented in PASCAL and is shown to perform well in all cases. The algorithm is based on the idea of transforming a set of equations into a solved form, where each equation is of the form x = t, with x being a variable and t a term. The algorithm uses a set of transformations to simplify the equations and find a most general unifier (mgu). The algorithm is non-deterministic and can be implemented efficiently by grouping equations with common variables. The authors also introduce the concept of multiequations, which allow for the grouping of multiple equations into a single equation. This allows for more efficient processing of equations and reduces the number of transformations needed. The algorithm is then extended to handle multiequations, and it is shown that this approach leads to a more efficient unification algorithm. The algorithm is implemented in PASCAL and is shown to have a time complexity of O(n log n), where n is the number of variables. The algorithm is also shown to handle cases where variables are not unifiable, and it correctly identifies these cases by checking for cycles in the variable dependencies. The authors conclude that their algorithm is efficient and performs well in all cases, and that it is a significant improvement over previous unification algorithms. The algorithm is implemented in PASCAL and is available for further study and use.This paper presents an efficient unification algorithm for first-order predicate logic. The unification problem is described as finding a substitution that makes two terms equal. The authors propose a new algorithm that incorporates an efficient acyclicity test and compare it with other well-known unification algorithms. The algorithm is implemented in PASCAL and is shown to perform well in all cases. The algorithm is based on the idea of transforming a set of equations into a solved form, where each equation is of the form x = t, with x being a variable and t a term. The algorithm uses a set of transformations to simplify the equations and find a most general unifier (mgu). The algorithm is non-deterministic and can be implemented efficiently by grouping equations with common variables. The authors also introduce the concept of multiequations, which allow for the grouping of multiple equations into a single equation. This allows for more efficient processing of equations and reduces the number of transformations needed. The algorithm is then extended to handle multiequations, and it is shown that this approach leads to a more efficient unification algorithm. The algorithm is implemented in PASCAL and is shown to have a time complexity of O(n log n), where n is the number of variables. The algorithm is also shown to handle cases where variables are not unifiable, and it correctly identifies these cases by checking for cycles in the variable dependencies. The authors conclude that their algorithm is efficient and performs well in all cases, and that it is a significant improvement over previous unification algorithms. The algorithm is implemented in PASCAL and is available for further study and use.