This paper investigates the inherent computational complexity of various word problems in automata theory and logic, showing that some problems require exponential time or space. The authors demonstrate that the equivalence problem for regular expressions is inherently computationally inefficient, as it can be reduced to the membership problem in context-sensitive languages. They define several word problems, including those involving regular-like expressions, integer expressions, and Boolean expressions, and analyze their complexity in terms of deterministic and nondeterministic Turing machine time and space requirements. The paper introduces log-space reducibility and log-linear reducibility, and shows that these relations are transitive. It also establishes that certain word problems, such as the inequivalence of regular-like expressions over {0,1}, are log-linear complete in exponential space. The authors prove that the family of sets recognizable in nondeterministic exponential time is log-linear reducible to the inequivalence problem for regular-like expressions over {0,1}. The paper also discusses the complexity of integer expressions and Boolean expressions, showing that certain decision problems, such as the inequivalence of integer expressions, are complete in polynomial space. It further explores the relationship between the polynomial time hierarchy and polynomial space, and shows that the set of disjunctive (conjunctive) form formulas in B_k is log-space complete when k is even (odd). The authors conclude that the inherent complexity of these word problems can be precisely characterized in terms of time and space requirements on Turing machines. They also identify several open problems, including the computational complexity of the inequivalence problem for regular-like expressions over {0} and the equivalence problem for polynomial expressions over finite sets of integers. The results suggest that many decision problems in logic and automata theory require exponential or greater time to solve.This paper investigates the inherent computational complexity of various word problems in automata theory and logic, showing that some problems require exponential time or space. The authors demonstrate that the equivalence problem for regular expressions is inherently computationally inefficient, as it can be reduced to the membership problem in context-sensitive languages. They define several word problems, including those involving regular-like expressions, integer expressions, and Boolean expressions, and analyze their complexity in terms of deterministic and nondeterministic Turing machine time and space requirements. The paper introduces log-space reducibility and log-linear reducibility, and shows that these relations are transitive. It also establishes that certain word problems, such as the inequivalence of regular-like expressions over {0,1}, are log-linear complete in exponential space. The authors prove that the family of sets recognizable in nondeterministic exponential time is log-linear reducible to the inequivalence problem for regular-like expressions over {0,1}. The paper also discusses the complexity of integer expressions and Boolean expressions, showing that certain decision problems, such as the inequivalence of integer expressions, are complete in polynomial space. It further explores the relationship between the polynomial time hierarchy and polynomial space, and shows that the set of disjunctive (conjunctive) form formulas in B_k is log-space complete when k is even (odd). The authors conclude that the inherent complexity of these word problems can be precisely characterized in terms of time and space requirements on Turing machines. They also identify several open problems, including the computational complexity of the inequivalence problem for regular-like expressions over {0} and the equivalence problem for polynomial expressions over finite sets of integers. The results suggest that many decision problems in logic and automata theory require exponential or greater time to solve.