Graph Isomorphism is PSPACE-complete

Combining the the results of A.R. Meyer and L.J. Stockmeyer “The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space”, and K.S. Booth “Isomorphism testing for graphs, semigroups, and finite automata are polynomiamlly equivalent problems” shows that graph isomorphism is PSPACE-complete.

💡 Research Summary

The manuscript entitled “Graph Isomorphism is PSPACE‑complete” attempts to establish that the graph isomorphism problem (GI) belongs to the class of PSPACE‑complete problems by stitching together two earlier results: the PSPACE‑hardness of regular‑expression equivalence with squaring proved by Meyer and Stockmeyer (1972), and the polynomial‑time equivalence between graph isomorphism and finite‑automaton isomorphism shown by Booth (1978). While the idea of leveraging known hardness results to raise the complexity of GI is attractive, the paper fails to provide a rigorous reduction and misinterprets the cited literature in several critical ways.

First, Meyer and Stockmeyer demonstrated that the equivalence problem for regular expressions that allow the squaring operation requires exponential space. Their result shows that this restricted version of regular‑expression equivalence is PSPACE‑hard, but it does not claim PSPACE‑completeness for the unrestricted regular‑expression equivalence problem. Moreover, the proof is specific to the presence of the squaring operator; it does not automatically extend to other formalisms such as finite automata or graphs.

Second, Booth proved that testing isomorphism of deterministic finite automata (DFA) can be reduced to graph isomorphism and vice versa in polynomial time. This establishes a polynomial‑time equivalence between the two decision problems, implying that they lie in the same complexity class. However, Booth’s result does not say anything about the space complexity of DFA isomorphism. In fact, DFA isomorphism is known to be in NP, and no PSPACE‑hardness result for DFA isomorphism is known. Consequently, the chain “regular‑expression equivalence is PSPACE‑hard → DFA isomorphism is PSPACE‑hard → graph isomorphism is PSPACE‑hard” is broken at the second step.

Third, the paper invokes the well‑known equivalence between regular expressions, right‑linear grammars, and finite automata. While these formalisms generate the same class of regular languages, the equivalence is language‑theoretic, not complexity‑theoretic. A reduction that preserves language recognition does not necessarily preserve the amount of space required to solve the associated decision problem. To transfer PSPACE‑hardness from regular‑expression equivalence to graph isomorphism, one would need an explicit space‑preserving reduction, which the manuscript does not provide.

The bibliography further reveals inaccuracies. Reference