Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

Abstract

The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem:Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) > T(n)+T(m)).Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is 0(T(|a;| + \v\) 2 )-Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(T 0 )) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(T D )).In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function < D (where D(n) > n), then the word problem of S has nondeterministic time complexity 0(D).The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T)°' 1 ')) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) 0 ' 1 ') isoperimetric function.An algorithmic problem L is in NP (or more generally, in NTIME^T) 0^) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) 0 ' 1 ') isoperimetric function.In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

Isoperimetric and Isodiametric Functions of Groups

“…Since f (n) is equivalent to g(n) 4 , we conclude that f (n) is equivalent to the Dehn function of a finitely presented group. 2…”

“…By that time the semigroup analog of this conjecture was proved by Birget. It appeared later in preprint [4]. In this preprint, Birget also proved semigroup versions of the main results of this paper about isoperimetric functions: analogs of Theorems 1.1, 1.2, 1.3.…”

“…This is like decomposing a snowman into small snow balls. Then we show that the total area of the diagrams without discs is bounded by |W | 4 and the sum of the lengths of boundaries of discs is bounded roughly by O(|W |). The area of a disc with perimeter n is bounded by the area of the corresponding computation of S, that is by a function equivalent to T (n) 4 and its diameter is equivalent to T 3 (n).…”

“…By an area of a word w = 1 in G we mean the smallest area of a van Kampen diagram with boundary label w. Then the function d G,H (n) shows how distorted the areas of elements in the subgroup G of H are. Theorem 1.1 shows that the function d G,H (n) cannot be lower than the computational complexity T (n) of the word problem for G. Corollary 1.1 shows that the distortion can reach as low as T (n) 4 . We are not going to prove Corollary 1.1 in this paper because in the next paper (joint with A. Yu.…”

“…The area of a disc with perimeter n is bounded by the area of the corresponding computation of S, that is by a function equivalent to T (n) 4 and its diameter is equivalent to T 3 (n). Using the fact that T 4 is superadditive, we conclude that the total area of the maximal discs inside ∆ is bounded from above by a function equivalent to T (n) 4 . This gives us the upper bound of the Dehn function.…”