On cogrowth function of algebras and its logarithmical gap

Let A ≅ k 〉X〈/I be an associative algebra. A finite word over alphabet X is I -reducible if its image in A is a k-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal I - reducibleword. If the number of obstructions is finite then I has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.