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 k-linear combination of length-lexicographically lesser words. An obstruction subword-minimal word. If the number obstructions then I has Gröbner basis, and problem for algebra decidable. cogrowth function length ≤n. We show that finitely presented either bounded or at least logarithmical. also uniformly recurrent logarithmical cogrowth.