  We introduce two new types of Dehn functions of group presentations which
seem more suitable (than the standard Dehn function) for infinite group
presentations and prove the fundamental equivalence between the solvability of
the word problem for a group presentation defined by a decidable set of
defining words and the property of being computable for one of the newly
introduced functions (this equivalence fails for the standard Dehn function).
Elaborating on this equivalence and making use of this function, we obtain a
characterization of finitely generated groups for which the word problem can be
solved in nondeterministic polynomial time. We also give upper bounds for these
functions, as well as for the standard Dehn function, for two well-known
periodic groups one of which is an infinite 2-group of intermediate growth and
the other is a free Burnside group of sufficiently large exponent.
