  To study the distributed task solvability, Goubault, Ledent, and Rajsbaum
devised a model of dynamic epistemic logic that is equivalent to the
topological model for distributed computing. In the logical model, the
unsolvability of a particular distributed task can be proven by finding a
formula, called logical obstruction. This logical method is very appealing
because the concrete formulas that prevent to solve task would have
implications of intuitive factors for the unsolvability. However, it has not
been well studied when a logical obstruction exists and how to systematically
construct a concrete logical obstruction formula, if any. In addition, it is
proved that there are some tasks that are solvable but do not admit logical
obstructions.
  In this paper, we propose a method to prove the non-existence of logical
obstructions to the solvability of distributed tasks, based on the technique of
simulation. Moreover, we give a method to determine whether a logical
obstruction exists or not for a finite protocol and a finite task, and if it
exists, construct a concrete obstruction. Using this method, we demonstrate
that the language of the standard epistemic logic, without distributed
knowledge, does not admit logical obstruction to the solvability of $k$-set
agreement tasks. We also show that there is no logical obstruction for
multi-round immediate snapshot even in the language of epistemic logic with
distributed knowledge. In addition, for the know-all model, we provide a
concrete obstruction formula that shows the unsolvability of the $k$-set
agreement task.
