Directed st-Connectivity Is Not Expressible in Symmetric Datalog

We show that the directed st-connectivity problem cannot be expressed in symmetric Datalog, a fragment of Datalog introduced in [5]. It was shown there that symmetric Datalog programs can be evaluated in logarithmic space and that this fragment of Datalog captures logspace when augmented with negation, and an auxiliary successor relation S together with two constant symbols for the smallest and largest elements with respect to S. In contrast, undirected st-connectivity is expressible in symmetric Datalog and is in fact one of the simplest examples of the expressive power of this logic. It follows that undirected non-st-connectivity can be expressed in restricted symmetric monotone Krom SNP, whereas directed non-st-connectivity is only definable in the more expressive restricted monotone Krom SNP. By results of [8], the inexpressibility result for directed st-connectivity extends to a wide class of homomorphism problems that fail to meet a certain algebraic condition.