NoPE: The Counting Power of Transformers with No Positional Encodings

Positional Encodings (PEs) seem to be indispensable for ensuring expressiveness of transformers; without them attention transformers reduce to a bag-of-word model. NoPE-transformers (i.e. with No PEs) with unique hard attention mechanisms were very recently shown to only be able to express regular languages, i.e., with limited counting ability. This paper shows that, with average hard attention mechanisms, NoPE-transformers are still surprisingly expressive: they can express counting languages corresponding to nonnegative integer solutions to multivariate polynomial equations (i.e. Diophantine equations), reasoning about which is well-known to be undecidable. In fact, we provide a precise characterization of languages expressible by Average Hard Attention NoPE-Transformers (NoPE-AHATs): they correspond precisely to what we call \emph{semi-algebraic sets}, i.e., finite unions of sets of nonnegative integer solutions to systems of multivariate polynomial inequations. We obtain several interesting consequences of our characterization. Firstly, NoPE-transformers can express counting properties that are far more complex than established models like simplified counter machines and Petri nets, but cannot express a very simple counting property of PARITY. Secondly, the problem of analyzing NoPE-transformers is undecidable, e.g., whether a given NoPE transformer classifies all input strings in one class. To complement our results, we exhibit a counting language that is not expressible by average hard attention transformers even with arbitrary PEs but is expressible in the circuit complexity class TC$^0$, answering an open problem.

Directed Regular and Context-Free Languages

We study the problem of deciding whether a given language is directed. A language $L$ is \emph{directed} if every pair of words in $L$ have a common (scattered) superword in $L$. Deciding directedness is a fundamental problem in connection with ideal decompositions of downward closed sets. Another motivation is that deciding whether two \emph{directed} context-free languages have the same downward closures can be decided in polynomial time, whereas for general context-free languages, this problem is known to be coNEXP-complete. We show that the directedness problem for regular languages, given as NFAs, belongs to $AC^1$, and thus polynomial time. Moreover, it is NL-complete for fixed alphabet sizes. Furthermore, we show that for context-free languages, the directedness problem is PSPACE-complete.