Approximating Petri Net Reachability Along Context-free Traces

We investigate the problem asking whether the intersection of a context-free language (CFL) and a Petri net language (PNL) is empty. Our contribution to solve this long-standing problem which relates, for instance, to the reachability analysis of recursive programs over unbounded data domain, is to identify a class of CFLs called the finite-index CFLs for which the problem is decidable. The k-index approximation of a CFL can be obtained by discarding all the words that cannot be derived within a budget k on the number of occurrences of non-terminals. A finite-index CFL is thus a CFL which coincides with its k-index approximation for some k. We decide whether the intersection of a finite-index CFL and a PNL is empty by reducing it to the reachability problem of Petri nets with weak inhibitor arcs, a class of systems with infinitely many states for which reachability is known to be decidable. Conversely, we show that the reachability problem for a Petri net with weak inhibitor arcs reduces to the emptiness problem of a finite-index CFL intersected with a PNL.Comment: 16 page

The Non-Hierarchical Nature of the Chomsky Hierarchy-Driven Artificial-Grammar Learning

Recent artificial-grammar learning (AGL) paradigms driven by the Chomsky hierarchy paved the way for direct comparisons between humans and animals in the learning of center embedding ([A[AB]B]). The AnBn grammars used by the first generation of such research lacked a crucial property of center embedding, where the pairs of elements are explicitly matched ([A1 [A2 B2] B1]). This type of indexing is implemented in the second-generation AnBn grammars. This paper reviews recent studies using such grammars. Against the premises of these studies, we argue that even those newer AnBn grammars cannot test the learning of syntactic hierarchy. These studies nonetheless provide detailed information about the conditions under which human adults can learn an AnBn grammar with indexing. This knowledge serves to interpret recent animal studies, which make surprising claims about animals’ ability to handle center embedding

On the Commutative Equivalence of Algebraic Formal Series and Languages

The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated