Fast Computations on Ordered Nominal Sets

Abstract

We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon. In contrast to previous approaches, we work directly at the level of orbits, which allows for an accurate complexity analysis. The approach is implemented as the library Ons (Ordered Nominal Sets).Our main motivation is nominal automata, which are models for recognising languages over infinite alphabets. We evaluate Ons in two applications: minimisation of automata and active automata learning. In both cases, Ons is competitive compared to existing implementations and outperforms them for certain classes of inputs.

Fast computations on ordered nominal sets

“…The current paper extends the conference version (ICTAC 2018 [1]) with proofs of all results, new experiments for evaluating Ons based on randomly generated formulas, and an implementation in Haskell, Ons-hs.…”

“…). But if we choose x corresponding to x as in the previous paragraph, then by (1) we obtain that the least support of x equals the least support of x . Hence x = x .…”

“…In the second line, we create the set B containing the elements of {(a, b) | a < b}. To do this, we instruct the constructor of B to create the minimal nominal set containing the element (1,2). This creates the nominal set with the orbit of (1, 2), which is exactly the set {(a, b) | a < b}.…”

“…For each orbit in the set of states, its dimension is chosen uniformly at random between 0 and k, inclusive. Each orbit has a probability 1 2 of consisting of accepting states.…”

“…Notably, Nλ [19] and Lois [20,21] provide a general purpose programming language to manipulate infinite sets. 1 Both tools are based on SMT solvers and use logical formulas to represent the infinite sets. These implementations are very flexible, and the SMT solver does most of the heavy lifting, which makes the implementations themselves relatively straightforward.…”