LTK.Decide.FO2

This module implements an algorithm to decide whether a given FSA is representable in two-variable logic based on the semigroup characterization as reported by Thérien and Wilke in their 1998 STOC article: https://doi.org/10.1145/276698.276749

Two-variable logic with general precedence is a strict superclass of PT while still being a strict subclass of star-free. It represents exactly the class of properties expressible in temporal logic using only the "eventually in the future/past" operators.

The section regarding betweenness is built on Krebs et al. (2020): https://doi.org/10.23638/LMCS-16(3:16)2020

Since: 1.0

Documentation

isFO2 :: (Ord n, Ord e) => FSA n e -> Bool

True iff the automaton recognizes a stringset representable in \(\mathrm{FO}^{2}[<]\).

isFO2B :: (Ord n, Ord e) => FSA n e -> Bool

True iff the automaton recognizes a stringset representable in \(\mathrm{FO}^{2}[<,\mathrm{bet}]\). Labelling relations come in the typical unary variety \(\sigma(x)\) meaning a \(\sigma\) appears at position \(x\), and also in a binary variety \(\sigma(x,y)\) meaning a \(\sigma\) appears strictly between the positions \(x\) and \(y\).

isFO2S :: (Ord n, Ord e) => FSA n e -> Bool

True iff the automaton recognizes a stringset representable in \(\mathrm{FO}^{2}[<,+1]\).

isFO2M :: (Ord n, Ord e) => SynMon n e -> Bool

True iff the monoid represents a language in \(\mathrm{FO}^{2}[<]\).

isFO2BM :: (Ord n, Ord e) => SynMon n e -> Bool

True iff the monoid represents a stringset that satisfies isFO2B.

isFO2SM :: (Ord n, Ord e) => SynMon n e -> Bool

True iff the local subsemigroups are in \(\mathrm{FO}^{2}[<]\). This means the whole is in \(\mathrm{FO}^{2}[<,+1]\).