2019  OriginalPaper  Buchkapitel Open Access
Coalgebra Learning via Duality
1 Introduction
2 Learning by Example
3 Preliminaries
4 Subformula Closed Collections of Formulas
5 Reachability and the Base
6 Learning Algorithm

\((S,\hat{\gamma }, \hat{s}_0)\) is correct w.r.t. the collection \(\varPhi \) of all tests, i.e., the theory of \((X,\gamma )\) and \((S,\hat{\gamma })\) coincide on the initial states \(s_0\) and \(\hat{s}_0\), (Definition 25);

\((S,\hat{\gamma },\hat{s}_0)\) is minimal w.r.t. logical equivalence;

\((S,\hat{\gamma },\hat{s}_0)\) is reachable.

for any subobject \(S \rightarrowtail X\) and subformula closed subobject \(\varPsi \) of \(\varPhi \), the composite theory map

for \((S, \hat{\gamma }, \hat{s}_0)\) a pointed coalgebra, whether or not it is correct w.r.t. the collection \(\varPhi \) of all tests;

in case of a negative answer to the previous question, a counterexample, which essentially is a subobject \(\varPsi '\) of \(\varPhi \) representing some tests on which the learned coalgebra is wrong (defined more precisely below);

for a given subobject S of X, the ‘next states’; formally, the computation of the Bbase of the composite arrow

that we deal with coalgebras over the base category \(\mathcal {C} = \mathsf {Set}\);

a functor \(B :\mathcal {C} \rightarrow \mathcal {C}\) that preserves preimages and wide intersections;

a category \(\mathcal {D}\) with an initial object 0 s.t. arrows with domain 0 are monic;

a functor \(L :\mathcal {D} \rightarrow \mathcal {D}\) with an initial algebra \(L \varPhi {\mathop {\rightarrow }\limits ^{\cong }} \varPhi \);

an adjunction \(P\dashv Q:\mathcal {C} \leftrightarrows \mathcal {D}^\mathsf {op}\), and a logic \(\delta :L P\Rightarrow PB\).