Active Learning of Nondeterministic Systems from an ioco Perspective

This document contains the proofs to the theorems of the original paper. Some lemmas are also introduced. Theorem 1. Let qδ ∈ LT S(LI , LU ∪ {δ}) be a valid suspension automaton. Then, there exists a labelled transition system q ∈ LT S(LI , LU ) such that Straces(q) = traces(qδ). Proof. Follows directly from Theorem 2 in [18]. u t Algorithm 1 Construct Hypothesis H Input: A closed and consistent observation table (S,E, T ). Output: A labelled transition system H = 〈Q,LI , LU ∪ {δ},→, q0〉. 1: Q = {row(s) | s ∈ S} 2: q0 = row( ) 3: for each row(s) ∈ Q do 4: for each λ ∈ LI do 5: add row(s) λ −→ row(s·λ) 6: for each λ ∈ (LU ∪ {δ}) do 7: if λ ∈ T (s, ) then 8: add row(s) λ −→ row(s·λ) Lemma 1. Let (S,E, T ) be a closed and consistent observation table such that S is prefix closed and let H be the hypothesis obtained by running Algorithm 1 on (S,E, T ), then ∀s ∈ (S ∪ S·Lδ) . (H after s) is equal to either {row(s)}, if row(s) is defined, or ∅, if not. ∗ This research is supported by the Dutch Technology Foundation STW, which is part of the Netherlands Organisation for Scientific Research (NWO), and which is partly funded by the Ministry of Economic Affairs. Proof. We prove it by induction on the length of s. For length 0, trivially (H after ) = {row( )}. Let us assume it is true for any trace of length at most k ≥ 0. Let s ∈ (S ∪ S·Lδ) be a trace of length k + 1. We can decompose s into s1λ where s1 is a trace of length k and λ ∈ Lδ. The trace s1 must be in S, because, either s is in S·Lδ (thus s1 ∈ S), or s is in S, which is prefix closed, and thus s1 ∈ S. (H after s) = (H after s1λ) = ((H after s1) after λ) property of after = (row(s1) after λ) induction hypothesis = {row(s1λ)} Algorithm 1, lines 5 and 8 = {row(s)} If λ is enabled in row(s1) then row(s1λ) is defined and it is equal to the row of a valid state, because (S,E, T ) is closed and consistent. If λ is not enabled in row(s1) then row(s1λ) is not defined and (H after s1λ) = ∅ as expected. u t Theorem 2. If an observation table (S,E, T ) is closed and consistent, S is prefix closed and E is suffix closed, then the hypothesis H, obtained by running Algorithm 1 on it, is compatible with the function T , i. e., ∀s ∈ (S ∪S·Lδ),∀e ∈ E . out(H after s·e) = T (s·e). Proof. We prove it by induction on the length of e. Consider e = and s any element of (S ∪ S·Lδ). out(H after s·e) = out(H after s) = out({row(s)}) Lemma 1 = T (s) Algorithm 1, line 8 = T (s·e) Let us assume it is true for any e ∈ E of length at most k ≥ 0. Let e′ ∈ E be a trace of length k + 1. E is suffix closed, thus we can decompose e′ into a·e for some a ∈ Lδ and some e ∈ E of length k. The observation table is closed and s ∈ (S ∪ S·Lδ), thus ∃s1 ∈ S . row(s) = row(s1). (Note that s1a ∈ (S ∪ S·Lδ)). out(H after s·e′) = out(H after sa·e) = out((H after s) after a·e) property of after = out({row(s)} after a·e) Lemma 1 = out({row(s1)} after a·e) row(s) = row(s1) = out((H after s1) after a·e) Lemma 1 = out(H after s1a·e) property of after = T (s1a·e) induction hypothesis = T (s1·e) = T (s·e′) row(s) = row(s1)