(Finite State Automaton (FSA)) An FSA is a 5-tuple A = (Q, Σ, δ, i, F) where Q is a ﬁnite set of states, Σ is the ﬁnite input alphabet, δ ⊆ Q × (Σ ∪ { }) × Q is the transition relation, i ∈ Q is the initial state, and F ⊆ Q are the ﬁnal states. Deﬁnition 21 (Extended Transition Relation) Deﬁne the extended transition relation ¯ δ ⊆ Q × Σ∗ × Q inductively: 1 if (s, a, t) ∈ δ, then (s, a, t) ∈ ¯ δ; 2 if (s, a, q) ∈ δ and (q, w, t) ∈ ¯ δ, then (s, aw, t) ∈ ¯ δ. Deﬁnition 22 (Language Accepted by an FSA) Deﬁne L(A) := {w ∈ Σ∗ | ∃q ∈ F with (i, w, q) ∈ ¯ δ}. Call A deterministic whenever δ is functional on Q × Σ ∪ { }, and without -transitions (no (s, , t) ∈ δ for any s, t ∈ Q). Graham Campbell School of Mathematics, Statistics and Physics, Newcastle University, UK Formal Languages and Groups