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When an FSA is deterministic the set of triples encoding its edges represents a relation that is functional in its ?rst and third components: for every q and σ there is exactly one state p such that hq, p, σi ∈ T. This function is called the transition function of the automaton and is usually denoted δ:
For any state q and symbol σ, then, δ(q, σ) is the state reached from q by following a single edge labeled σ. This can be extended to the path function, a function taking a state q and any string w ∈ Σ∗ which returns the statereached from q by following a path labeled w:
Note that ˆ δ is total (has some value for all q and w) and functional (that value is unique) as a consequence of the fact that δ is, which, in turn, is a consequence of the fact that the automaton is deterministic. In terms of the transition graph, this means that for any string w and any node q, there will always be exactly one path labeled w from q (which leads to δ(q,w)) and this is a consequence of the fact that there is always exactly one edge labeled σ from each node q of the graph and every σ ∈ Σ (which leads to δ(q, σ)).
Normal forms are important because they give us a 'standard' way of rewriting and allow us to compare two apparently different grammars G1 and G2. The two grammars can be shown to
These assumptions hold for addition, for instance. Every instance of addition has a unique solution. Each instance is a pair of numbers and the possible solutions include any third
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The path function δ : Q × Σ* → P(Q) is the extension of δ to strings: This just says that the path labeled ε from any given state q goes only to q itself (or rather never l
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