An isomorphism of complete degrees of finite-state transformability

Abstract

Let ( Q, ∑, δ, ω ) be a finite-state automaton with input-output alphabet ∑ . Let ∑ + be the free semigroup generated by ∑ and ∑ N be the set of all infinite sequences of elements of ∑ . Rayna (1974) Inform. Contr. 24 calls a sequence x ∑ ∑ N complete if and only if every A ∑ ∑ + is contained in x . Huffman (1959) IRE Trans. Circuit Theory Suppl. CT-6 calls a finite-state automaton information-lossless iff there exist no two states q i and q f and no two different equal-length input blocks A, B ∑ ∑ + and an output block C ∑ ∑ + such that δ ( q i , A ) = q f = δ ( q i , B ) and ω ( q i , A ) = C = ω ( q i , B ). Gordon (1976) Inform Contr. 32 calls a degree of finite-state transformability complete iff it contains a complete sequence. In that paper the properties of an information-lossless finite-state automaton are studied when a complete sequence is the input. In this paper we use the properties discovered by Gordon (1976) to study the set U [ x ] of upper bounds for a complete degree of finite-state transformability. The main result states that, for any two complete degrees [ x ] and [ y ], we have U [ x ] is orderisomorphic to U [ y ].