WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS Merve Nur Cakir, Mehwish Saleemi, DOI: 10.37394/23209.2021.18.6 Karl-Heinz Zimmermann On the Decomposition of Generalized Semiautomata MERVE NUR CAKIR, MEHWISH SALEEMI, KARL-HEINZ ZIMMERMANN Department of Computer Engineering Hamburg University of Technology 20171 Hamburg GERMANY Abstract: - Semiautomata are abstractions of electronic devices that are deterministic finite-state machines having inputs but no outputs. Generalized semiautomata are obtained from stochastic semiautomata by dropping the restrictions imposed by probability. It is well-known that each stochastic semiautomaton can be decomposed into a sequential product of a dependent source and deterministic semiautomaton making partly use of the celebrated theorem of Birkhoff-von Neumann. It will be shown that each generalized semiautomaton can be partitioned into a sequential product of a generalized dependent source and a deterministic semiautomaton. Key-Words: - Semiautomaton, stochastic automaton, monoid, Birkhoff-von Neumann Received: May 31, 2021. Revised: May 4, 2021. Accepted: May 22, 2021. Published: May 29, 2021 1 Introduction ity [5, 16]. It is well-known that each stochastic au- The theory of discrete stochastic systems has been tomaton can be decomposed into a sequential product initiated by the work of Shannon [14] and von Neu- of a dependent source and deterministic semiautoma- mann [10]. While Shannon has considered memory- ton [2]. This result makes use in part of the celebrated less communication channels and their generalization theorem of Birkhoff-von Neumann that each doubly by introducing states, von Neumann has studied the stochastic matrix can be represented as a convex com- synthesis of reliable systems from unreliable compo- bination of permutation matrices. In this paper, it will nents. The fundamental work of Rabin and Scott [12] be shown that each generalized semiautomaton can be about deterministic finite-state automata has led to partitioned into a sequential product of a generalized two generalizations. First, the generalization of tran- dependent source and a deterministic semiautomaton. sition functions to conditional distributions studied by Carlyle [3] and Starke [15]. This in turn yields a gen- Notation. Let X be a set. The set of all mappings eralization of discrete-time Markov chains in which on X, T (X) = ff j f : X ! Xg, forms a monoid the chains are governed by more than one transition under function composition (fg)(x) = g(f(x)), x 2 probability matrix. Second, the generalization of reg- X, and the identity function idX : X ! X : x 7! x ular sets by introducing stochastic automata as de- is the identity element. The monoid T (X) is called scribed by Rabin [11]. the full transformation monoid of X. By the work of Turakainen [16], stochastic accep- tors can be viewed equivalently as generalized au- 2 Semiautomata tomata in which the ”probability” is neglected. This Semiautomata are abstractions of electronic devices leads to a more accessible approach to stochastic au- which are deterministic finite-state machines having tomata [5]. input but no output [7, 9]. On the other hand, the class of nondeterministic A (deterministic) semiautomaton (SA) is a triple automata [13] can be generalized to monoidal au- tomata, where the input alphabet corresponds to an A = (S; Σ; fδx j x 2 Σg) arbitrary monoid instead of a free monoid [8, 9, 17]. This leads to the class of monoidal automata whose where languages are closed under a smaller set of operations • S is the non-empty finite set of states, when compared with regular languages. A first step into the study of automata theory are • Σ is the set of input symbols, semiautomata which are abstractions of electronic de- • δ : S ! S is a (partial) mapping for each x 2 Σ. vices that are deterministic finite-state machines hav- x ing inputs but no outputs [7, 9]. Generalized semi- Let Σ∗ denote the free monoid over the alphabet Σ. automata are obtained from stochastic semiautomata By the universal property of free monoids [4, 9], the by dropping the restrictions imposed by probabil- mapping δ :Σ ! T (S): x 7! δx extends uniquely to E-ISSN: 2224-3402 34 Volume 18, 2021 WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS Merve Nur Cakir, Mehwish Saleemi, DOI: 10.37394/23209.2021.18.6 Karl-Heinz Zimmermann ∗ a monoid homomorphism δ :Σ ! T (S): u 7! δu • S is the non-empty finite set of states, such that for each word u = x : : : x 2 Σ∗, 1 k • Σ is the input alphabet, and δ = δ ··· δ (1) u x1 xk • Q is a collection of n × n nonnegative matrices Qx, x 2 Σ, where n is the number of states. and particularly δϵ = idS. The mapping δ is called the transition function of A. Its image T (A) = fδu j In view of the universal property of free u 2 Σ∗g is a submonoid of the full transformation monoids [4, 9], the mapping Q :Σ ! Rn×n : monoid T (S) generated by fδx j x 2 Σg. The semi- x 7! Qx extends uniquely to a monoid homomor- automaton A is also denoted by A = (S; M; δ) or phism Q :Σ∗ ! Rn×n such that for each word A A A ∗ A = (S ;M ; δ ). u = x1 : : : xk 2 Σ , A semiautomaton A = (S; Σ; δ) serves as a skele- Q = Q ··· Q (2) ton of a deterministic finite-state machine that is ex- u x1 xk actly in one state at a time. If the semiautomaton A is and particularly Qϵ = In is the n × n identity ma- in state s and reads the word u 2 Σ∗, it transits into trix. The mapping Q is called the transition func- 0 ∗ the state s = δu(s). tion of A. Its image T (A) = fQu j u 2 Σ g is a submonoid of the full transformation monoid T (S) Example 1. Consider the semiautomaton A = f j 2 g f g generated by Qx x Σ . The generalized semi- (S; Σ; δ) with state set S = 1; 2; 3 , input alpha- automaton A is also denoted by A = (S; Σ;Q) or bet Σ = fx; yg, and transition function δ given by the A = (SA; ΣA;QA). automaton graph in Fig. 1. The associated transfor- The state set S = fs1; : : : ; sng can be viewed as mation monoid is generated by the transformations Rn ( ) ( ) the standard basis for the Euclidean vector space , 1 2 3 1 2 3 where si is the basis vector whose ith coordinate is 1 δ = δ = : x 1 1 1 and y 2 2 3 and all others are 0. In this way, the (i; j)the entry of (u) (u) T the matrix Qu = (sij ) is given by sij = si Qusj. We have ( ) ( ) Proposition 1. Each deterministic semiautomaton is 1 2 3 1 2 3 a generalized automaton. δxx = ; δxy = ; ( 1 1 1 ) ( 2 2 2 ) Proof. Let A = (S; Σ; δ) be a deterministic semiau- 1 2 3 1 2 3 f g δ = ; δ = : tomaton and let S = s1; : : : ; sn . Define the gener- yx 1 1 1 yy 2 2 3 alized semiautomaton B = (S; Σ;Q), where for each x 2 Σ, the (i; j)th entry of Qx is 1 if δx(si) = sj Hence, the transformation monoid T (A) is given by and otherwise 0. Then the mapping T (A) ! T (B): f g } idS; δx; δy; δxy . δu 7! Qu is a monoid isomorphism. y A generalized semiautomaton A = (S; Σ;P ) is + x ?>=<89:;k ?>=<89:; y called stochastic if the matrices Px, x 2 Σ, are 6 1 >^ > 2 h > x stochastic, i.e., P is a matrix of nonnegative real >> x >> numbers such that each row sum is equal to 1. The x >> ?>=<89:; product of stochastic matrices is again a stochastic 3 h y matrix and so the transition monoid T (A) consists of ∗ the stochastic matrices Pu, u 2 Σ . In particular, the (i; j)th element p(sj j u; si) of the matrix Pu is the Figure 1: Semiautomaton. transition probability that the automaton enters state sj when started in state si and reading the word u. 3 Generalized Semiautomata Example 2. Let m ≥ 2 be an integer. Put Σ = f − g A Stochastic automata are a generalization of non- 0; : : : ; m 1 . The stochastic semiautomaton = (fs1; s2g; Σ;P ) given by deterministic finite state automata [5]. Generalized ( ) automata can be obtained from stochastic automata 1 m − x x P = ; x 2 Σ; by dropping the restrictions imposed by probabil- x m m − x − 1 x + 1 ity [5, 16, 17]. A generalized semiautomaton (GSA) is a triple is called m-adic semiautomaton. For each word u = ∗ x1 : : : xk 2 Σ , A = (S; Σ; fQx j x 2 Σg); ( ) k 1 m − wk wk Pu = k k ; where m m − wk − 1 wk + 1 E-ISSN: 2224-3402 35 Volume 18, 2021 WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS Merve Nur Cakir, Mehwish Saleemi, DOI: 10.37394/23209.2021.18.6 Karl-Heinz Zimmermann k−1 where wk = xkm + ::: + x2m + x1 and the en- A permutation matrix P is a square binary matrix 1 try mk wk corresponds in the m-adic representation which has exactly one entry of 1 in each row and each to 0:xk : : : x1. } column and 0’s elsewhere. By the Birkhoff-von Neu- mann theorem [6], for each n × n doubly stochas- A generalized semiautomaton A = (S; Σ;D) is ≥ tic matrixPP there exist real numbers α1; : : : ; αN called doubly stochastic if the matrices Dx, x 2 Σ, N 0 with αi = 1 and permutation matrices are doubly stochastic, i.e., Dx is a matrix of nonnega- i=1 tive real numbers such that each row and column sum P1;:::;PN such that is equal to 1.