A Finite Field Framework for Modeling, Analysis and Control of Finite State Automata

ing from the latter exemplary viewpoint, in general we obtain a multilinear transition function f : F2 ×F2 ×F2 → F2 f (x[k+1],x[k],u[k]) = 0 = ∑ S1∈2In ∑ S2∈2In ∑ S3∈2Im δS1,S2,S3 ( ∏ j∈S1 x j[k+1] )( ∏ l∈S2 xl[k] )( ∏ m∈S3 um[k] ) , (21) with x ∈ F2 and u ∈ F2 , where the sets In = {1,2, . . . ,n}, Im = {1,2, . . . ,m} are index sets, 2In denotes the (possibly empty) power set of In and δS1,S2,S3 are constants in F2. If for fixed x[k] and u[k] we focus on x[k+1], we might observe multiple successors x[k+1]. Strictly speaking, we then should call (21) not a function, but a relation. As a consequence, the finite field representation (21) is capable of modeling non-deterministic finite state automata as well. Leaving aside the details in the following paragraphs, for brevity, we will only bring to light the ideas of some straight forward extensions of the setting.