Deciding One to One property of Boolean maps: Condition and algorithm in terms of implicants
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This paper addresses the computational problem of deciding invertibility (or one to one-ness) of a Boolean map F in n-Boolean variables. This problem has a special case of deciding invertibilty of a map F:š½_2^nāš½_2^n over the binary field š½_2. Further the problem can be extended and stated over a finite field š½ instead of š½_2. Algebraic condition for invertibility of F in this special case over a finite field is well known to be equivalent to invertibility of the Koopman operator of F as shown in <cit.>. In this paper a condition for invertibility is derived in the special case of Boolean maps F:B_0^nā B_0^n where B_0 is the two element Boolean algebra in terms of implicants of Boolean equations. This condition is then extended to the case of general maps in n variables. Hence this condition answers the special case of invertibility of the map F defined over the binary field š½_2 alternatively, in terms of implicants instead of the Koopman operator. The problem of deciding invertibility of a map F (or that of finding its GOE) over finite fields appears to be distinct from the satisfiability problem (SAT) or the problem of deciding consistency of polynomial equations over finite fields. Hence the well known algorithms for deciding SAT or of solvability using Grobner basis for checking membership in an ideal generated by polynomials is not known to answer the question of invertibility of a map. Similarly it appears that algorithms for satisfiability or polynomial solvability are not useful for computation of GOE(F) even for maps over the binary field š½_2.
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