3.4.2  Knot automata

A class of circuit automata based on knot theory is considered by Kauffman (1994). The basic circuit element for these automata has an equation of the form z=xRy or z=xLy with box depictions as shown in the following figure. These equations correspond to "left" or "right" crossings of a KL, i.e., to the crossings with the sign 1 or -1, respectively. Translated into equations, Reidemeister moves become:

1. aRa=a, aLa=a;

2. (aRb)Lb=a, (aLb)Rb=a;

3. (aRb)Rc=(aRc)R(bRc), (aLb)Lc=(aLc)L(bLc).

The resulting algebraic structure, a quandle, can be obtained by different solutions of the equations defined by Reidemeister moves. The simplest example of a quandle is the structure aRb=aLb=2b-a, where a and b are elements of an additive abelian group G. In the case of a trefoil knot automaton the feedback loop forces the conclusion 3(b-a)=0. Hence, 3 must divide the order of G in order for the trefoil automaton to have any balanced states. If G=({0,1,2}, +₃) is the abelian group with addition modulo 3, for a=1, b=2 we obtain a stable state of the trefoil automaton, i.e., the three-coloring of a trefoil. It distinguishes a trefoil from the unknot, or from a figure-eight knot.

If the values for the knot automaton lie in a module over the ring Z[t,t^-1], then for aRb=ta+(1-t)b, aLb=sa+(1-s)b with s=t^-1 we obtain the Alexander polynomial of a KL. The quandle considered before, defined by aRb=aLb=2b-a, is a particular case (t=-1) of this general solution.

In a digital circuit model the basic element is a NAND gate, or a simple inverter. From an input p it produces , and p₁p₂...p_n gives as the output . NOR gates work in the same way, producing for p, and for p₁Úp₂Ú... In a circuit diagram, a state is the edge-coloring of the directed graph in which the colors are chosen from the discrete set {0,1}. All edges emanating from a given inverter have the same color in a given state. If denotes the equation that defines the operation of a given inverter, a state is called balanced if the equation is satisfied at every inverter in the diagram. For example, the circuit defined by equations and  has exactly two balanced states in the discrete set {0,1}. In general case, if we take values from a continuous interval [0,1], all the values (x,y)=(x,1-x) (x Î [0,1]) will give balanced states of this circuit. A transition consists of reassigning the value of z for the outgoing edges z of one inverter that is unbalanced. Transition may or may not result in a balanced state (Kauffman, 1994).

The equation  describes the circuit that embodies the Liar paradox. In bivalent Boolean logic, it has no solutions for a balanced state, so it demands a polyvalent logic in order to achieve a stable state. Such a solution is x=¹/₂. In a similar way, the automaton defined by equations has no stable states for the discrete values from the set {0,1}, yet stabilizes for (x,y)=(1-j, j) from the continuous set [0,1].