Translation between the NP-complete problem of halting in a non-deterministic Turing machine and the classic NP-complete problem of satisfiability. In satisfiability one sets up a collection of rows of black, white and gray squares, then asks whether there exists any sequence of black and white squares that satisfies the constraint that on every row the color of at least one square agrees with the color of the corresponding square in the sequence. Each row can be viewed as a term in a conjunctive normal form Boolean expression, with each column corresponding to a different variable. When a given square on a particular row is black or white it indicates that a variable or its negation appear in that term. The translation from the Turing machine problem is achieved by representing the behavior of the Turing machine by saying which of a sequence of elementary statements are true about it at each step: whether the head is in one state or another, whether the cell under the head is black or white, and whether the head is at each of the possible positions it can be in. The Boolean expression then gives constraints on which of these statements can simultaneously be true. In the first two pictures, for example, the first row corresponds to the constraint that on the first step of Turing machine evolution, the head cannot simultaneously be in an up and a down state. About the first half of the terms in each Boolean expression correspond to similar general constraints about the operation of Turing machines. There are then a few terms that specify the particular initial conditions used here, followed by terms that give the rule for the Turing machine that is used. The very last term makes the statement that the Turing machine halts. As the pictures indicate, each possible path of evolution for the Turing machine then corresponds to a possible assignment of truth values to the variables associated with each elementary statement. And if there is any path that leads the Turing machine to halt the Boolean expression will be satisfiable. This is the case in the first and fourth examples shown, but not in the other two. In general, it is possible to represent t steps in the evolution of a non-deterministic Turing machine by a Boolean expression with at most t3 terms in t2 variables. A version of the translation shown here was what launched the study of NP completeness in the early 1970s.
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