By David Eppstein

The concentration of this ebook is a mathematical constitution modeling a actual or organic method that may be in any of a few `states.' every one country is characterised through a collection of binary positive factors, and differs from another neighbor kingdom or states by means of only one of these function. an easy instance of a `state’ is a partial answer of a jigsaw puzzle, that are remodeled into one other partial resolution or into the ultimate resolution simply by including or removal a unmarried adjacent piece. The evolution of this kind of process over the years is taken into account. this kind of constitution is analyzed from algebraic and probabilistic (stochastic) standpoints.

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**Example text**

Proof. (i) Suppose that Q, M ∈ [S], with Qτ = R, M τ = W , Q ∼ R and M ∼ W . We have to show that [R] = [W ]. 4 (with τ = m), Qτ = R and M τ = W yield {τ } = R \ Q = W \ M ⊆ R ∩ W . 15) that R W = Q M ⊆ T \ U ; so R ∼ W , and [R] = [W ] follows. (ii) This follows immediately from (i) and the fact that U = ∅. (iii) For τ ∈ (T \ U) and S ∼ T , we have S τ|U = T ⇔ T τ|U = S ⇔ T τ = S ⇔ S τ˜ = Q ⇔ S τ˜|U = T. The other case is also straightforward. Except for details of formulation, the next result is due to Cavagnaro (2006).

2 Verify formally that, in a token system, any token has at most one reverse, and that if τ˜ is the reverse of τ , then τ˜ ˜ = τ , that is, τ and τ˜ are mutual reverses. 2). 7. 5 Verify formally that the token system formed by the family of all strict partial orders on a ﬁnite set Y satisﬁes Axioms [Ma] and [Mb] of a medium (with the tokens of the two types: P → P ∪ {xy} and P → P \ {xy}). ) Describe the contents of the states in this example, and compare such a description with the standard deﬁnition of a binary relation.

5). 1 Lemma. For any three sets P , Q and S, we have P Consequently, if both P Q ⊆ (P S and S S) ∪ (S Q). 1) Q; thus d(P, Q) ≤ d(P, S) + d(S, Q), with moreover d(P, Q) = d(P, S) + d(S, Q) ⇐⇒ P ∩ Q ⊆ S ⊆ P ∪ Q. 2 Deﬁnition. Let F be a family of subsets of a set X, and let k be a natural number. A k-graded path between two distinct sets P and Q (or from P to Q) in F is a sequence P0 = P, P1 , . . , Pn = Q in F such that d(Pi , Pi+1 ) = k for 0 ≤ i ≤ n − 1, and d(P, Q) = kn. A 1-graded path is called a tight path.