Tuple

In general, a vector in an n-space can be represented by an n-tuple of numbers for example, a vector in 3-space can be represented as a number triplet. 

Since each site can take one any of fc values, p is completely defined by specifying the value assigned to each of the possible (2r -f l)-tuple configurations... 

The generalization of this simple formula to the case when p 1/2 is straightforward. In this case, each of the eight possible 3-tuple neighborhoods a = 010203 ... 

Two blocks, B and S, are of the same 2" -order block type if Pn B) = P[y B ). Thus, B and B must share three characteristics (1) the same number of sites (excluding the right-most and left-most sites) with value T (= m), (2) the same number of 11 sub-blocks (= nn), and (3) the same total number of OT and 10 sub-blocks (= Tiio+oi)- Accordingly, 2 -order blocks of length N types are characterized by the 3-tuple = ( i ii, riio+oi) -... 

Before reviewing some of the more common approaches to constructing reversible CA from scratch, it is first of all important to understand why we should expect any difficulty at all in casually defining them. The difficulty stems mainly from the different cardinalities of the site value set and the set of A -tuples (where J f is the neighborhood size) defining the rule table. That is to say, since a general CA map (j) maps —> Zk, an obvious but important consequence is that (except for the... 

Define to be the set of all possible 2r-tuples (excluding the center-site). 

Finally, using the physical interpretation of the quantum site-state coefficients ai , we can write down an explicit form for the probability functions p . Since 0 is defined by the list n, we must simply write down the probability that a measurement of the 2r states around a given site F will yield a 2r-tuple which is an element of n. We therefore get sums of products of absolute squares, with individual list elements contributing the terms and list elements... 

All next-nearest neighbors i and j , linking them only if the 2-tuple (q,/ ) = (1,3). By next-nearest neighbor we mean those pairs which are themselves unlinked but which share at least one other linked neighbor a,g), h,r) and w,y), for example, are all next-nearest neighbors at t = l. For c and g we find... 

A Deval transition function is a listing of descriptions of possible neighborhoods of a specified length, c(). These rules are tuples of the form ... 

Under any free interpretation I, the computation (P,I,X) never applies the same n-placed test T twice to the same n-tuple of members of U(P). ... 

DEFINITION A recursion scheme S is free if for every free interpretation I and every m-place test T in S, T is never applied twice to the same m-tuple (t-, ...,t ) of members of U(S) during the computation (S,I, X). ... 

The unique attribute means that no other object of this owner type has the same attribute value, unique can apply to a tuple of attributes. 

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