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H.3.5 Tight satisfaction with local variables</div>
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A local variable context is a function that assigns values to local variable names. If L is a local variable context, </p>
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then dom(L) denotes the set of local variable names that are in the domain of L. If D dom(L), then L|D </p>
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means the local variable context obtained from L by restricting its domain to D. </p>
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In the presence of local variables, tight satisfaction is a four-way relation defining when a finite word w over </p>
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the alphabet Σ together with an input local variable context L0 satisfies an unclocked sequence R and yields an </p>
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output local variable context L1. This relation is denoted </p>
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w, L0, L1 R . </p>
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and is defined below. It can be proved that the definition guarantees that w, L0, L1 R implies </p>
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dom(L1) = flow (dom(L0), R) . </p>
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? w, L0, L1 ( 1, v = e ) iff |w| = 1 and w 0 1 and L1 = {(v, e[L0, w 0])} U L0|D , where e[L0, w 0] denotes the </p>
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value obtained from e by evaluating first according to L0 and second according to w 0 and D = dom(L0) – </p>
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{v}. In case w 0 {T, }, e[L0,T] and e[L0, ] can be any constant values of the type of e. </p>
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? w, L0, L1 b iff |w| = 1 and w 0 b[L0] and L1 = L0. Here b[L0] denotes the expression obtained from b by </p>
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substituting values from L0 . </p>
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? w, L0, L1 ( R ) iff w, L0, L1 R . </p>
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? w, L0, L1 ( R1 ##1 R2 ) iff there exist x, y, L’ such that w = xy and x, L0, L’ R1 and y, L’, L1 R2 . </p>
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? w, L0, L1 ( R1 ##0 R2 ) iff there exist x, y, z, L’ such that w = xyz and |y| = 1, and xy, L0, L’ R1 and </p>
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yz, L’, L1 R2 . </p>
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? w, L0, L1 ( R1 or R2 ) iff there exists L’ such that both of the following hold: </p>
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— either w, L0, L’ R1 or w, L0, L’ R2, and </p>
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— L1 = L’ |D, where D = flow (dom(L0), ( R1 or R2 )) . </p>
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? w, L0, L1 ( R1 intersect R2 ) iff there exist L’, L" such that </p>
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w, L0, L’ R1 and w, L0, L" R2 and L1 = L’ |D’ U L" |D’’ , where </p>
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D’ = flow (dom(L0), R1) – (block (( R1 intersect R2 )) U sample (R2)) </p>
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D’’ = flow (dom(L0), R2) – (block (( R1 intersect R2 )) U sample (R1)) </p>
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Remark: It can be proved that if w, L0, L’ R1 and w, L0, L" R2 , then L’ |D’ U L" |D’’ is a function. </p>
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? w, L0, L1 first_match ( R ) iff both </p>
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— w, L0, L1 R and </p>
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— if there exist x, y, L’ such that w = xy and x, L0, L’ R, then y is empty. </p>
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? w, L0, L1 R [*0] iff |w| = 0 and L1 = L0. </p>
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? w, L0, L1 R [*1:$] iff there exist L(0) = L0, w1, L(1), w2, L(2),..., wj, L( j) = L1 ( j > 1) such that w = w1w2...wj </p>
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and for every i such that 1 < i < j, wi, L(i –1), L(i) R . </p>
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If S is a clocked sequence, then w, L0, L1 S iff w, L0, L1 S’, where S’ is the unclocked sequence that results </p>
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from S by applying the rewrite rules. </p>
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An unclocked sequence R is non-degenerate iff there exist a non-empty finite word w over Σ and local variable </p>
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contexts L0, L1 such that w, L0, L1 R. A clocked sequence S is non-degenerate iff the unclocked sequence S’ </p>
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that results from S by applying the rewrite rules is non-degenerate.</p></div>
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