, Cm. Seq(S) is defined to be the set of all (finite) sequences whose elements are in the set S (with possible repetitions of elements in the sequence). Moreover, we use the term c-seq when referring to a finite sequence of commands—that is, an element of Seq(Cmd). We can now extend the operator • to a c-struct v and a c-seq 〈C1, , Cm 〉 as follows: v • 〈C , . . . , C 〉 = . v if m = 0, 1 m (v • C1) • 〈C2, . . . , Cm 〉 otherwise ± H ± ± H ± ± • ± We say that c-struct w extends c-struct v or v is a prefix of w , and note as v w , iff there exists a c-seq σ such that w = v σ. Given a set T of c-structs, we say that c-struct v is a lower bound of T iff v w for all w in T . A greatest lower bound (glb) of T is a lower bound v of T such that w v for every lower bound w of T , and we represent it by T . Similarly, we say that v is an upper bound of T iff w v for all w in T . A least upper bound (lub) of T is an upper bound v of T such that v w for every upper bound w of T , and we represent it by T . If is a reflexive partial order on the set of c-structs and a glb or lub of T exists, then it is unique. For simplicity of notation, we use v w and v w to represent v , w and v , w , respectively. Two c-structs v and w are defined to be compatible iff they have a common upper bound, and a set S of c-structs is compatible iff its elements are pairwise compatible. ⊥ • We say that c-struct v is constructible from a set P of commands if v = σ, for some c-seq σ containing all elements of P . Moreover, we say that v contains ∈ ∆ command C if v is constructible from some set P of commands such that C P . We define Str(P) to be the set of all c-structs constructible from subsets of P for some set P of commands—that is, Str(P) = {⊥ • σ : σ ∈ Seq(P)}. A c-struct set CStruct must satisfy the five axioms below. CS0. ∀C ∈ Cmd, w ∈ CStruct : w • C ∈ CStruct CS1. CStruct = Str (Cmd ) CS2. ± is a reflexive partial order on CStruct.
, Cm. Con- struct the formula ϕ from ϕ as follows. The variables of ϕ are • ϕ’s variables x1, x2, . . . , xn and j j j • for each clause Cj of ϕ, a number of additional variables y1, y2, . . . , ykj as needed, where kj depends on the structure of Cj according to the case distinction below. ^ ^ ^ ^ ^ ^ Now, define ϕ = C1 ∧ C2 ∧ · · · ∧ Cm, where each clause Cj of ϕ is constructed from the clause Cj of ϕ as follows. Suppose that Cj = (z1 ∨ z2 ∨ · · · ∨ zk), where each zi is a literal over {x1, x2, . . . , xn}. Distinguish the following four cases. • If k = 1, define 1 2 1 2 C^j = (z1 ∨ yj ∨ yj) ∧ (z1 ∨ ¬yj ∨ yj) ∧ (z1 ∨ yj ∨ ¬yj) ∧ (z1 ∨ ¬yj ∨ ¬yj) . 1 • If k = 2, define C^j = (z1 ∨ z2 ∨ yj) ∧ (z1 ∨ z2 ∨ ¬yj). ^ • If k = 3, define Cj = Cj = (z1 ∨ z2 ∨ z3), i.e., the jth clause remains unchanged. • If k ≥ 4, define (¬yj C^j = (z1 ∨ z2 ∨ yj) ∧ (¬yj ∨ z3 ∨ yj) ∧ (¬yj ∨ z4 ∨ yj) ∧ · · · ∧ k−4 ∨ zk−2 ∨ y 1 2 j k−3 k−3 ) ∧ (¬yj 2 3 ∨ zk−1 ∨ zk) . It remains to show that (a) the reduction r is polynomial-time computable, and
