## Algebraic and Geometrical Methods in Topology by L.F. McAuley

By L.F. McAuley

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Tn . φ1 where x is a variable and φ1 is a ﬁrst-order formula. The semantics of terms and ﬁrst-order formulas is determined by the interpretation of the symbols used. An interpretation I is a pair U, A formed by a non-empty set U and a mapping A. A maps every n-ary function constant f to an n-ary function A(f ) : U n → U and every n-ary relation constant P to an n-ary relation A(P ) ⊆ U n . , A(f (t1 , . . , tn )) = A(f )(A(t1 ), . . , A(tn )). A variable assignment α assigns variables with elements of U.

A(tn )). A variable assignment α assigns variables with elements of U. α[x/d] means that α assigns the variable x with d ∈ U. The truth value of a ﬁrst-order formula φ under an interpretation I and a variable assignment α is deﬁned inductively on the structure of φ. If φ is true under I and α, then I under α is a model of φ, written as I, α |= φ: I, α |= P (t1 , . . , tn ) iﬀ A(t1 ), . . 3 Computational Complexity Meanwhile computers have entered almost every part of our life and we expect them to work accurately and eﬃciently—real-time.

If revise(i, k, j) then 5. if Rij = ∅ then return fail 6. else Q ← Q ∪{(i, j, k), (k, i, j) | k = i, k = j}; Function: revise(i, k, j) Input: three variables i, k and j Output: true, if Rij is revised; false otherwise. Side eﬀects: Rij and Rji revised using the operations ∩ and ◦ over the constraints involving i, k, and j. 1. 2. 3. 4. 5. oldR := Rij ; Rij := Rij ∩ (Rik ◦ Rkj ); if (oldR = Rij ) then return false; Rji := Rij ; return true. Fig. 2. Van Beek’s Path-consistency algorithm are sometimes called atomic, basic, or base relations.