CSC2429, MAT1304: Circuit Complexity
November 22, 2016
Handout on relations (projection, restriction, join, density)
Instructor: Benjamin Rossman
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Relations
We consider relations over a fixed universe [n] (= {1, . . . , n}). Rather than k-tuples and k-ary
relations over [n] (i.e. elements and subsets of [n]k where k ∈ N), we will speak of tuples and
relations over [n] indexed by finite sets. Below, let V, W, S, T represent arbitrarily finite sets.
Definition 1 (V -Tuples).
• A V -tuple is an element x ∈ [n]V . For v ∈ V , the v-th coordinate of x is denoted xv ∈ [n].
• There is a unique ∅-ary tuple, which we denote by ◦. That is, [n]∅ = {◦}.
• For x ∈ [n]V and S ⊆ V , we write xS ∈ [n]S for the restriction of x to coordinates in S.
• For y ∈ [n]S and z ∈ [n]V \S , we write yz for the V -tuple x ∈ [n]V with xS = y and xV \S = z.
(This concatenation operation is unordered, that is, yz = zy.)
Definition 2 (V -ary Relations, Projection, Restriction).
• A V -ary relation is a set A ⊆ [n]V of V -tuples.
• For A ⊆ [n]V and S ⊆ V , the S-projection of A is the S-ary relation
projS (A) := {xS : x ∈ A}.
• For A ⊆ [n]V and S ⊆ V and z ∈ [n]V \S , the S-restriction of A at z is the S-ary relation
A|zS := {y ∈ [n]S : yz ∈ A}.
Note that A|zS ⊆ projS (A).
Definition 3 (Join).
• The join of relations A ⊆ [n]V and B ⊆ [n]W is the V ∪ W -ary relation defined by
A ./ B := {x ∈ [n]V ∪W : xV ∈ A and xW ∈ B}.
The join operation ./ is a hybrid of intersection ∩ and product ×. If V = W , then we have
A ./ B = A ∩ B. On the opposite extreme, if V ∩ W = ∅, then we have A ./ B = A × B (i.e. the set
{xy : x ∈ A and y ∈ B}). As an operation on relations, note that ./ is associative, commutative
and idempotent (i.e. A ./ (B ./ C) = (A ./ B) ./ C and A ./ B = B ./ A and A ./ A = A). Also
note that ∅-ary relation {◦} is an identity under ./ (i.e. A ./ {◦} = A).
Exercise 4. Suppose A ⊆ [n]V and B ⊆ [n]W and C ⊆ [n]V ∪W such that C ⊆ A ./ B.
i. Show that projV (C) ⊆ A.
ii. For S ⊆ V , show that projS (C) ⊆ projS (A).
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Density
Definition 5 (Density, Projection Density, Restriction Density).
• The density of a relation A ⊆ [n]V , denoted µ(A), is defined by
|A| µ(A) := |V |
= P [x∈A] .
n
x∈[n]V
• For S ⊆ V , the S-projection density of A is the quantity µ(projS (A)). Note that
h
i
_
µ(projS (A)) = P
yz ∈ A .
y∈[n]S
z∈[n]V \S
• The S-restriction density of A, denoted µS (A), is defined by
µS (A) := max µ(A|zS )
= max
P [ yz ∈ A ] .
z∈[n]V \S y∈[n]V
z∈[n]V \S
Note that µS (A) ≤ µ(projS (A)) and µV (A) = µ(A) and µ∅ (A) = 1[A6=∅] .
Exercise 6. Let A ⊆ [n]V and S ⊆ V .
iii. Show that µ(A) ≤ µ(projS (A)) · µV \S (A).
iv. Show that inequality (iii) cannot be strengthened to µ(A) ≤ µ(projS (A)) · µ(projV \S (A)).
That is, find an example of A and S where this inequality is false.
v. For S 00 ⊆ S 0 ⊆ S, show that µS 00 (projS (A)) ≤ µS 00 (projS 0 (A)).
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Bounding the density of a sub-relation of a join
The following lemma will play a role in our lower bound on the AC0 formula size of SUB(Pathk ).
Lemma 7. Suppose A ⊆ [n]V and B ⊆ [n]W and C ⊆ [n]V ∪W such that C ⊆ A ./ B. Then for all
S ⊆ V and T ⊆ W , we have
µ(C) ≤ µ(projS (A)) · µT \S (projT (B)) · µ(V ∪W )\(S∪T ) (C).
Proof. Lemma 7 is mainly derived by two applications of inequality (iii). First we apply (iii) with
respect to C and S ∪ T :
µ(C) ≤ µ(projS∪T (C)) · µ(V ∪W )\(S∪T ) (C).
Next we apply (iii) with respect to projS∪T (C) and S:
µ(projS∪T (C)) ≤ µ(projS (C)) · µT \S (projS∪T (C)).
Now observe that
(ii)
µ(projS (C)) ≤ µ(projS (A)),
(v)
(ii)
µT \S (projS∪T (C)) ≤ µT \S (projT (C)) ≤ µT \S (projT (B)).
Combining the four above inequalities finishes the proof.
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