comput. complex. 16 (2007), 211–244
1016-3328/07/030211-34
DOI 10.1007/s00037-007-0229-6
c Birkhäuser Verlag, Basel 2007
computational complexity
THE COMPLEXITY OF MEMBERSHIP
PROBLEMS
FOR CIRCUITS OVER SETS OF NATURAL
NUMBERS
Pierre McKenzie and Klaus W. Wagner
Abstract. The problem of testing membership in the subset of the natural numbers produced at the output gate of a {∪, ∩,− , +, ×} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {∪, ∩, +, ×} is shown
NEXPTIME-complete, the cases {∪, ∩,− , ×}, {∪, ∩, ×}, {∪, ∩, +} are
shown PSPACE-complete, the case {∪, +} is shown NP-complete, the
case {∩, +} is shown C= L-complete, and several other cases are resolved.
Interesting auxiliary problems are used, such as testing nonemptyness for
union-intersection-concatenation circuits, and expressing each integer,
drawn from a set given as input, as powers of relatively prime integers
of one’s choosing. Our results extend in nontrivial ways past work by
Stockmeyer and Meyer (1973), Wagner (1984) and Yang (2000).
Keywords. Computational complexity, circuit problems.
Subject classification. 68Q25, 68Q15, 03D15.
1. Introduction
Combinational circuits permeate complexity theory. Countless lower bounds,
complexity class characterizations, and completeness results involve circuits
over the Boolean semiring (see (29) among many others). Circuits and formulas
over more general structures have been studied as well (see for a few examples
(1; 4; 8; 9; 11; 13)).
In this paper, we study circuits operating on sets of natural numbers. Our
universe of natural numbers includes 0 and is simply called the set N of numbers from now on. Next to the Boolean semiring, the semiring of numbers is
certainly the most fundamental, and two results involving number arithmetic
have appeared recently: integer division and iterated number multiplication
Research supported by the NSERC of Canada and the (Québec) FQRNT.
212 McKenzie & Wagner
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were finally shown to belong to uniform TC0 (3; 15; 19) and (∪, +, ×)-circuit
evaluation was shown PSPACE-hard (31).
In the Boolean setting, the AND and the OR operations combine to capture
alternation (in, say, simulations of alternating Turing machines by circuits). In
the setting of numbers, perhaps the closest analogs to the AND and the OR
become the ∩ and the ∪ operations. To make sense, this requires an adjustment:
since gates will now compute sets of numbers, a +-gate and a ×-gate having
input gates computing S1 ⊆ N and S2 ⊆ N then compute {a+b : a ∈ S1 , b ∈ S2 }
and {a × b : a ∈ S1 , b ∈ S2 } respectively.
Another reason to study {∪, ∩, +, ×}-circuits over N is that they obey some
form of monotonicity condition: if the set S ⊂ N carried by an input gate is
replaced by a larger set S ⊃ S, then the set computed at the output gate
of the circuit can only become larger (never smaller). This is reminescent
of monotone Boolean functions (a Boolean function is monotone if flipping
an input from 0 to 1 can never change the output from 1 to 0), for which
significant complexity bounds are known (such as (25)). Are the circuit and
formula evaluation problems over subsets of {∪, ∩, +, ×} related to monotone
Boolean function complexity?
The problem studied in this paper is the following, which we think of as
combining number arithmetic with some form of alternation: given a number b
and a {∪, ∩,− , +, ×}-circuit C with number inputs, where “ − ” denotes complementation, does b belong to the set computed by the output gate of C? We
call this problem MC(∪, ∩,− , +, ×), and we call MF(∪, ∩,− , +, ×) the same
problem restricted to formulas. Note that a complement gate − applied to a
finite set S ⊂ N computes the infinite set N \ S. Hence, in the presence of
complement gates, the brute force strategy which would exhaustively compute
all the sets encountered in the circuit fails. The notation for restricted versions
of these problems, for instance MC(∪, +), is self-explanatory (see Section 2).
Beyond the results mentioned above, it was known prior to this work
that the problem MF(∪, +) is NP-complete (26), that MF(∪, ∩,− , +) and
MF(∪,− , +) are PSPACE-complete (26), and that MC(∪, +) is in PSPACE
(30).
Adding results from the present paper, we obtain Table 10.1. We highlight
here some of the interesting results or techniques:
◦ The problem MC(∪, ∩, +, ×) is NEXPTIME-complete. As an intermediate step, we prove that determining whether a union-intersection-concatenation circuit over a finite alphabet produces a nonempty set is also
NEXPTIME-complete.
cc 16 (2007)
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213
◦ The problem MC(∪, ∩, ×) is PSPACE-complete.
◦ The problem MC(∪, ∩, ×) reduces in polynomial time to MC(∪, ∩, +),
which is thus also PSPACE-complete. This reduction is possible because
the following problem is solvable in polynomial time (6): given as input
a set S of numbers excluding 0, compute a set T of pairwise relatively
prime numbers and express each m ∈ S as a product of powers of the
numbers in T .
◦ The problem MC(∪, +) is NP-complete. This is a nontrivial improvement
over the former PSPACE upper bound.
◦ The problem MC(∩, +) is C= L-complete, and so is the problem of testing
whether two MC(+)-circuits are equivalent.
The hardest problem we consider is MC(∪, ∩,− , +, ×). As will be seen, we
do not have an upper bound for this problem and it may well be undecidable. In
fact, we will see that Goldbach’s conjecture is easily expressible as an instance
of MC(∪, ∩,− , +, ×). On the other hand, Table 10.1 shows that its various
restrictions hit upon a wealth of complexity classes.
2. Definitions and known results
We assume familiarity with basic complexity theory, such as can be found in
Papadimitriou’s book on the subject (24). We will encounter the complexity classes NEXPTIME, PSPACE, NP, R, P, C= L, NL, L, NC1 , TC0 , AC0
and #L. Here L and NL stand for deterministic and nondeterministic logarithmic space respectively. The class #L is the class of functions f : Σ∗ → N
such that there is an NL machine whose number of accepting paths on any
input w is f (w). The class C= L is the set of languages A for which there exist
functions f1 , f2 ∈ #L such that for all x, x ∈ A ⇔ f1 (x) = f2 (x); for a survey
on these and other counting classes see (2). The circuit classes NC1 , TC0 and
AC0 play an incidental role in this paper: these classes correspond, respectively, to the sets of langages accepted by DLOGTIME-uniform logarithmicdepth bounded-fan-in {∧, ∨, ¬}-circuits, DLOGTIME-uniform unbounded-fanin constant-depth polynomial-size {MAJORITY}-circuits, and DLOGTIMEuniform unbounded-fan-in constant-depth polynomial-size {∧, ∨, ¬}-circuits.
We adopt the definitions of (11) for constant-depth circuit uniformity (see
also (7)). Finally, we will make use of polynomial-space many-one reducibility, of polynomial-time many-one reducibility, of logarithmic-space many-one
reducibility and of AC0 -reducibility.
214 McKenzie & Wagner
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If not otherwise stated, the hardness results in this paper are in terms
of many-one logspace reducibility. Indeed, although several of our reductions
could be refined into AC0 -reductions, for complexity classes C1 ⊆ C2 where C2
is believed to be strictly larger than L, we feel that the emphasis of our C2 completeness results is in delineating the frontier between C1 and C2 rather
than in refining the reductions used.
We assume any circuit and formula encoding in which the description of a
gate and its immediate predecessors, if any, are readily available (say in AC0 ).
Viewed as graphs, circuits and formulas are not necessarily connected; our
choice of encoding is motivated by our wish to preserve L-hardness of all our
problems (see the remark at the end of Section 9). Numbers are encoded in
binary notation.
A circuit C = (G, E, gC ) is a finite directed acyclic graph (G, E) with a
specified node gC , the output gate. The gates with indegree 0 are called the
input gates.
We consider different types of arithmetic circuits. Let O ⊆ {∪, ∩,− , +, ×}.
An O-circuit C = (G, E, gC , α) is a circuit (G, E, gC ) whose gates have indegree
0, 1, or 2 and are labelled by the function α : G → O ∪ N in the following way:
Every input gate g has a label α(g) ∈ N, every gate g with indegree 1 has
the label α(g) = − , and every gate g with indegree 2 has a label α(g) ∈
{∪, ∩, +, ×}. For each of its gates g the arithmetic circuit C computes a set
I(g) ⊆ N inductively defined as follows:
◦ If g is an input gate with label a, then I(g) =def {a}.
◦ If g is +-gate with predecessors g1 , g2 , then
I(g) =def k+m : k ∈ I(g1 ) ∧ m ∈ I(g2 ) .
◦ If g is ×-gate with predecessors g1 , g2 , then
I(g) =def k·m : k ∈ I(g1 ) ∧ m ∈ I(g2 ) .
◦ If g is a ∪-gate with predecessors g1 , g2 , then I(g) =def I(g1 ) ∪ I(g2 ).
◦ If g is a ∩-gate with predecessors g1 , g2 , then I(g) =def I(g1 ) ∩ I(g2 ).
◦ If g is a − -gate with predecessor g1 , then I(g) =def N \ I(g1 ).
The set computed by C is I(C) =def I(gC ). If I(g) = {a} then we also
write I(g) = a. An O-formula is an O-circuit with maximal outdegree 1. For
cc 16 (2007)
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215
O ⊆ {∪, ∩,− , +, ×} the membership problems for O-circuits and O-formulae
are
MC(O) =def (C, b) : C is an O-circuit and b ∈ N such that b ∈ I(C)
and
MF(O) =def (F, b) : F is an O-formula and b ∈ N such that b ∈ I(F ) .
For simplicity we write MC(o1 , . . . or ) instead of MC({o1 , . . . or }), and we
write MF(o1 , . . . or ) instead of MF({o1 , . . . or }).
Example 2.1 (Prime numbers). A circuit PRIMES such that I(PRIMES) is
the set of prime numbers is obtained by defining the subcircuit GE2 as 0 ∪ 1
and defining PRIMES as GE2 ∩ (GE2 × GE2). Hence the problem of primality
testing easily reduces to MF(∪, ∩,− , ×).
♦
Example 2.2 (Goldbach’s conjecture). Consider the circuit GOLDBACH defined as (GE2 × 2) ∩ (PRIMES + PRIMES). Then I(GOLDBACH) is empty iff
every even number greater than 2 is expressible as a sum of two primes. Hence
Goldbach’s conjecture holds iff “0 ∈ 0 × GOLDBACH” is a positive instance of
MC(∪, ∩,− , +, ×).
♦
The following results are known from the literature:
Theorem 2.3.
(i) (26) The problem MF(∪, +) is NP-complete.
(ii) (26) The problems MF(∪, ∩,− , +) and MF(∪,− , +) are PSPACE-complete.
(iii) (30) The problem MC(∪, +) is in PSPACE.
(iv) (31) The problem MC(∪, +, ×) is PSPACE-complete.
In (30) it is shown that the problem MC(∪, +) restricted to (∪, +)-circuits
for which every +-gate has at least one input gate as predecessor is NPcomplete.
By De Morgan’s laws we have
216 McKenzie & Wagner
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Proposition 2.4. For every O ⊆ {+, ×},
−
log
−
(i) MC({∪, ∩,− } ∪ O) ≡log
m MC({∪, } ∪ O) ≡m MC({∩, } ∪ O).
−
log
−
(ii) MF({∪, ∩,− } ∪ O) ≡log
m MF({∪, } ∪ O) ≡m MF({∩, } ∪ O).
Hence we can omit MC({∪,− } ∪ O), MC({∩,− } ∪ O), MF({∪,− } ∪ O),
and MF({∩,− } ∪ O) from our exhaustive study.
3. Multiplication versus addition
In this section we will establish a relationship between the complexity of the
membership problems for (O ∪ {×})-circuits and (O ∪ {+})-circuits, for O ⊆
{∪, ∩,− }. To this end we need the following problem. Let gcd(a, b) be the
greatest common divisor of the numbers a, b ≥ 1.
Gcd-Free Basis (GFB)
Given: Numbers a1 , a2 , . . . , an ≥ 1.
Compute: Numbers m ≥ 1, q1 , . . . , qm ≥ 2, and
e11 ,eij. . . , enm ≥ 0 such that
gcd(qi , qj ) = 1 for i = j and ai = m
j=1 qj for i = 1, . . . , n.
Despite the fact that factoring may not be possible in polynomial time, the
following is known (see (6)):
Proposition 3.1. Gcd-Free Basis can be computed in polynomial time.
As an auxiliary tool we need the generalized membership problems MC∗ (O)
and MF∗ (O) for arithmetic circuit and formulae, resp., with addition. These
problems deal with elements of Nm ∪ {∞}, for an m ≥ 1 given as part of the
input, where the addition on m-tuples is defined componentwise and a + ∞ =
∞ + a = ∞ + ∞ = ∞ for every a ∈ Nm . Note that polynomial-space many-one
reducibility is understood to be performed by polynomial space computable
polynomially bounded functions, where the bounds are in terms of the length
of the input.
Lemma 3.2.
(i) For O ⊆ {∪, ∩}, the problem MC(O ∪ {×}) is polynomial time many-one
reducible to the problem MC∗ (O ∪ {+}).
(ii) For O ⊆ {∪, ∩}, the problem MF(O ∪ {×}) is polynomial time many-one
reducible to the problem MF∗ (O ∪ {+}).
cc 16 (2007)
Circuits over sets of natural numbers
217
(iii) For O ⊆ {∪, ∩,− }, the problem MC(O ∪ {×}) is polynomial space manyone reducible to the problem MC∗ (O ∪ {+}).
(iv) For O ⊆ {∪, ∩,− }, the problem MF(O ∪ {×}) is polynomial space manyone reducible to the problem MF∗ (O ∪ {+}).
In each of the 4 cases above, if we make the added hypotheses that the prime
number decompositions of the inputs and of the “target number” b are known,
then the complexity of the reduction drops down to logspace many-one.
Proof. 1. Let O ⊆ {∪, ∩}, let C be a (O ∪ {×})-circuit with the input gates u1 , . . . , us , and let b ∈ N. Observe that the absence of + in C
entails that any number in I(C) is expressible as a monomial in the inputs.
Compute in polynomial time (Proposition 3.1) numbers q1 , . . . , qm ≥ 2 and
eij
e11 , . . . , esm , e1 , . . . , em ≥ 1 such that gcd(qi , qj ) = 1 for i = j, α(ui ) = m
j=1 qj
m ej
for any i = 1, . . . , s with α(ui) > 0 and b = j=1 qj if b > 0. Let M =def
dj
m
{ m
j=1 qj : d1 , . . . , dm ≥ 0 } ⊆ N and let σ : M ∪ {0} → N ∪ {∞} be defined
m dj
by σ( j=1 qj ) =def (d1 , . . . , dm ) and σ(0) = ∞. Obviously, σ is a monoid
isomorphism between (M ∪ {0}, ×) and (Nm ∪ {∞}, +). Because M ∪ {0} is
closed under ×, the set of numbers computed by any gate in C is included in
M ∪ {0}. Furthermore, the following holds for any S1 , S2 ⊆ M ∪ {0}:
σ(S1 × S2 ) = σ(S1 ) + σ(S2 ) ,
σ(S1 ∪ S2 ) = σ(S1 ) ∪ σ(S2 ) ,
σ(S1 ∩ S2 ) = σ(S1 ) ∩ σ(S2 ) .
The reduction therefore consists of converting C into a (O ∪ {+})-circuit C which has the same structure as C where a ×-gate in C becomes a +-gate in C and an input gate ui gets label σ(α(ui)). An induction using the three identities
above shows that for all a ∈ M ∪ {0} the following holds: a ∈ I(v) in C ⇔
σ(a) ∈ I(v) in C . This concludes the proof because b ∈ M ∪ {0}.
2. Same as above because the construction preserves the circuit structure.
3. and 4. If we have complementation then it is no longer true that every number computed within C has a decomposition into q1 , . . . , qm . In order to nonetheless make use of the above construction, the isomorphism σ
must therefore be extended to convey information about N \ M. A slick
way to do this is to begin from the full prime decomposition of the numbers u1 , . . . , us , b and to trade the former isomorphism for a homomorphism.
Indeed let q1 , . . . , qm , qm+1 , qm+2 , . . . be the sequence of all primes in natural order, and let q1 , . . . , qm exhaust at least all the distinct prime divisors of
218 McKenzie & Wagner
cc 16 (2007)
dj
α(u1 ), . . . , α(us ), b. For every number ∞
j=1 qj ∈ N \ {0} define
∞
d
σ
qj j =def d1 , . . . , dm ,
dj ,
j=1
j>m
and define σ(0) =def ∞. Let
M =def
m
d
qj j
: d1 , . . . , dm ≥ 0
⊆N
j=1
d
and Ki =def { j>m qj j :
j>m dj = i} for i ≥ 0. Because the full prime
decomposition was used, σ is a well-defined monoid homomorphism from (N, ×)
onto (Nm+1 ∪ {∞}, +), where σ(0) = ∞ and σ(M × Ki ) = (Nm ⊗{i}) for i ≥ 0
(⊗ denotes direct product). Notice that M = M × K0 and that σ is one-one
on this part, so that the following holds for any S1 , S2 ⊆ M:
σ(S1 × S2 ) = σ(S1 ) + σ(S2 ) ,
σ(S1 ∪ S2 ) = σ(S1 ) ∪ σ(S2 ) ,
σ(M \ S1 ) = σ(M) \ σ(S1 ) .
The reduction then again consists of converting C into a (O ∪ {+})-circuit C having the same structure as C where a ×-gate in C becomes a +-gate in C and
an input gate ui gets label σ(α(ui)). Let ei =def (0, . . . , 0, i) ∈ Nm+1 for i ≥ 0.
Claim. For every
0 , S1 , S2 , · · · ⊆ M such
gate v in C there exist T ⊆ {0} and S
that I(v) = T ∪ i≥0 (Si × Ki ) in C and I(v) = σ(T ) ∪ i≥0 (σ(Si ) + ei ) in C .
Proof of the claim. The proof is by induction on the structure of C.
Without loss of generality assume that C has no ∩-gates. Let v be an input gate. If
α(v) = 0 then I(v) = {0} ∪ i≥0 (∅ × Ki ) in C
and I(v) = {∞} = σ({0}) ∪ i≥0 (σ(∅) + ei ) in C .
For a ×-gate or a ∪-gate v let v1 and v2 be the predecessor gates, and
for a − -gate v let v1 be the predecessor gate. By the induction hypothesis
thereare T1 , T2 ⊆ {0} and S01 , S11 , S21 , . . . S02 , S12 , S22 , · · · ⊆ M such that I(vr ) =
r
Tr ∪ i≥0 (Si × Ki ) in C and I(vr ) = σ(Tr ) ∪ i≥0 (σ(Sir ) + ei) in C for r = 1, 2.
For a ∪-gate v we obtain in C:
I(v) = I(v1 ) ∪ I(v2 ) = T1 ∪ (Si1 × Ki ) ∪ T2 ∪ (Si2 × Ki )
= (T1 ∪ T2 ) ∪
i≥0
i≥0
(Si1
∪
Si2 )
× Ki
i≥0
cc 16 (2007)
Circuits over sets of natural numbers
and in C :
I(v) = I(v1 ) ∪ I(v2 )
σ(Si1 ) + ei
σ(Si2 ) + ei
= σ(T1 ) ∪
∪ σ(T2 ) ∪
i≥0
= σ(T1 ∪ T2 ) ∪
σ(Si1
∪
Si2 )
i≥0
+ ei .
i≥0
For a − -gate v we obtain in C:
I(v) = I(v1 ) = T1 ∪
(M \ Si1 ) × Ki
(Si1 × Ki ) = {0} \ T1 ∪
i≥0
i≥0
and in C :
I(v) = I(v1 ) = σ(T1 ) ∪
σ(Si1 ) + ei
i≥0
m
N ⊗ {i} \ σ(Si1 ) + ei
= {∞} \ σ(T1 ) ∪
i≥0
m
N ⊗ {0} \ σ(Si1 ) + ei
= σ {0} \ T1 ∪
i≥0
σ(M \ Si1 ) + ei .
= σ {0} \ T1 ∪
i≥0
For a ×-gate v we obtain in C:
I(v) = I(v1 ) × I(v2 ) =
=T∪
i≥0
where
and in C :
(Si1
i≥0
i
(Sj1
j=0
T =
T1 ∪
×
(T1 × T2 ) ∪
T1 ×
(Si2
i≥0
×
T2 ∪
2
Si−j
)
× Ki )
(Si2
i≥0
× Ki )
× Ki
× Ki )
∪
T2 ×
(Si1
i≥0
× Ki )
,
219
220 McKenzie & Wagner
cc 16 (2007)
I(v) = I(v1 ) + I(v2 )
σ(Si1 ) + ei
σ(Si2 ) + ei
= σ(T1 ) ∪
+ σ(T2 ) ∪
=
i≥0
σ(T1 ) + σ(T2 ) ∪
i≥0
σ(T1 ) +
σ(Si2 )
i≥0
∪
σ(Si1 ) + ei
σ(T2 ) +
∪
i≥0
= σ(T ) ∪
i
= σ(T ) ∪
i≥0
= σ(T ) ∪
2
σ(Sj1 × Si−j
) + ei
i
×
2
Si−j
)
i≥0
i
×
2
Si−j
)
σ(Sj1 )
+
2
σ(Si−j
)
+ ei
j=0
+ ei
(Sj1
j=0
i
σ(Sj1
j=0
σ
i≥0
i≥0 j=0
+ ei
+ ei
.
This completes the proof of the claim.
Now we prove that a ∈ I(v) in C ⇔ σ(a) ∈ I(v) in C for every gate v and
every a ∈ N. Byour claim there exist T ⊆ {0} and S0, S1 , S2 , · · · ⊆ M such
that I(v) = T ∪ i≥0 (Si × Ki ) in C and I(v) = σ(T ) ∪ i≥0 (σ(Si ) + ei ) in C .
We consider two cases. In the case a = 0 we obtain:
0 ∈ I(v) in C ⇔ T = {0} ⇔ σ(T ) = {∞}
⇔ ∞ ∈ I(v) in C ⇔ σ(0) ∈ I(v) in C .
d
In the case a = j≥1 qj j we obtain (define d =def j>m dj ):
d
qj j ∈ I(v) in C
j≥1
⇔
d
qj j ∈
j≥1
⇔
m
i≥0
d
qj j
m
j=1
∈ Sd ⇔ σ
j=1
⇔ σ
(Si × Ki ) ⇔
d
qj j
d
qj j ∈ Sd × Kd
j≥1
m
d
qj j
∈ σ(Sd )
j=1
+ ed ∈ σ(Sd ) + ed ⇔ σ
j≥1
d
qj j
∈ σ(Sd ) + ed
cc 16 (2007)
Circuits over sets of natural numbers
⇔ σ
j≥1
d
qj j
221
d
j
σ(Si ) + ei ⇔ σ
qj
∈
∈ I(v) in C .
i≥0
j≥1
Applying the equivalence a ∈ I(v) in C ⇔ σ(a) ∈ I(v) in C to the output
gate yields the desired reduction. Polynomial space is needed to perform the
prime decomposition (a possibly weaker reducibility, like a many-one polynomial time reduction with an NP oracle, would suffice).
In some cases the generalized membership problems used above are logspace
many-one equivalent to their standard versions:
Lemma 3.3. Let {∪} ⊆ O ⊆ {∪, ∩}.
(i) MC∗ (O ∪ {+}) ≡log
m MC(O ∪ {+}).
(ii) MF∗ (O ∪ {+}) ≡log
m MF(O ∪ {+}).
Proof. 1. Let C be a (O ∪ {+})-circuit processing m-tuples of natural
numbers. For every gate v of C and every (a1 , . . . , am ) ∈ I(v) we have N =def
2|C| > ai for i = 1, . . . , m. Here |C| is the length of a simple description of
the circuit C where the input numbers are written in binary. We consider an
(O ∪ {+})-circuit C which has the
as C but an input with
same structure
i−1
a
·
N
in
C
, and an input u with
label (a1 , . . . , am ) in C gets label m
i=1 i
label ∞ is fed with an (O ∪ {+})-formula F (C) such that I(u) = {N m , N m +
1, N m + 2, . . . , N m + N m+1 } in C . (For example, define G(0) =def (0 ∪ 1),
G(n + 1) =def (0 ∪ 2n ) + G(n), and F (C) =def (N m + G((m + 1) · |C|)).) Now
we obtain for every gate v:
(a1 , . . . , am ) ∈ I(v) in C ⇔
m
ai ·N i−1 ∈ I(v) in C ,
i=1
m
where (a1 , . . . , am ) ∈ I(v) in C implies i=1 ai ·N i−1 < N m .
Moreover, if v is a level-i-gate in C (input gates being level-0-gates) then
(by induction on i):
∞ ∈ I(v) in C ⇔ 2i · N m , N m +N m+1 ⊆ I(v) in C .
Hence, we have for every gate v:
∞ ∈ I(v) in C ⇔ N m +N m+1 ∈ I(v) in C .
This yields one of the desired reductions. The other is obvious.
2. Here we can proceed in the same way.
222 McKenzie & Wagner
cc 16 (2007)
4. NP-complete membership problems
Lemma 4.1. The problem MF(∪, ∩, +, ×) is in NP.
Proof. Let F = (G,
E, gF , α) be a {∪, ∩, +, ×}-formula, and let b ∈ N.
Observe that
M =def v input gate of F (α(v) + 1) is an upper bound on every
number in g gate of F I(g). The NP-algorithm just works as follows: Guess
nondeterministically a number ≤ M to every gate of F . Then check whether
these numbers really prove that b is in I(gF ).
The following is a nontrivial improvement over the PSPACE upper bound
for MC(∪, +) in (30):
Lemma 4.2. The problems MC(∪, +) and MC(∪, ×) are in NP.
Proof. Let C = (G, E, gC , α) be a {∪, +}-circuit such that gC is the only
gate of C with outdegree 0, and let TC be the result of unfolding C into a tree.
By analogy with Venkateswaran’s proof trees (28), a subtree T of TC is called
computation tree of C iff
◦ the output gate of TC is in T ,
◦ both predecessors of a +-gate of T are in T , and
◦ exactly one predecessor of a ∪-gate of T is in T .
Hence T describes one of the many ways to compute a number from I(C).
A gate g in C corresponds to several copies of it in TC (and hence also in
a computation tree T of C). Let βC,T (g) be the number of copies of g in T .
In the same way, an edge in C corresponds to several copies of it in TC (and
hence also in a computation tree T of C). Let βC,T (e) be the number of copies
of the edge e in T .
Defining
s(C, T ) =def
α(g)·βC,T (g) ,
input gate g of C
we obtain immediately I(C) = {s(C, T ) : T is a computation tree of C}.
A function β : G ∪E → N is a valuation function of C if the following holds:
◦ β(gC ) = 1,
◦ if g is a +-gate with the incoming edges e1 and e2 then β(g) = β(e1 ) =
β(e2 ),
cc 16 (2007)
Circuits over sets of natural numbers
223
◦ if g is a ∪-gate with the incoming edges e1 and e2 then β(g) = β(e1 )+β(e2 ),
◦ if g is a gate with the outgoing edges e1 , . . . ek then β(g) = β(e1 )+· · ·+
β(ek ).
α(g)·β(g).
For a valuation function β of C define s(C, β) =def
input gate g of C
Claim 1. If T is a computation tree of C then there exists a valuation function β of C such that s(C, β) = s(C, T ).
Proof.
This is easy because βC,T has the desired properties.
Claim 2. If β is a valuation function of C then there exists a computation
tree T of C such that s(C, T ) = s(C, β).
Proof. For a valuation function β of C there can be many computation
trees T of C such that βC,T = β and hence s(C, T ) = s(C, β). We construct
one of them by applying the following replacement rules to the gates v of C
(starting with the predecessors of the output gate and going upward). The
labels at the gates and edges represent the β-values.
If v is a +-gate then apply:
u1
u1
u2
u2
r
r
...
v r
r copies of v
...
...
r
r
If v is a ∪-gate then apply:
u1
u2
u1
u2
...
...
...
...
r
s
s
r
v r+s
...
r+s
If v is an input gate then apply:
r+s copies of v
224 McKenzie & Wagner
cc 16 (2007)
...
v r
...
...
r
r copies of v
r
After the application of a rule to a gate v the following holds:
◦ the definition of a computation tree applies to the copies of v,
◦ the number of copies of v is β(v), and
◦ for every incoming edge e to v, the number of copies of e is β(e).
In such a way we obtain βC,T = β and hence s(C, T ) = s(C, β). This completes
the proof of Claim 2.
Now we obtain I(C) = {s(C, β) : β is a valuation function of C}, and hence
a ∈ I(C) ⇔ ∃β(β is a valuation function of C and s(C, β) = a). However, the
latter property is in NP. This completes the proof of MC(∪, +) ∈ NP. From
this, Lemma 3.2, and Lemma 3.3 we obtain MC(∪, ×) ∈ NP.
Lemma 4.3. MF(∪, ×) is NP-hard.
log
Proof. We prove the NP-hardness
n of MF(∪, ×) by showing 3-SAT ≤m
MF(∪, ×). Let H(x1 , . . . , xm ) = j=1 Hj where every Hj is a clause with
three literals. Further, for j ≥ 1, let pj be the j-th prime. For i = 1, . . . , m
define ai =def xi in Hj pj and bi =def xi in Hj pj . Obviously, H ∈ 3-SAT ⇔
m
n
n 3
2
j=1 pj ∈ I( i=1 (ai ∪ bi )· j=1 (1 ∪ pj ∪ pj )).
As an immediate consequence of the preceding lemmas and Theorem 2.3.1
we obtain:
Theorem 4.4. The problems MF(∪, ∩, +, ×), MF(∪, ∩, +), MF(∪, ∩, ×),
MF(∪, +, ×), MF(∪, ×), MC(∪, +), and MC(∪, ×) are NP-complete.
cc 16 (2007)
Circuits over sets of natural numbers
225
5. PSPACE-complete membership problems
Lemma 5.1. The problems MF(∪, ∩,− , ×), MC(∪, ∩, ×), and MC(∪, ∩, +) are
PSPACE-hard.
Proof. We present a logspace many-one reduction from the PSPACEcomplete quantified Boolean 3-CNF formula problem to MF(∪, ∩,− , ×) and
MC(∪, ∩, ×). Since in these constructions we use only numbers whose prime
number decompositions are known, the latter reduction can be converted by
Lemma 3.2 and Lemma 3.3 into a logspace many-one reduction to MC(∪, ∩, +).
Let Q1 x1 . . .Qm xm H(x1 , . . . , xm ) be a quantified Boolean 3-CNF formula
such that H = nj=1 Hj , the formulae H1 , . . . Hn are clauses with three literals
each, and Q1 . . . Qm ∈ {∃, ∀}.
For k = m, m − 1, . . . 1, 0 we construct a (∪, ∩, ×)-circuit (a (∪, ∩,− , ×)formula, resp.) Ck such that for all α1 , . . . , αk ∈ {0, 1} the following holds:
Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk , xk+1 , . . . , xm )
(5.2)
n
k
m
i
p3j ·
pαn+i
·
pn+i ∈ I(Ck )
⇔
j=1
i=1
i=k+1
where pi is the i-th prime. Then, for k = 0, we obtain
Q1 x1 . . . Qm xm H(x1 , . . . , xm ) ⇔
n
j=1
p3j
·
m
pn+i ∈ I(C0 ) ,
i=1
the desired reduction.
For k = m we define ai =def pn+i · xi in Hj pj and bi =def
xi in Hj pj
m
n
2
for i = 1, . . . , m, and Cm =def i=1 (ai ∪ bi ) · j=1(1 ∪ pj ∪ pj ). It is easy to see
αi
that H(α1 , . . . , αm ) ⇔ nj=1 p3j · m
i=1 pn+i ∈ I(Cm ).
Now, for the step from k to k−1, assume (5.2). In the case Qk = ∃ we have:
∃xk Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk−1 , xk , xk+1 , . . . , xm )
⇔ Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk−1 , 0, xk+1, . . . , xm )
∨ Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk−1 , 1, xk+1, . . . , xm )
⇔
n
p3j ·
k−1
j=1
∨
n
j=1
i=1
p3j
·
m
i
pαn+i
·
k−1
i=1
pn+i ∈ I(Ck )
i=k+1
i
pαn+i
·
m
i=k
pn+i ∈ I(Ck )
226 McKenzie & Wagner
⇔
n
p3j
·
j=1
k−1
cc 16 (2007)
i
pαn+i
·
m
i=1
pn+i ∈ I (Ck × pn+k ) ∪ Ck = I(Ck−1 )
i=k
where Ck−1 =def Ck × (pn+k ∪ 1). Since Ck appears only once in the definition
of Ck−1 , this works for the circuit case as well as for the formula case.
In the case Qk = ∀ the reasoning becomes:
∀xk Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk−1 , xk , xk+1 , . . . , xm )
⇔ Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk−1 , 0, xk+1, . . . , xm )
∧ Qk+1 xk+1 . . . Qm xm H(α1 , . . . , αk−1 , 1, xk+1, . . . , xm )
⇔
n
p3j
·
j=1
∧
k−1
n
p3j ·
n
k−1
k−1
pn+i ∈ I(Ck )
i=k+1
i
pαn+i
·
i=1
p3j ·
j=1
m
·
i=1
j=1
⇔
i
pαn+i
m
pn+i ∈ I(Ck )
i=k
i
pαn+i
·
m
i=1
pn+i ∈ I (Ck × pn+k ) ∩ Ck = I(Ck−1 )
i=k
where Ck−1 =def (Ck × pn+k ) ∩ Ck . Since Ck appears twice in the definition of
Ck−1 this works only for the circuit case. For the formula case we continue as
follows:
n
k−1
m
n
k−1
m
αi
αi
3
3
pj ·
pn+i ·
pn+i ∈ I(Ck ) ∨
pj ·
pn+i ·
pn+i ∈ I(Ck )
⇔¬
j=1
i=1
i=k+1
n
k−1
m
⇔ ¬
p3j ·
j=1
⇔
n
j=1
p3j ·
i
pαn+i
·
i=1
k−1
i=1
i
pαn+i
·
j=1
i=1
pn+i ∈ I Ck × (pn+k ∪ 1)
i=k
i=k
m
pn+i ∈ I Ck × (pn+k ∪ 1) = I(Ck−1) ,
i=k
where Ck−1 =def Ck × (pn+k ∪ 1).
Recall that a generalized circuit has + as its only arithmetic operation and
that it operates on subsets of Nm ∪ {∞}, where ∞ is an absorbing element
for + (see definition before Lemma 3.2).
cc 16 (2007)
Circuits over sets of natural numbers
227
Lemma 5.3. For every gate g of a generalized (∪, ∩,− , +)-circuit, if I(g) = ∅
then I(g) ∩ ({0, 1, . . . , 2|C| + 1}m ∪ {∞}) = ∅.
Proof. Let M, M ⊆ Nm and A, A ∈ {{∞}, ∅}. Since (N∪{∞})\(M ∪A) =
(N \ M) ∪ ({∞} \ A), (M ∪ A) ∪ (M ∪ A ) = (M ∪ M ) ∪ (A ∪ A ), and
(M ∪ A) + (M ∪ A ) = (M + M ) ∪ (A ∪ A ), the presence of ∞ in a circuit
does not influence the generation of the other elements. Hence it suffices to
consider a generalized (∪, ∩,− , +)-circuit C = (G, E, gC , α) over Nm ∪ {∞}
where an input ∞ is replaced with ∅ and to prove that I(g) = ∅ implies
I(g) ∩ {0, 1, . . . , 2|C| + 1}m = ∅, for every gate g.
By De Morgan’s law we can assume that C has no ∩-gates. For the gates g
of C we define inductively σ(g) =def max{ai : i = 1, . . . , m} for an input gate g
with α(g) = (a1 , . . . , am ), σ(g) =def 0 for an input gate g with α(g) = ∞,
σ(g) =def σ(g1 ) + σ(g2 ) + 1 for ∪-gates and +-gates g with predecessors g1
and g2 , and σ(g) =def σ(g1 ) for − -gates g with predecessor g1 . Obviously we
have σ(g) ≤ 2|C| for every gate g.
For s ∈ N let [s] =def {0, 1, . . . , s}. A set M ⊆ N is called an s-set if and
only if M is a finite union of sets M1 × M2 × · · · × Mm where Mk ⊆ [s] or
Mk ⊆ [s] for every k = 1, 2, . . . , m. The following two claims are easy to prove.
Claim 1. For every s ≥ 0, the class of all s-sets from 2N
union, intersection and complement.
m
is closed under
Claim 2. If s1 ≤ s2 , then every s1 -set is an s2 -set. The following claim
implies the lemma because σ(g) ≤ 2|C| for any g. So any nonempty σ(g)-set
contains an (i1 , i2 , . . . , im ) such that for every k = 1, 2, . . . , m, either ik ≤ 2|C|
or ik = 2|C| + 1.
Claim 3. For every gate g of C, the set I(g) is a σ(g)-set.
Proof of Claim 3. By induction.
1. If g is an input gate then I(g) ⊆ [σ(g)]m .
2. Let g be a − -gate with predecessor g1 . By the assumption of the induction, I(g1 ) is a σ(g1 )-set. By Claim 1, I(g1 ) is a σ(g1 )-set too, but σ(g1 ) = σ(g).
3. Let g be a ∪-gate with predecessors g1 and g2 . By the assumption of the
induction, I(g1 ) is a σ(g1 )-set and I(g2 ) is a σ(g2 )-set. By Claim 2, I(g1 ) and
I(g2 ) are σ(g)-sets. By Claim 1, I(g) = I(g1 ) ∪ I(g2 ) is a σ(g)-set.
4. Let g be a +-gate with predecessors g1 and g2 . By the assumption
of the induction, I(g1 ) is a σ(g1 )-set and I(g2 ) is a σ(g2 )-set. Let I(g1 ) =
228 McKenzie & Wagner
· · × Mim )
i∈I (Mi1 × · and I(g2 ) = j∈J (Mj1
cc 16 (2007)
where Mik ⊆ [σ(g1 )] of Mik ⊆ [σ(g1 )] for k = 1, 2, . . . , m
× · · · × Mjm
) where Mjk
⊆ [σ(g2 )] of Mjk
⊆ [σ(g2 )] for
k = 1, 2, . . . , m.
We conclude
I(g1 ) + I(g2 ) =
(Mi1 × · · · × Mim ) +
i∈I
=
i∈I j∈J
=
(Mj1
× · · · × Mjm
)
j∈J
(Mi1 × · · · × Mim ) + (Mj1
× · · · × Mjm
)
(Mi1 + Mj1
) × · · · × (Mim + Mjm
) .
i∈I j∈J
. If Mik = ∅ or Mjk
= ∅ then Mik + Mjk
=∅⊆
Now we consider Mik + Mjk
[σ(g)]. Otherwise we conclude as follows.
Case 1. If Mik ⊆ [σ(g1 )] and Mjk
⊆ [σ(g2 )] then Mik + Mjk
⊆ [σ(g1 ) + σ(g2 )] ⊆
[σ(g)].
Case 2. If Mik ⊆ [σ(g1 )] and Mjk
⊆ [σ(g2 )] then Mik +Mjk
⊇ [σ(g1 )]+[σ(g2 )] =
[σ(g1 ) + σ(g2 ) + 1] = [σ(g)] and hence Mik + Mjk ⊆ [σ(g)].
⊆ [σ(g2 )] then Mik +Mjk
⊇ [minMik + σ(g2 )] ⊇
Case 3. If Mik ⊆ [σ(g1 )] and Mjk
[σ(g1 ) + σ(g2 )] ⊇ [σ(g)] and hence Mik + Mjk ⊆ [σ(g)].
Lemma 5.4. The problem MC∗ (∪, ∩,− , +) is in PSPACE.
Proof. For a generalized {∪, ∩,− , +}-circuit C = (G, E, gC , α) and b ∈
Nm ∪ {∞}, the following alternating polynomial time algorithm defines on each
of its accepting subtrees a proof that b ∈ I(F ). Moreover, in every step on
every path the algorithm defines g ∈ G, a ∈ Nm ∪ {∞}, and σ(g) ∈ {0, 1}.
Afterwards, i.e., in the computation tree above this configuration, the algorithm
checks whether a ∈ I(g) if σ = 1 and a ∈ I(g) if σ = 0.
For a gate g of indegree 2 let g1 and g2 be the predecessors of g, for a gate g
of indegree 1 let g1 be the predecessor of g. The justification for choosing c1
and c2 below from M =def {0, 1, . . . , 2|C|+1}m ∪ {∞} is provided by Lemma 5.3.
1. g := gC ; a := b; σ = 1
2. while g is not an input gate do:
2.1. if g is a +-gate and σ = 1 then: choose existentially c1 , c2 ∈ M
such that a = c1 + c2 ; choose universally i ∈ {1, 2}; g := gi ; a := ci
cc 16 (2007)
Circuits over sets of natural numbers
229
2.2. if g is a +-gate and σ = 0 then: choose universally c1 , c2 ∈ M such
that a = c1 + c2 ; choose existentially i ∈ {1, 2}; g := gi ; a := ci
2.3. if g is a ∪-gate with σ = 1 then: choose existentially i ∈ {1, 2};
g := gi
2.4. if g is a ∪-gate with σ = 0 then: choose universally i ∈ {1, 2};
g := gi
2.5. if g is a ∩-gate with σ = 1 then: choose universally i ∈ {1, 2};
g := gi
2.6. if g is a ∩-gate with σ = 0 then: choose existentially i ∈ {1, 2};
g := gi
2.7. if g is a − -gate then: g := g1 ; σ = 1 − σ
3. if g is an input gate then: if (σ = 1 ⇔ a = α(g)) then accept else
reject .
Since every proof of b ∈ I(F ) is considered on one accepting subtree of this
algorithm, the latter really checks whether b ∈ I(F ).
As a direct consequence of Lemma 5.1, Lemma 5.4, and Lemma 3.2 we
obtain:
Theorem 5.5. The problems MC∗ (∪, ∩,− , +), MC(∪, ∩,− , +), MC(∪, ∩, +),
MC(∪, ∩,− , ×), MC(∪, ∩, ×), and MF(∪, ∩,− , ×) are PSPACE-complete. The
problem MF(∪, ∩,− , +, ×) is PSPACE-hard.
6. Beyond PSPACE
As an auxiliary tool to prove the main results of this section we introduce the
following (∪, ∩, ·)-circuits that compute finite sets of words. Such a circuit C
only has gates of indegrees 0 and 2. Every input gate (i.e., gate of indegree 0)
g is labelled with a word from a given alphabet Σ∗ , and every gate of indegree 2
is labelled with ∪, ∩, or · (concatenation). For each of its gates g, the circuit
computes a set I(g) ⊆ Σ∗ inductively defined as follows: If g is an input
gate with label v then I(g) =def {v}. If g is an ω-gate (ω ∈ {∪, ∩, ·}) with
predecessors gl , gr then I(g) =def I(gl ) ω I(gr ). Finally, I(C) =def I(gC ) where
gC is the output gate of C. A (∪, ∩, ·)-circuit C is called special if for every
gate g of C there exists a σ(g) ∈ N such that I(g) ⊆ Σσ(g) and the following
holds for every gate g with predecessors g1 and g2 : If g is a ∩-gate or a ∪-gate
then σ(g) = σ(g1 ) = σ(g2 ), and if g is a ·-gate then σ(g) = σ(g1 ) + σ(g2 ).
230 McKenzie & Wagner
cc 16 (2007)
Let NE(∪, ∩, ·) be the nonemptyness problem for special (∪, ∩, ·)-circuits, i.e.,
NE(∪, ∩, ·) =def {C : C is a special (∪, ∩, ·)-circuit such that I(C) = ∅}.
Lemma 6.1. The problem NE(∪, ∩, ·) is NEXPTIME-hard.
Proof. We describe a logarithmic space many-one master reduction from
NEXPTIME to NE(∪, ∩, ·), inspired by (26). More specifically, given a nondeterministic one-tape Turing machine M which runs in exponential time, we
construct in logarithmic space for every input x a special (∪, ∩, ·)-circuit Cx
such that: M accepts x if and only if I(Cx ) = ∅.
Let B be the tape alphabet of M including the blank symbol , let A ⊆ B
be the input alphabet, and let S be the set of states including the initial state s0
and the accepting state s1 . Without loss of generality we may assume that M
starts with the head in tape cell 1 which carries the leftmost input symbol,
that M never moves its head left from tape cell 1, that it accepts with the head
on tape cell 1 and with an empty tape, and that it loops from then on in its
accepting state without moving its head. Furthermore, let p be a polynomial
such that every computation path of M on input x has at most 2p(|x|) steps.
Next we see how to describe configurations and computations of M on an
input x of length n as words over the alphabet Σ =def B ∪ S ∪ {} where is a new symbol. A situation which occurs during M’s work on input x can be
completely described by the configuration b1 . . . bi−1 sbi bi+1 . . . b2p(n) meaning
that b1 , . . . b2p(n) are the symbols in the tape cells 1, . . . , 2p(n), resp., and that
the head scans tape cell i with the state s. An accepting computation of M
on input x can be described by the sequence c0 c1 . . . c2p(n) of such configuratons
p(n)
p(n)
where c0 = s0 x2 −n , c2p(n) = s1 2 , and ci is a successor configup(n)
p(n)
ration of ci−1 for i = 1, . . . 2p(n) . Let compM (x) ⊆ Σ(2 +1)·(2 +3) be the set
of all such descriptions of accepting computations of M on input x.
In what follows we will construct in logarithmic space a special (∪, ∩, ·)circuit Cx such that I(Cx ) = compM (x). Then, M accepts x ⇔ compM (x) =
∅ ⇔ I(Cx ) = ∅, and we are done.
Obviously, if c is a successor configuration of c then c and c can differ by at
most three symbols, i.e., there are u, v ∈ Σ∗ and a, b, d, a , b , d ∈ Σ such that
c = uabdv, c = ua b d v, and abda b d ∈ PM where PM ⊆ Σ6 is defined by the
program of M. Let RM =def {(uabdv, uab d v) : u, v ∈ Σ∗ ∧ abda b d ∈ PM }.
cc 16 (2007)
Circuits over sets of natural numbers
231
Now we can write down compM (x) as
compM (x) = firstM (x) ∩ nextM (x) ∩ lastM (x), where
p(n)
p(n)
p(n)
firstM (x) =def {s0 x2 −n } · Σ2 (2 +3)
(all strings of length (2p(n) + 1)(2p(n) + 3) which start
with the correct configuration)
nextM (x) =def {c0 c1 . . . c2p(n) : |ci| = 2p(n) +3 ∧ (ci−1 , ci ) ∈ RM }
(all strings of length (2p(n) + 1)(2p(n) + 3) which
have the correct next configuration property)
p(n)
p(n)
p(n)
lastM (x) =def Σ2 (2 +3) · {s1 2 }
(all strings of length (2p(n) + 1)(2p(n) + 3) which end
with the correct configuration)
We now define sets E(k) and Ka (k) for a ∈ Σ:
E(k) =def Σk ,
Ka (k) =def {ak } for a ∈ Σ .
These sets are easily seen to have logspace-constructible (in the length of the parameter k) special (∪, ∩, ·)-circuits, as the following recursive definitions show.
E(0) = {ε} , E(2k) = E(k)·E(k) ,
and E(2k+1) = E(k)·E(k)·Σ .
Ka (0) = {ε} , Ka (2k) = Ka (k)·Ka(k) , and Ka (2k+1) = Ka (k)·Ka (k)·{a}.
Now we can write:
firstM (x) = {s0 x} · K(2p(n) −n) · {} · E 2p(n) · (2p(n) +3) ,
lastM (x) = E 2p(n) · (2p(n) +3) · {s1 } · K(2p(n) ) · {} ,
which shows that special (∪, ∩, ·)-circuits can be constructed in logarithmic
space (in the length of x) which compute firstM (x) and lastM (x). To do the
same with nextM (x) we define some auxiliary sets with parameter k ∈ N, and
in each case we show how to construct in logarithmic space (in the value of the
parameter k) special (∪, ∩, ·)-circuits to compute these sets.
k
m
D(k, m) =def wvw : |w| = 2m ∧ |v| = 2k −2m +3 ⊆ Σ2 +2 +3
for m = 0, 1, . . . , k − 1 constructed using D(k, 0) = a∈Σ {a} · E(2k +2) · {a}
and
D(k, m+1) = D(k, m) · E(2m ) ∩ E(2m ) · D(k, m) .
S(k, m) =def wvw : |w| = |w | = 2m +3 ∧ |v| = 2k −2m
k
m
∧ (w, w ) ∈ RM ⊆ Σ2 +2 +6
232 McKenzie & Wagner
cc 16 (2007)
for m = 0, 1, . . . , k constructed using
(aw · E(2k −1) · aw ) ∪ (wa · E(2k −1) · w a) ,
S(k, 0) =
a∈Σ ww ∈PM
D(k, m) · E(2m +3) ∩ E(2m ) · S(k, m)
∪ S(k, m) · E(2m ) ∩ E(2m +3) · D(k, m) .
S(k) =def ww : w, w ∈ Σ∗ ∧ |w| = |w | = 2k +3 ∧ (w, w ) ∈ RM
S(k, m+1) =
k+1 +6
⊆ Σ2
constructed using S(k) = S(k, k).
R(k, m) =def w0 w1 . . . w2m : wi−1 wi ∈ S(k) for i = 1, . . . , 2m
m +1)(2k +3)
⊆ Σ(2
constructed using R(k, 0) = S(k), and
R(k, m+1) = R(k, m) · E (2k +3)2m ∩ E (2k +3)2m · R(k, m) .
Finally we obtain nextM (x) = R(p(n), p(n)).
Theorem 6.2. The problem MC(∪, ∩, +, ×) is NEXPTIME-complete.
Proof.
1. To prove MC(∪, ∩, +, ×) is NEXPTIME-hard we show
NE(∪, ∩, ·) ≤log
m MC(∪, ∩, +, ×) .
For w ∈ {0, 1}∗ let bin−1 (w) be that natural number whose binary description is w (possibly with leading zeros), and for L ⊆ {0, 1}∗ let bin−1 (L) =def
{bin−1 (w) : w ∈ L}.
Given a special (∪, ∩, ·)-circuit C we construct in logarithmic space a
(∪, ∩, +, ×)-circuit C such that I(C ) = bin−1 (I(C)). The circuit C basically has the same structure as C: An input gate in C with label w becomes
an input gate in C with label bin−1 (w), a ∪-gate in C becomes a ∪-gate in C ,
and a ∩-gate in C becomes a ∩-gate in C . A · -gate g in C with predecessor
g1 , g2 such that I(g2 ) ⊆ {0, 1}k is replaced in C by a subcircuit which computes bin−1 (I(g1 ) · I(g2 )) = (2k × bin−1 (I(g1 ))) + bin−1 (I(g2 )). The 2k can
be computed by an (∪, ∩, ×)-circuit C which has the same structure as C
but instead of an input w it has the input 2|w| , and instead of a ·-gate it
cc 16 (2007)
Circuits over sets of natural numbers
233
has a ×-gate. Here it is important that C is a special (∪, ∩, ·)-circuit. Now,
I(C) = ∅ ⇔ I(C ) = ∅ ⇔ 0 ∈ ({0} × I(C )).
2. To see that MC(∪, ∩, +, ×) is in NEXPTIME, simply unfold a given
(∪, ∩, +, ×)-circuit into a (possibly exponentially larger) (∪, ∩, +, ×)-formula
and apply the NP-algorithm from Lemma 4.1.
Corollary 6.3. The problem NE(∪, ∩, ·) is NEXPTIME-complete.
As an immediate consequence of Theorem 6.2 we obtain also:
Theorem 6.4. The problem MC(∪, ∩,− , +, ×) is NEXPTIME-hard.
−
Remark 6.5. Since MC(∪, ∩,− , +, ×) ≡log
m MC(∪, ∩, , +, ×) these problems
cannot be in NEXPTIME unless NEXPTIME = co-NEXPTIME. In fact,
there is evidence suggesting that MF(∪, ∩,− , +, ×) might not be decidable.
Indeed, Christian Glaßer in Würzburg was the first to observe that there is a
simple {∪, ∩,− , +, ×}-formula G (see the Example 2.2 in Section 2) having the
property that (G, 0) ∈ MF(∪, ∩,− , +, ×) if and only if Goldbach’s Conjecture
is true. Hence, a decision procedure for MF(∪, ∩,− , +, ×) would provide a
terminating algorithm to test Goldbach’s Conjecture; this would be surprising.
7. P-complete membership problems
Theorem 7.1. The problem MC(+, ×) is P-complete.
Proof. To prove MC(+, ×) ∈ P, let C be a (+, ×)-circuit and b ∈ N. Any
gate in C computes a singleton. If I(C) = {b} then I(g) = {a} such that a > b
for some gate g can only occur if this a is multiplied with 0 in a subsequent
gate. Hence b + 1 does the same job as a in this computation. Therefore a
polynomial time algorithm to decide I(C) = {b} consists in just evaluating C
but replacing every a > b with b + 1.
The hardness proof is by showing that the P-complete monotone Boolean
circuit value problem (17) can be reduced to MC(+, ×). To do so we convert
a monotone Boolean circuit C into a (+, ×)-circuit C of exactly the same
structure where every Boolean input 0 (1, resp.) in C becomes the integer
input 0 (1, resp.) in C , every ∨-gate in C becomes a +-gate in C , and every
∧-gate in C becomes a ×-gate in C . It is easy to see that v evaluates to 0 in C
if and only if 0 ∈ I(v) in C . Hence, C evaluates to 0 if and only if 0 ∈ I(C ). 234 McKenzie & Wagner
cc 16 (2007)
Theorem 7.2. The problems MC(∪, ∩) and MC(∪, ∩,− ) are P-complete.
Proof. To prove MC(∪, ∩,− ) ∈ P, let C be a (∪, ∩,− )-circuit and b ∈ N.
Define S =def v input gate I(v). We prove that the claim that for every gate v
there are sets P, N ⊆ S such that I(v) = P ∪ N . Then we can compute in
polynomial time all I(v) from the inputs down to the output by storing only
the sets P and N.
To see the claim, let P1 , P2 , N1 , N2 ⊆ S and observe
(P1 ∪ N1 ) ∪ (P2 ∪ N2 ) = (P1 ∪ P2 ) ∪ (N1 ∩ N2 ) ,
(P1 ∪ N1 ) ∩ (P2 ∪ N2 ) = (P1 ∩ P2 ) ∪ (P1 \ N2 ) ∪ (P2 \ N1 ) ∪ (N1 ∪ N2 ) ,
P1 ∪ N1 = N1 \ P1 .
The hardness proof is by showing that the P-complete monotone Boolean circuit
value problem can be reduced to MC(∪, ∩). To do so we convert a monotone
Boolean circuit C into a (∪, ∩)-circuit C of almost the same structure where
every input gate in C with Boolean value 1 becomes an input gate in C with
integer value 1, every input gate in C with Boolean value 0 becomes an ∩-gate
in C with two input gates with labels 0 and 1, resp., as predecessors, every
∨-gate in C becomes a ∪-gate in C , and every ∧-gate in C becomes a ∩-gate
in C . It is easy to see that v evaluates to 0 in C if and only if I(v) = ∅ in C ,
and v evaluates to 1 in C if and only if I(v) = {1} in C . Hence, C evaluates
to 1 if and only if 1 ∈ I(C ).
8. Circuits with intersection as the only set operation
Circuits having intersection as their only set operation are special in the sense
that every node computes a singleton or the empty set. Thus these circuits
bear some relationship to circuits of the same type without intersection. For
O ⊆ {+, ×} define
EQ(O) =def (C1 , C2 ) : C1 , C2 are O-circuits such that I(C1 ) = I(C2 ) .
Lemma 8.1.
(i) MC(∩, +) ≤log
m EQ(+),
(ii) MC(∩, +, ×) ≡log
m EQ(+, ×).
cc 16 (2007)
Circuits over sets of natural numbers
235
Proof. 1. Let C be a (∩, +)-circuit with output gate gC . Without loss of
generality, every predecessor of a ∩-gate is a +-gate having outdegree 1. For a
gate g of C, let Cg be that (+)-circuit which arises from C by defining g as the
output gate and by removing all ∩-gates h where all gates fed with the output
of h are now fed with output of the right predecessor of h. Further, let C be
the (+)-circuit which arises from C by replacing every ∩-gate by a +-gate and
every input by 0. Finally, let Cg be the (+)-circuit constructed from C and Cg
as follows: Cut the vertices going into g in C , and identify gate g of C with
gate g of Cg . The output gate of Cg is the output gate gC of C . If n(g) is the
number of paths in C from g to gC then we have I(Cg ) = n(g) · I(Cg ). Let A be
the set of all ∩-gates from which gC is accessible in C. Observe that encodings
of the circuits Cg and Cg can be computed in logarithmic space and that, if
I(left predecessor of h) = I(right predecessor of h) for every gate h ∈ A then
I(g) = I(Cg ) for every gate g of C from which gC is accessible. For b ∈ N
let C(b) be the circuit which consists only of an input gate with label b. We
obtain:
b ∈ I(C)
⇔ b ∈ I(CgC ) ∧
h∈A with predecessors h1 ,h2
I(Ch1 ) = I(Ch2 )
I(Ch 1 ) = I(Ch 2 )
⇔ b ∈ I(CgC ) ∧
h is a ∩-gate with predecessors h1 ,h2
⇔ I(CgC ) = I C(b) ∧
I(Ch 1 ) = I(Ch 2 ) .
h is a ∩-gate with predecessors h1 ,h2
Thus it is sufficient to show that there exists a logarithmic space algorithm converting the (+)-circuits C0 , . . . Cr , C0 , . . . Cr into (+)-circuits D, D such that
I(D) = I(D ) ⇔ I(Ci ) = I(Ci ) for i = 0, 1, . . . r .
r
r
r
i=0 |Ci |+ i=0 |Ci | we get a ≤ N −1 for a ∈
2
Defining
N
=
def
i=0 I(Ci ) ∪
r
I(C
).
Now
it
is
easy
to
construct
(+)-circuits
D,
D
such that
i
i=0
i
)·N
.
However,
for
I(D) = ri=0 I(Ci )·N i and I(D ) = ri=0 I(C
ri
r a0 , . .i. , ar ,
i
a
·
N
=
⇔
b0 , . . . , br ∈ {0, 1, . . . N − 1} there holds
i=0 i
i=0 bi · N
(ai = bi for i = 0, 1, . . . r).
2. The ≤log
m -part is very similar to the proof of Item 1. Here it is sufficient
to show that there exists an logarithmic space algorithm converting the (+, ×)circuits C0 , . . . Cr , C0 , . . . Cr into (+, ×)-circuits C, C such that
I(C) = I(C ) ⇔ I(Ci ) = I(Ci ) for i = 0, 1, . . . r .
236 McKenzie & Wagner
cc 16 (2007)
r
r
r
i=0 |Ci |+ i=0 |Ci | we get a ≤ 2N −3 for a ∈
Defining
N
=
2
def
i=0 I(Ci ) ∪
r
). Now it is easy to construct (+, ×)-circuits C, C such that
i=0 I(Ci r
r
Ni
Ni
and I(C ) =
However, for
I(C) =
i ) + 2) .
i=0 (I(Ci ) + 2)
i=0 (I(C
i
r
N
Ni
⇔
a0 , . . . , ar , b0 , . . . , br ∈ {2, 3, . . . 2 − 1} there holds i=0 ai = ri=0 bN
i
(ai = bi for i = 0, 1, . . . r).
For the ≥log
m -part observe I(C) = I(C ) ⇔ 0 ∈ ((I(C) ∩ I(C )) × {0}). Theorem 8.2. MC(∩, +), MC(+), and EQ(+) are ≤log
m -complete for C= L.
Proof. 1. For the membership in C= L it is sufficient to prove that EQ(+) ∈
C= L (Lemma 8.1). For a (+)-circuit C, let C be that circuit which arises
from C by adding a new input gate gI labelled with 1 and replacing every
old input gate with label a by a (+)-circuit computing a from the sole input
gate gI . Consequently,
I(C) = I(C ) = the number of paths in C from the input to the output ,
and hence I is a #L-function. For given (+)-circuits C1 , C2 define the #Lfunctions f, g by f (C1, C2 ) =def I(C1 ) and g(C1, C2 ) =def I(C2 ). Now, (C1 , C2 ) ∈
EQ(+) ⇔ f (C1 , C2 ) = g(C1 , C2).
2. For the C= L-hardness of MC(+) let A ∈ C= L. By definition there
exist a function f ∈ #L and a logarithmic space computable function g such
that x ∈ A ⇔ f (x) = g(x). Let M be a nondeterministic logspace Turing
machine such that f (x) is the number of accepting paths of M on input x. Let
further Cx be the transition graph of M on input x. If we consider Cx to be a
(+)-circuit whose accepting and rejecting nodes are inputs with labels 1 and 0
respectively, then we have I(Cx ) = {f (x)}. Consequently
x ∈ A ⇔ f (x) = g(x) ⇔ g(x) ∈ I(Cx ) ⇔ Cx , g(x) ∈ MC(+) .
Theorem 8.3.
(i) The problem MC(∩, +, ×) is in co-R.
(ii) The problem MC(∩, +, ×) is P-hard.
Proof. 1. By Lemma 8.1 it is sufficient to prove EQ(+, ×) ∈ co-R. We
can do this by feeding the two circuits forming the EQ(+, ×) instance into
a subtraction gate, and then testing whether the resulting circuit over the
integers computes the 0 function. Now Ibarra and Moran show how to perform
the latter in co-R (20).
2. This is a direct consequence of Theorem 7.1.
cc 16 (2007)
Circuits over sets of natural numbers
237
Theorem 8.4.
(i) The problem MC(∩, ×) is in P.
(ii) The problem MC(∩, ×) is C= L-hard.
Proof. 1. By Lemma 3.2 it suffices to prove that MC∗ (∩, +) is in P. Let C
be a {∩, +}-circuit processing values from Nm ∪ {∞}. There is at most one
m-tuple or ∞ in I(v) for every gate v. The size of these m-tuples is polynomial
in the size of the input. So we can evaluate C step by step in polynomial time.
2. For the C= L-hardness of MC(∩, ×) let A ∈ C= L. Hence there exist
functions f1 , f2 ∈ #L such that x ∈ A ⇔ f1 (x) = f2 (x). For i = 1, 2, let Mi
be a nondeterministic logspace Turing machine such that fi (x) is the number
of accepting paths of Mi on input x. Let further Cix be the transition graph
of Mi on input x. If we consider Cix to be a (×)-circuit whose rejecting nodes
are inputs with label 1 and whose accepting nodes are inputs with label 2 then
we have I(Cix ) = {2fi (x) }. Consequently
x ∈ A ⇔ f1 (x) = f2 (x) ⇔ I(C1x ) = I(C2x ) ⇔ 0 ∈ I(C1x ) ∩ I(C2x ) × {0} .
Theorem 8.5. The problem MC(×) is NL-complete.
Proof. 1. To prove the hardness we reduce the NL-complete graph accessibility problem GAP for directed acyclic graphs to the problem MC(×). Let
G = (V, E) be a directed acyclic graph and s, t ∈ V be the source and target
vertices, respectively. Recall that our circuits are not necessarily connected.
Without loss of generality assume that every vertex has indegree 0 or 2 and
that s has indegree 0. Now convert G into a (×)-circuit C by labelling every
vertex with indegree 2 by ×, the source vertex s by 0 and all other vertices
with indegree 0 by 1. Let further gC =def t. We obtain
(G, s, t) ∈ GAP
⇔ there is a path in G from s to t
⇔ 0 ∈ I(C) ⇔ (C, 0) ∈ MC(×) .
2. Now we describe a many-one reduction from MC(×) to the iterated
multiplication
decision problem IMD =def {(a1 , . . . ar , b) : a1 , . . . ar , b ∈ N ∧
r
i=1 ai = b} via a function which is logspace computable with an oracle from
NL. Since IMD ∈ L ((15), or better yet, uniform TC0 (3; 19)) we obtain
MC(×) ∈ LNL . However, LNL = NL.
238 McKenzie & Wagner
cc 16 (2007)
Let C be a (×)-circuit and b ∈ N. Let g1 , . . . gr be the input gates of C,
let gC be the output gate of C, and let n(C, g, g ) be the number of different
paths
from gate g to gate g . Obviously, n ∈ #L, and we obtain I(C) =
r in Cn(C,g
i ,gC )
. Defining s(i) =def min{n(C, gi, gC ), |b|+1} for i = 1, . . . , r
i=1 I(gi )
we obtain
b ∈ I(C) ⇔ b = ri=1 I(gi )n(C,gi ,gC ) ⇔ b = ri=1 I(gi )s(i)
⇔ I(g1 ), . . . , I(g1), . . . , I(gr ), . . . , I(gr ), b ∈ IMD .
s(1)
s(r)
The latter tuple of numbers is generated as follows: for i = 1, . . . r and
k = 1, . . . , |b| + 1 ask n(C, gi, gC ) ≥ k, which are queries to a NL-set (5).
(This is owing to the fact that only small k, i.e., k whose values are polynomially bounded in the length of the input, are considered.) If such a query
n(C, gi, gC ) ≥ k is answered in the affirmative, output I(gi), and finally output b.
9. Further results
Theorem 9.1. The problems MC(∪) and MC(∩) are NL-complete.
Proof. Upper bounds. For a ∪-circuit C and b ∈ N , it holds that b ∈ I(C)
if and only if there exists a path in C from an input gate with label b to the
output gate. This can be checked by an NL-algorithm.
For a ∩-circuit C and b ∈ N , it holds that b ∈ I(C) if and only if all input
gates from which there exists a path to the output gate have label b. This can
be checked by an co-NL-algorithm. However, co-NL = NL (21; 27).
Lower bounds. We reduce the NL-complete graph accessibility problem
GAP for directed acyclic graphs to the problems MC(∪) and MC(∩). Let
G = (V, E) be a directed acyclic graph and s, t ∈ V be the source and target
vertices, resp. Without loss of generality assume every vertex has indegree 0
or 2 and that s has indegree 0. Now convert G into a (∪)-circuit C by labelling
every vertex with indegree 2 by ∪, the source vertex s by 0 and all other vertices
with indegree 0 by 1. Let further gC =def t. We obtain
(G, s, t) ∈ GAP
⇔ there is a path in G from s to t
⇔ 0 ∈ I(C) ⇔ (C, 0) ∈ MC(∪) .
Furthermore, convert G into a (∩)-circuit C by labelling every vertex with
indegree 2 by ∩, the source vertex s by 0 and all other vertices with indegree 0
cc 16 (2007)
Circuits over sets of natural numbers
239
by 1. Let further gC =def t. We obtain
(G, s, t) ∈ GAP ⇔
⇔
there is no path in G from s to t
1 ∈ I(C) ⇔ (C, 1) ∈ MC(∩) .
Consequently, GAP ≤log
m MC(∩). However, because of NL = co-NL we have
GAP.
GAP ≤log
m
Theorem 9.2. The problems MF(∩, +,×), MF(+,×), MF(∪, ∩,− ), MF(∪, ∩),
MF(∪), MF(∩), MF(∩, +), MF(∩, ×), MF(+) and MF(×) are L-complete under AC0 -reducibility.
Proof. Hardness. Any one of the associative binary operations ∪, ∩, +
and × is able to simulate the Boolean ∨ operation once an appropriate representation for the truth values is chosen. For example, representing σ(true) = ∅
and σ(false) = {0} allows ∩ to simulate ∨ since a ∨ b is true iff σ(a) ∩ σ(b) = ∅,
and a∨b is false iff σ(a)∩σ(b) = {0}. Hence, to prove all eight hardness claims,
it suffices to prove that evaluating a Boolean formula (in our encoding, as a
possibly disconnected outdegree-one circuit) with ∨ as the only operation is
L-hard. But this is clear, and is explicitly stated under NC1 -reducibility in (8)
(where the formulas involve both ∨ and ∧, but ∧ is used only to enforce the (8)
requirement that every formula be connected). To justify the more restrictive
AC0 -reducibility, consider a deterministic logspace Turing machine M. An AC0
circuit with input w is capable of computing whether two arbitrary polynomialsize configurations of M running on w follow one another by a transition of M.
Hence an AC0 circuit can compute the relevant part of the (outdegree-one)
configuration graph of M running on w. It can also ensure acyclicity by adding
time stamps to duplicate the graph appropriately. Finally, it can view the
graph as a ∨-circuit whose output gate is the initial configuration of M, whose
final configuration becomes an input gate carrying the Boolean value 1, whose
other outdegree-zero nodes become input gates carrying the Boolean value 0,
and all of whose other nodes are assigned a second input wire carrying the
value 0. The resulting circuit outputs 1 iff M accepted w, and this circuit was
constructed in AC0 .
Upper bounds. It suffices to prove that MF(∪, ∩,− ) and MF(∩, +, ×) are
in L.
One way to solve MF(∪, ∩,− ) is to reduce in logspace the problem to that
of evaluating a Boolean (∨, ∧, ¬)-formula, then transforming in logspace (as
mentioned in (8, Proposition 3.2, p. 445)) the connected part of this Boolean
formula into infix notation, and finally evaluating the infix Boolean formula in
240 McKenzie & Wagner
cc 16 (2007)
logspace (22) (or in NC1 ⊆ L using Buss’s algorithm (12)). To see the first
step, namely reducing a (∪, ∩,− )-formula F with b ∈ N to a Boolean (∨, ∧, ¬)formula F , let F have exactly the same structure as F , with every input in
F labelled b (resp. having label = b) giving rise in F to an input labelled 1
(resp. 0), and with every ∪-gate in F (resp. ∩-gate, − -gate) giving rise to a
∨-gate in F (resp. ∧-gate, ¬-gate). Then for every gate g, (b ∈ I(g) in F ) ⇔
(I(g) = 1 in F ). Hence b ∈ I(F ) ⇔ I(F ) = 1.
The case MF(∩, +, ×) is reduced in logspace to EQ(+, ×) in Lemma 8.1.
The EQ(+, ×) instance now involves two {+, ×}-formulas. We first transform
each {+, ×}-formula into infix notation as explained in the above paragraph.
Then the formulas are transformed into logarithmic-depth formulas by applying
the techniques of (11). Now, following Ben-Or and Cleve (10), we can reduce
formula evaluation to the problem of computing a product of 3-by-3 integer
matrices. Finally, checking equality of two such matrix products can be done in
logspace by carrying out the checks modulo (a polynomial number of) distinct
small primes.
The L-hardness of the MF problems considered in this section result from
our choice of formula encoding. At such low complexity levels, a more appropriate choice is infix notation; then MF(∪, ∩,− ) becomes NC1 -complete by equivalence with the Boolean formula value problem (12), and some of the other
restrictions considered in Theorem Theorem 9.2 drop down into yet smaller
classes.
10. Conclusion
Table 10.1 summarizes the known complexity status of the membership problems for arithmetic circuits over subsets of N. Several open questions are apparent from the table, most notably the intriguing question of finding an upper
bound on the complexity of MC(∪, ∩,− , +, ×) or proving the problem undecidable.
We observe that the problems MC(×) and MC(+), complete for NL and
for C= L respectively, offer an interesting new perspective on these two classes.
If one could reduce MC(+) to MC(×), then it would follow that NL = C= L.
Finally, some cases of interest were left out of our analysis, namely those
in which complementation is the only set operation. Klaus Reinhardt (private
communication) has shown that MF(− , +, ×) is already powerful enough to
express a weak form of Goldbach’s conjecture. He also extended our techniques
to prove that MC(− , ×), MC(− , +), MF(− , ×) and MF(− , +) are PSPACEcomplete, and he observed that MC(− ) is L-complete.
cc 16 (2007)
Circuits over sets of natural numbers
241
Table 10.1: State of the art. The results on MF(∪, +), MF(∪, ∩,− , +) as well
as on MC(∪, +, ×) were already known from the literature, please refer to the
relevant sections for the appropriate credit. Lower bounds of course refer to
hardness results.
MC(O)
MC(O)
MF(O)
MF(O)
O
lower bound upper bound Th. low. bound upp. bound Th.
∪, ∩,− , +, × NEXPTIME
?
6.4 PSPACE
?
5.5
∪, ∩,
+, × NEXPTIME NEXPTIME 6.2
NP
NP
4.4
∪,
+, × PSPACE
PSPACE 2.3
NP
NP
4.4
∩,
+, ×
P
co-R
8.3
L
L
9.2
+, ×
P
P
7.1
L
L
9.2
∪, ∩,− , +
PSPACE
PSPACE 5.5 PSPACE PSPACE 2.3
∪, ∩,
+
PSPACE
PSPACE 5.5
NP
NP
4.4
∪,
+
NP
NP
4.4
NP
NP
2.3
∩,
+
C= L
C= L
8.2
L
L
9.2
+
C= L
C= L
8.2
L
L
9.2
−
∪, ∩, , ×
PSPACE
PSPACE 5.5 PSPACE PSPACE 5.5
∪, ∩,
×
PSPACE
PSPACE 5.5
NP
NP
4.4
∪,
×
NP
NP
4.4
NP
NP
4.4
∩,
×
C= L
P
8.4
L
L
9.2
×
NL
NL
8.5
L
L
9.2
∪, ∩,−
P
P
7.2
L
L
9.2
∪, ∩
P
P
7.2
L
L
9.2
∪
NL
NL
9.1
L
L
9.2
∩
NL
NL
9.1
L
L
9.2
Acknowledgements
We are grateful to Eric Allender for his help with the upper bound in
Theorem 8.5, to Heribert Vollmer, Christian Glaßer, Daniel Meister, Stephen
Travers, and Hans-Georg Breunig for very useful discussions, and to anonymous referees for valuable suggestions, including the need to correct our choice
of formula encoding and to clarify its ramifications, the need to correct a former
proof of Lemma 5.3 and the nice observation that MF(∩, +, ×) belongs to L.
Finally, we gratefully acknowledge the role played by the Dagstuhl seminar at
which some of the work reported here was performed.
242 McKenzie & Wagner
cc 16 (2007)
References
[1] M. Agrawal, E. Allender, and S. Datta, On TC0 , AC0 , and arithmetic
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Pierre McKenzie
Informatique et recherche opérationnelle
Université de Montréal
C.P. 6128, Succ. Centre-Ville
Montréal (Québec), H3C 3J7, Canada
kckenzie@iro.umontreal.ca
Klaus W. Wagner
Theoretische Informatik
Julius-Maximilians-Universität
Würzburg
Am Hubland
D-97074 Würzburg, Germany
wagner@informatik.uni-wuerzburg.
de