- Email: [email protected]
The graph of multiplication is equivalent to counting
Information Processing North-Holland
Letters
18 March
41 (1992) 199-201
The graph of multiplication to counting Samuel
1992
is equivalent
R. Buss *
Department of Mathematics, Uniclersity of California, San Diego, La Jolla, CA 92093, USA Communicated by F. Dehne Received 22 June 1991 Revised 12 November 1991
Abstract Buss, S.R., The graph Counting reductions Keywords:
of multiplication
is equivalent
is AC”-reducible to the graph to majority and to the function Constant
depth
circuits,
to counting,
of multiplication. Hence form of multiplication.
threshold
circuits,
circuit
Chandra, Stockmeyer and Vishkin [41 shows that MULTIPLICATION and BINARY-COUNT and four other problems are equivalent with respect to constant-depth circuit reducibility (see also [5]). In this note we prove that the graph of multiplication is also equivalent to these six problems, by proving that BINARY-COUNT is AC”reducible to the graph of multiplication. The problems we are most interested in are: MULTIPLICATION: Input: Two integers X, y in binary notation Output: The product x . y in binary notation
Science
Publishers
complexity,
graph
Letters
41 (1992) 199-201.
of multiplication
is equivalent
of multiplication,
computational
BINARY-COUNT: Inpuf: x ,,..., z,, E (0, 1) Output: The binary representation of x,‘s equal to 1
under
AC”
complexity
of the number
The prior constant-depth circuit reduction of BINARY-COUNT to MULTIPLICATION is based on a method of Furst, Saxe and Sipser. For inputs xi, . . . , x,, construct the integers _i?,j with binary representations
(
Correspondence to: Professor S.R. Buss, Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA. Email: [email protected]. * Supported in part by NSF Grant DMS-8902480. 0 1992 - Elsevier
the graph
Processing
x= x,5x,&.x
MULTIPLICATION-GRAPH: Input: Three integers x, y and z in binary notation Output: Accept if and only if x . y = z.
0020-0190/92/$05.00
Information
n-&n)2
and y = (l&G. . . 161) 2 where G denotes log(n) many 0’s. In the middle of the binary representation of the product X . j will be a block of log(n) many bits containing the binary representation of the sum of the xi’s. However, the binary representation of X . J contains a lot of additional information; and it is easy to see that there is a constant depth circuit which,
B.V. All rights reserved
199
INFORMATION
Volume 41. Number -1
PROC‘ESSINCi LETTERS
given .r , , . . . x,, and given an integer Z. detcrmines if z =X y. Thus, the prior work left open the possibility that there are constant depth circuits for recognizing the graph of multiplication -this paper shows that such circuits do not exist. Part of the reason we are interested in the non-existence of constant depth circuits for the graph of multiplication is that this implies the J$prcdicates of bounded arithmetic [2] are not in AC”; this should be contrasted with the work of Mancivi on A$formulas in one free variable [7]. Recall the following theorem: Theorem
1 (Chandra.
Stockmeyer
[Jl). (a)
MULTIPLICATION COUNT. (b) BINARY-COUNT CA TION.
and
Vishkin
< ,\(.” BINARY< ,,.(I MULTIPLI-
Obviously MULTIPLICATION-GRAPH d ,,.I~MULTIPLICATION. For the converse duction we shall prove the following theorem: Theorem 2. BINARY-COUNT CATION-GRAPH.
re-
< ,,(.KMULTIPLI-
and I ~‘1()X,’ 2-‘1 . ..()A’
,~=(lO~,,
‘1)2_
where 0’ denotes k occurrences of the symbol 0. As before the product X 7 will contain a block of bits encoding the sum of the x,‘s; however. the values of k, will be chosen that the rest of the product is easily predictable and thus cannot aid in determining the sum of the x,‘s. Let z be the (binary representation of the) product 2 j. The computation of z by the usual multiplication algorithm reduces to computing the summation of the values X 2”‘) where nr, = C:__, k, for 0
Recall that TC” is the set of predicates which can be computed by a family of unbounded fanin. constant depth circuits formed from NOT gates, AND gates and MAJORITY gates. In [4] it is also shown that MAJORITY is AC”-equivalent to MULTIPLICATION, etc.
Since x, occurs in column C:‘,‘k, of S and since _i; has a “1” in precisely columns C{‘i_;‘k, for 1
g( ,j, m) = Corollary 3. The decision problem MULTIPLICATION-GRAPH is AC” equillalent to MULTIPLICATION and to BINARY-COUNT. Unbounded fanin, constant depth circuits with NOT gates, AND gates and gates for the graph qf multiplication capture exactly TC”. Proof. The proof of Theorem cation of the Furst-Saxe-Sipser let
2 involves a modifimethod. Now we
for I < rrr
J’-
I
kc+
1
c ,=,
k,,
Volume
41. Number
INFORMATION
4
PROCESSING
/‘-I c n’+i.
1=1Tl’
IX March
1992
linear time (see [6] or [8]) and because the computation of g(j, m) involves addition and multiplication of numbers of length O(log n).
i.e., that rn- I c n2+i#
LETTERS
I =,
But since C”K 1-I ‘i < n2, it must be that the two of terms; summations have the same number which is impossible. We have shown that for X and jj as above, the product z contains the number of nonzero xi’s in columns C through C + log n; the other bits of z are zero except for that for all j f m, z has bit value x, in column gtj, m). Thus BINARYCOUNT can be reduced to MULTIPLICATION-GRAPH by in parallel trying every possible number of nonzero values of x,‘s and using graph-of-multiplication gates to determine if the numbers are correct. q Let LH denote the logtime hierarchy, i.e., the set of predicates accepted by logtime, constant alternation Turing machines. LH was originally defined by Sipser and is a uniform version of AC”; see [1,3] for more information. To conclude, we observe that the constant depth circuits reducing BINARY-COUNT to the graph of multiplication are LH-uniform. To see this, it suffices to note that since g( j, m) =n’(n
+ k -j) n2-n
k’-k
j2-j
+---_
2
+
2
2
’
g( j, m> can be computed by constant depth, polynomial size, uniform circuits. This is because the multiplication function is in constant alternation,
Acknowledgment This construction is based, in part, in an idea due to P. Pudlak. The final form was constructed during conversations with E. Allender and H. Straubing.
References [l] D.A.M. Barrington. N. Immerman and H. Straubing, On uniformity within NC’. J. Cornput. System Sci. 41 (1990) 274-306. [2] S.R. Buss. Bounded Arithmetic (Bibliopolis. Napels, 1986): Revision of Ph.D. Thesis. Princeton University. 1985. [3] S.R. Buss, The Boolean formula value problem is in ALOGTIME. in: Proc. 19th Ann. ACM Symp. on Theory ofCompufing (1987) 1233131. [4] A.K. Chandra, L. Stockmeyer and U. Vishkin, Constant depth reducibility, SIAM J. Comput. 13 (I 984) 423-439. [5] M. Furst, J.B. Saxe and M. Sipser. Parity. circuits and the polynomial-time hierarchy, Math. Systems Theor?: 17 (1984) 13-27. [6] R.J. Lipton, Model theoretic aspects of computational complexity, in: Proc. 19th Annuui Symp. on Foundations of Compufrr Science (IEEE Computer Society Press, Silver Spring, MD, 1978) 193-200. (71 S.-G. Mancivi, Sharply bounded predicates do not exhaust the polynomial time computable predicates, parts I & 11, Typeset manuscript, 1990. [Xl A.J. Wilkie. Applications of complexity theory to I‘,,-definability problems in arithmetic, in: Model Theory of Algebra and Arithmetic, Lecture Notes in Mathematics 834 (Springer, Berlin, 1979) 363-369.
201
Letters
18 March
41 (1992) 199-201
The graph of multiplication to counting Samuel
1992
is equivalent
R. Buss *
Department of Mathematics, Uniclersity of California, San Diego, La Jolla, CA 92093, USA Communicated by F. Dehne Received 22 June 1991 Revised 12 November 1991
Abstract Buss, S.R., The graph Counting reductions Keywords:
of multiplication
is equivalent
is AC”-reducible to the graph to majority and to the function Constant
depth
circuits,
to counting,
of multiplication. Hence form of multiplication.
threshold
circuits,
circuit
Chandra, Stockmeyer and Vishkin [41 shows that MULTIPLICATION and BINARY-COUNT and four other problems are equivalent with respect to constant-depth circuit reducibility (see also [5]). In this note we prove that the graph of multiplication is also equivalent to these six problems, by proving that BINARY-COUNT is AC”reducible to the graph of multiplication. The problems we are most interested in are: MULTIPLICATION: Input: Two integers X, y in binary notation Output: The product x . y in binary notation
Science
Publishers
complexity,
graph
Letters
41 (1992) 199-201.
of multiplication
is equivalent
of multiplication,
computational
BINARY-COUNT: Inpuf: x ,,..., z,, E (0, 1) Output: The binary representation of x,‘s equal to 1
under
AC”
complexity
of the number
The prior constant-depth circuit reduction of BINARY-COUNT to MULTIPLICATION is based on a method of Furst, Saxe and Sipser. For inputs xi, . . . , x,, construct the integers _i?,j with binary representations
(
Correspondence to: Professor S.R. Buss, Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA. Email: [email protected]. * Supported in part by NSF Grant DMS-8902480. 0 1992 - Elsevier
the graph
Processing
x= x,5x,&.x
MULTIPLICATION-GRAPH: Input: Three integers x, y and z in binary notation Output: Accept if and only if x . y = z.
0020-0190/92/$05.00
Information
n-&n)2
and y = (l&G. . . 161) 2 where G denotes log(n) many 0’s. In the middle of the binary representation of the product X . j will be a block of log(n) many bits containing the binary representation of the sum of the xi’s. However, the binary representation of X . J contains a lot of additional information; and it is easy to see that there is a constant depth circuit which,
B.V. All rights reserved
199
INFORMATION
Volume 41. Number -1
PROC‘ESSINCi LETTERS
given .r , , . . . x,, and given an integer Z. detcrmines if z =X y. Thus, the prior work left open the possibility that there are constant depth circuits for recognizing the graph of multiplication -this paper shows that such circuits do not exist. Part of the reason we are interested in the non-existence of constant depth circuits for the graph of multiplication is that this implies the J$prcdicates of bounded arithmetic [2] are not in AC”; this should be contrasted with the work of Mancivi on A$formulas in one free variable [7]. Recall the following theorem: Theorem
1 (Chandra.
Stockmeyer
[Jl). (a)
MULTIPLICATION COUNT. (b) BINARY-COUNT CA TION.
and
Vishkin
< ,\(.” BINARY< ,,.(I MULTIPLI-
Obviously MULTIPLICATION-GRAPH d ,,.I~MULTIPLICATION. For the converse duction we shall prove the following theorem: Theorem 2. BINARY-COUNT CATION-GRAPH.
re-
< ,,(.KMULTIPLI-
and I ~‘1()X,’ 2-‘1 . ..()A’
,~=(lO~,,
‘1)2_
where 0’ denotes k occurrences of the symbol 0. As before the product X 7 will contain a block of bits encoding the sum of the x,‘s; however. the values of k, will be chosen that the rest of the product is easily predictable and thus cannot aid in determining the sum of the x,‘s. Let z be the (binary representation of the) product 2 j. The computation of z by the usual multiplication algorithm reduces to computing the summation of the values X 2”‘) where nr, = C:__, k, for 0
Recall that TC” is the set of predicates which can be computed by a family of unbounded fanin. constant depth circuits formed from NOT gates, AND gates and MAJORITY gates. In [4] it is also shown that MAJORITY is AC”-equivalent to MULTIPLICATION, etc.
Since x, occurs in column C:‘,‘k, of S and since _i; has a “1” in precisely columns C{‘i_;‘k, for 1
g( ,j, m) = Corollary 3. The decision problem MULTIPLICATION-GRAPH is AC” equillalent to MULTIPLICATION and to BINARY-COUNT. Unbounded fanin, constant depth circuits with NOT gates, AND gates and gates for the graph qf multiplication capture exactly TC”. Proof. The proof of Theorem cation of the Furst-Saxe-Sipser let
2 involves a modifimethod. Now we
for I < rrr
J’-
I
kc+
1
c ,=,
k,,
Volume
41. Number
INFORMATION
4
PROCESSING
/‘-I c n’+i.
1=1Tl’
IX March
1992
linear time (see [6] or [8]) and because the computation of g(j, m) involves addition and multiplication of numbers of length O(log n).
i.e., that rn- I c n2+i#
LETTERS
I =,
But since C”K 1-I ‘i < n2, it must be that the two of terms; summations have the same number which is impossible. We have shown that for X and jj as above, the product z contains the number of nonzero xi’s in columns C through C + log n; the other bits of z are zero except for that for all j f m, z has bit value x, in column gtj, m). Thus BINARYCOUNT can be reduced to MULTIPLICATION-GRAPH by in parallel trying every possible number of nonzero values of x,‘s and using graph-of-multiplication gates to determine if the numbers are correct. q Let LH denote the logtime hierarchy, i.e., the set of predicates accepted by logtime, constant alternation Turing machines. LH was originally defined by Sipser and is a uniform version of AC”; see [1,3] for more information. To conclude, we observe that the constant depth circuits reducing BINARY-COUNT to the graph of multiplication are LH-uniform. To see this, it suffices to note that since g( j, m) =n’(n
+ k -j) n2-n
k’-k
j2-j
+---_
2
+
2
2
’
g( j, m> can be computed by constant depth, polynomial size, uniform circuits. This is because the multiplication function is in constant alternation,
Acknowledgment This construction is based, in part, in an idea due to P. Pudlak. The final form was constructed during conversations with E. Allender and H. Straubing.
References [l] D.A.M. Barrington. N. Immerman and H. Straubing, On uniformity within NC’. J. Cornput. System Sci. 41 (1990) 274-306. [2] S.R. Buss. Bounded Arithmetic (Bibliopolis. Napels, 1986): Revision of Ph.D. Thesis. Princeton University. 1985. [3] S.R. Buss, The Boolean formula value problem is in ALOGTIME. in: Proc. 19th Ann. ACM Symp. on Theory ofCompufing (1987) 1233131. [4] A.K. Chandra, L. Stockmeyer and U. Vishkin, Constant depth reducibility, SIAM J. Comput. 13 (I 984) 423-439. [5] M. Furst, J.B. Saxe and M. Sipser. Parity. circuits and the polynomial-time hierarchy, Math. Systems Theor?: 17 (1984) 13-27. [6] R.J. Lipton, Model theoretic aspects of computational complexity, in: Proc. 19th Annuui Symp. on Foundations of Compufrr Science (IEEE Computer Society Press, Silver Spring, MD, 1978) 193-200. (71 S.-G. Mancivi, Sharply bounded predicates do not exhaust the polynomial time computable predicates, parts I & 11, Typeset manuscript, 1990. [Xl A.J. Wilkie. Applications of complexity theory to I‘,,-definability problems in arithmetic, in: Model Theory of Algebra and Arithmetic, Lecture Notes in Mathematics 834 (Springer, Berlin, 1979) 363-369.
201