The Complexity of Deciding if a Boolean Function Can Be Computed by Circuits over a Restricted Basis
We study the complexity of the following algorithmic problem: Given a Boolean function f and a finite set of Boolean functions B , decide if there is a circuit with basis B that computes f . We show that if both f and all functions in B are given by their truth-table, the problem is in quasipolynom...
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| Vydáno v: | Theory of computing systems Ročník 44; číslo 1; s. 82 - 90 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer-Verlag
01.01.2009
Springer Nature B.V |
| Témata: | |
| ISSN: | 1432-4350, 1433-0490 |
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| Abstract | We study the complexity of the following algorithmic problem: Given a Boolean function
f
and a finite set of Boolean functions
B
, decide if there is a circuit with basis
B
that computes
f
. We show that if both
f
and all functions in
B
are given by their truth-table, the problem is in quasipolynomial-size AC
0
, and thus cannot be hard for AC
0
(2) or any superclass like NC
1
, L, or NL. This answers an open question by Bergman and Slutzki (SIAM J. Comput.,
2000
). Furthermore we show that, if the input functions are not given by their truth-table but in a succinct way, i.e., by circuits (over any complete basis), the above problem becomes complete for the class coNP. |
|---|---|
| AbstractList | We study the complexity of the following algorithmic problem: Given a Boolean function f and a finite set of Boolean functions B, decide if there is a circuit with basis B that computes f. We show that if both f and all functions in B are given by their truth-table, the problem is in quasipolynomial-size AC0, and thus cannot be hard for AC0(2) or any superclass like NC1, L, or NL. This answers an open question by Bergman and Slutzki (SIAM J. Comput., 2000). Furthermore we show that, if the input functions are not given by their truth-table but in a succinct way, i.e., by circuits (over any complete basis), the above problem becomes complete for the class coNP. [PUBLICATION ABSTRACT] We study the complexity of the following algorithmic problem: Given a Boolean function f and a finite set of Boolean functions B , decide if there is a circuit with basis B that computes f . We show that if both f and all functions in B are given by their truth-table, the problem is in quasipolynomial-size AC 0 , and thus cannot be hard for AC 0 (2) or any superclass like NC 1 , L, or NL. This answers an open question by Bergman and Slutzki (SIAM J. Comput., 2000 ). Furthermore we show that, if the input functions are not given by their truth-table but in a succinct way, i.e., by circuits (over any complete basis), the above problem becomes complete for the class coNP. |
| Author | Vollmer, Heribert |
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| DOI | 10.1007/s00224-007-9030-9 |
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| Keywords | Post’s lattice Membership problem Clones Boolean circuit Computational complexity |
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| References | Immerman (CR11) 1999 Cohn (CR8) 1965 CR3 Böhler, Creignou, Reith, Vollmer (CR4) 2003; 34 Håstad (CR9) 1988 CR17 Vollmer (CR19) 1998; 207 Post (CR16) 1941; 5 Wegener (CR21) 1987 Böhler, Creignou, Reith, Vollmer (CR5) 2004; 35 Böhler, Reith, Schnoor, Vollmer (CR6) 2005; 96 CR22 CR10 Post (CR14) 1920; 26 Pippenger (CR12) 1997 Vollmer (CR20) 1999 Bergmann, Juedes, Slutzki (CR2) 1999; 9 Post (CR15) 1921; 43 Veith (CR18) 1998; 142 Pöschel, Kalužnin (CR13) 1979 Bergman, Slutzki (CR1) 2000; 30 Caussinus, McKenzie, Thérien, Vollmer (CR7) 1998; 57 C. Bergmann (9030_CR2) 1999; 9 E. Böhler (9030_CR6) 2005; 96 E. Böhler (9030_CR4) 2003; 34 H. Vollmer (9030_CR20) 1999 J. Håstad (9030_CR9) 1988 H. Vollmer (9030_CR19) 1998; 207 H. Caussinus (9030_CR7) 1998; 57 9030_CR10 E.L. Post (9030_CR15) 1921; 43 N. Pippenger (9030_CR12) 1997 H. Veith (9030_CR18) 1998; 142 9030_CR22 E. Böhler (9030_CR5) 2004; 35 N. Immerman (9030_CR11) 1999 P.M. Cohn (9030_CR8) 1965 9030_CR17 I. Wegener (9030_CR21) 1987 E.L. Post (9030_CR16) 1941; 5 E.L. Post (9030_CR14) 1920; 26 R. Pöschel (9030_CR13) 1979 9030_CR3 C. Bergman (9030_CR1) 2000; 30 |
| References_xml | – volume: 35 start-page: 22 issue: 1 year: 2004 end-page: 35 ident: CR5 article-title: Playing with Boolean blocks, part II: constraint satisfaction problems publication-title: ACM-SIGACT Newsl. doi: 10.1145/970831.970840 – volume: 207 start-page: 159 year: 1998 end-page: 170 ident: CR19 article-title: Relating polynomial time to constant depth publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(98)00061-9 – ident: CR22 – volume: 5 start-page: 1 year: 1941 end-page: 122 ident: CR16 article-title: The two-valued iterative systems of mathematical logic publication-title: Ann. Math. Stud. – volume: 26 start-page: 437 year: 1920 ident: CR14 article-title: Determination of all closed systems of truth tables publication-title: Bull. Am. Math. Soc. – year: 1979 ident: CR13 publication-title: Funktionen- und Relationenalgebren – ident: CR3 – year: 1965 ident: CR8 publication-title: Universal Algebra – volume: 57 start-page: 200 year: 1998 end-page: 212 ident: CR7 article-title: Nondeterministic NC computation publication-title: J. Comput. Syst. Sci. doi: 10.1006/jcss.1998.1588 – year: 1999 ident: CR11 publication-title: Descriptive Complexity – ident: CR17 – volume: 43 start-page: 163 year: 1921 end-page: 185 ident: CR15 article-title: Introduction to a general theory of elementary propositions publication-title: Am. J. Math. doi: 10.2307/2370324 – ident: CR10 – year: 1988 ident: CR9 publication-title: Computational Limitations of Small Depth Circuits – volume: 9 start-page: 113 year: 1999 end-page: 128 ident: CR2 article-title: Computational complexity of term equivalence publication-title: Int. J. Algebra Comput. doi: 10.1142/S0218196799000084 – year: 1997 ident: CR12 publication-title: Theories of Computability – volume: 142 start-page: 207 year: 1998 end-page: 236 ident: CR18 article-title: Succinct representation, leaf languages, and projection reductions publication-title: Inf. Comput. doi: 10.1006/inco.1997.2696 – volume: 96 start-page: 59 year: 2005 end-page: 66 ident: CR6 article-title: Bases for Boolean co-clones publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2005.06.003 – year: 1987 ident: CR21 publication-title: The Complexity of Boolean Functions – volume: 30 start-page: 359 issue: 2 year: 2000 end-page: 382 ident: CR1 article-title: Complexity of some problems concerning varieties and quasi-varieties of algebras publication-title: SIAM J. Comput. doi: 10.1137/S0097539798345944 – volume: 34 start-page: 38 issue: 4 year: 2003 end-page: 52 ident: CR4 article-title: Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory publication-title: ACM-SIGACT Newsl. doi: 10.1145/954092.954101 – year: 1999 ident: CR20 publication-title: Introduction to Circuit Complexity—A Uniform Approach – ident: 9030_CR17 – volume-title: Theories of Computability year: 1997 ident: 9030_CR12 – volume: 57 start-page: 200 year: 1998 ident: 9030_CR7 publication-title: J. Comput. Syst. Sci. doi: 10.1006/jcss.1998.1588 – ident: 9030_CR10 doi: 10.1109/SCT.1993.336526 – volume-title: The Complexity of Boolean Functions year: 1987 ident: 9030_CR21 – volume: 142 start-page: 207 year: 1998 ident: 9030_CR18 publication-title: Inf. Comput. doi: 10.1006/inco.1997.2696 – ident: 9030_CR22 doi: 10.1109/SFCS.1985.49 – volume-title: Introduction to Circuit Complexity—A Uniform Approach year: 1999 ident: 9030_CR20 doi: 10.1007/978-3-662-03927-4 – volume-title: Descriptive Complexity year: 1999 ident: 9030_CR11 doi: 10.1007/978-1-4612-0539-5 – volume-title: Funktionen- und Relationenalgebren year: 1979 ident: 9030_CR13 doi: 10.1007/978-3-0348-5547-1 – volume: 43 start-page: 163 year: 1921 ident: 9030_CR15 publication-title: Am. J. Math. doi: 10.2307/2370324 – volume: 5 start-page: 1 year: 1941 ident: 9030_CR16 publication-title: Ann. Math. Stud. – volume: 30 start-page: 359 issue: 2 year: 2000 ident: 9030_CR1 publication-title: SIAM J. Comput. doi: 10.1137/S0097539798345944 – volume-title: Computational Limitations of Small Depth Circuits year: 1988 ident: 9030_CR9 – volume: 9 start-page: 113 year: 1999 ident: 9030_CR2 publication-title: Int. J. Algebra Comput. doi: 10.1142/S0218196799000084 – volume: 35 start-page: 22 issue: 1 year: 2004 ident: 9030_CR5 publication-title: ACM-SIGACT Newsl. doi: 10.1145/970831.970840 – volume-title: Universal Algebra year: 1965 ident: 9030_CR8 – volume: 26 start-page: 437 year: 1920 ident: 9030_CR14 publication-title: Bull. Am. Math. Soc. – volume: 207 start-page: 159 year: 1998 ident: 9030_CR19 publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(98)00061-9 – ident: 9030_CR3 doi: 10.1007/s00224-006-1301-3 – volume: 96 start-page: 59 year: 2005 ident: 9030_CR6 publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2005.06.003 – volume: 34 start-page: 38 issue: 4 year: 2003 ident: 9030_CR4 publication-title: ACM-SIGACT Newsl. doi: 10.1145/954092.954101 |
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| Snippet | We study the complexity of the following algorithmic problem: Given a Boolean function
f
and a finite set of Boolean functions
B
, decide if there is a circuit... We study the complexity of the following algorithmic problem: Given a Boolean function f and a finite set of Boolean functions B, decide if there is a circuit... |
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| Title | The Complexity of Deciding if a Boolean Function Can Be Computed by Circuits over a Restricted Basis |
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