Does truth-table of linear norm reduce the one-query tautologies to a random oracle?
In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle A , with probability one, the degree of the one-query tautologies with respect to A is strictly higher than the degree of A .” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998;...
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description | In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle
A
, with probability one, the degree of the one-query tautologies with respect to
A
is strictly higher than the degree of
A
.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let
c
be a positive real number. We consider a concept of truth-table reduction whose norm is at most
c
times size of input, where for a relativized propositional formula
F
, the size of
F
denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that
c
is small enough. How small
c
can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all
c |
doi_str_mv | 10.1007/s00153-008-0076-4 |
format | Article |
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A
, with probability one, the degree of the one-query tautologies with respect to
A
is strictly higher than the degree of
A
.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let
c
be a positive real number. We consider a concept of truth-table reduction whose norm is at most
c
times size of input, where for a relativized propositional formula
F
, the size of
F
denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that
c
is small enough. How small
c
can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all
c
< 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65–76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes.</description><identifier>ISSN: 0933-5846</identifier><identifier>EISSN: 1432-0665</identifier><identifier>DOI: 10.1007/s00153-008-0076-4</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer-Verlag</publisher><subject>Algebra ; Mathematical Logic and Foundations ; Mathematics ; Mathematics and Statistics</subject><ispartof>Archive for mathematical logic, 2008-07, Vol.47 (2), p.159-180</ispartof><rights>Springer-Verlag 2008</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c381t-247f96488c48ee3b4e91dcf14538473b30b5b18a69b91432479d35d650adcd6d3</citedby><cites>FETCH-LOGICAL-c381t-247f96488c48ee3b4e91dcf14538473b30b5b18a69b91432479d35d650adcd6d3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00153-008-0076-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00153-008-0076-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>315,781,785,27926,27927,41490,42559,51321</link.rule.ids></links><search><creatorcontrib>Kumabe, Masahiro</creatorcontrib><creatorcontrib>Suzuki, Toshio</creatorcontrib><creatorcontrib>Yamazaki, Takeshi</creatorcontrib><title>Does truth-table of linear norm reduce the one-query tautologies to a random oracle?</title><title>Archive for mathematical logic</title><addtitle>Arch. Math. Logic</addtitle><description>In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle
A
, with probability one, the degree of the one-query tautologies with respect to
A
is strictly higher than the degree of
A
.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let
c
be a positive real number. We consider a concept of truth-table reduction whose norm is at most
c
times size of input, where for a relativized propositional formula
F
, the size of
F
denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that
c
is small enough. How small
c
can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all
c
< 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65–76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes.</description><subject>Algebra</subject><subject>Mathematical Logic and Foundations</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0933-5846</issn><issn>1432-0665</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2008</creationdate><recordtype>article</recordtype><recordid>eNp1kD1PwzAQhi0EEqXwA9gsdoMdf8SeECqfUiWWMluO7bSp0rjYztB_j6MwsDCcbrjnfe_uBeCW4HuCcf2QMCacIoxlqVogdgYWhNEKYSH4OVhgRSnikolLcJXSvtCVlGQBNs_BJ5jjmHcom6b3MLSw7wZvIhxCPMDo3Wg9zLsyGTz6Hn08wWzGHPqw7SZtgAZGM7hwgCEa2_vHa3DRmj75m9--BF-vL5vVO1p_vn2sntbIUkkyqljdKsGktEx6TxvmFXG2JYxTyWraUNzwhkgjVKOmX1itHOVOcGycdcLRJbibfY8xlMNS1vswxqGs1EQxqTiuVIHIDNkYUoq-1cfYHUw8aYL1lJ2es9MlOz1lp1nRVLMmFXbY-vjH-F_RD7hicKM</recordid><startdate>20080701</startdate><enddate>20080701</enddate><creator>Kumabe, Masahiro</creator><creator>Suzuki, Toshio</creator><creator>Yamazaki, Takeshi</creator><general>Springer-Verlag</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20080701</creationdate><title>Does truth-table of linear norm reduce the one-query tautologies to a random oracle?</title><author>Kumabe, Masahiro ; Suzuki, Toshio ; Yamazaki, Takeshi</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c381t-247f96488c48ee3b4e91dcf14538473b30b5b18a69b91432479d35d650adcd6d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2008</creationdate><topic>Algebra</topic><topic>Mathematical Logic and Foundations</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kumabe, Masahiro</creatorcontrib><creatorcontrib>Suzuki, Toshio</creatorcontrib><creatorcontrib>Yamazaki, Takeshi</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Archive for mathematical logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kumabe, Masahiro</au><au>Suzuki, Toshio</au><au>Yamazaki, Takeshi</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Does truth-table of linear norm reduce the one-query tautologies to a random oracle?</atitle><jtitle>Archive for mathematical logic</jtitle><stitle>Arch. Math. Logic</stitle><date>2008-07-01</date><risdate>2008</risdate><volume>47</volume><issue>2</issue><spage>159</spage><epage>180</epage><pages>159-180</pages><issn>0933-5846</issn><eissn>1432-0665</eissn><abstract>In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle
A
, with probability one, the degree of the one-query tautologies with respect to
A
is strictly higher than the degree of
A
.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let
c
be a positive real number. We consider a concept of truth-table reduction whose norm is at most
c
times size of input, where for a relativized propositional formula
F
, the size of
F
denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that
c
is small enough. How small
c
can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all
c
< 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65–76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer-Verlag</pub><doi>10.1007/s00153-008-0076-4</doi><tpages>22</tpages></addata></record> |
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title | Does truth-table of linear norm reduce the one-query tautologies to a random oracle? |
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