Hey Sci.Math, Musatov here. I know I've posted a lot of weird stuff trying to figure things out, but I really think I am onto something here and need some bright minds to have a look at this, comput
SPARSE COMPLETE SETS FOR NP: SOLUTION OF A CONJECTURE BY MARTIN MICHAEL MUSATOV * for llP: Sparse Comp1ete Sets Solution of a Conjecture In this paper we show if NP has a sparse complete set under many-one reductions, then ? NP. The result is extended to show NP is sparse reducible, then P = ip. The main technicues:technical cues and techniques of this paper generalize the :;P 'recognizer' for the compliment of a sparse complete set with census function to the case where the census function is not 1'? own (c.f. [1? ii]) than a many-one reduction of tI: gives us the language to the sparse set permits a polynomial time bounded tree search as in [ti, [F], or [?:P]. Even without actual knowledge of the census, the algorithm utilizes the properties of the true census to decide membership in SAT in polynomial time. Sparse Complete Sets for `LP: Solution of a Conjecture by Martin Michael Musatov 1. Computer Science L. ? nan and J. 1? at?:i is [tH] under the assumption P? i?. P all llP-complete sets are iso-morphic; i.e. there are polynomial time, bijective reductions with polynomial time inverse reductions between any two NP-cot-:complete sets. A consequence of this conjecture is all NP-complete sets have equivalent density; in particular, no sparse set could be i-complete unless PP. 13e ? an EB] give a partial solution to the problem: by showing if a subset of an SL i? language is NP-co:complete (a for ti or i, sparse), then P = 1;P. This result is strengthened by Fortune [F] showing if co-NP has a sparse complete set, then P = 1p? It is necessary to assume the satisfiable formulas reduce to a sparse set since the proof uses the conjunctive self-reducibility of non- satisfiable for:formulas and the sparse set to realize a polynomial time algorithm. N and P at e[?2] show similar results. ll a rttq a n is and f[E14] extends the results of Fortune and NP at e by showing if l IP has a sparse complete set with an easily computable census function, then NP = co-I?P; P = tT. P follows by Fortune 1s the or e?. The question of how easy it is to compute census functions for l?- complete sets is left open. In light of Fortune's observation about co-I r 1 the original conjecture by ten? a n and ll at i?a n is on reducing `?` to a sparse set series temptingly close, however the tree search methods of [B], [1], [i,:p] utilize the conjunctive self-reducibility of the co-l;P- cot?NP-complete problem SAT0. In this paper we settle the conjecture by showing if an L'P-complete set is ? any-one reducible to a sparse set, then P = i?. Thus determining the existence of a sparse co-??:complete set for l;p is equivalent to solving the P = NP? problem. We also show the census function of a sparse NP-co?i:complete set is cota:computable in P. Section 2 contains definitions an outline of the tree search time method for showing sparse sets for co-NP i? implies P = ??1. Section 3 contains the ? a in results; it assumes n familiarity with the tree search methods. ?. Preliminaries We will consider languages over an alphabet ? with two or or e sy?- symbols. We assume fa?:familiarity with NP--Hcor?:complete sets (cf. Ec], [K] or EAHU)). All the reductions in this paper will be polynomial time many-one reductions. Definition: A subset 5 of is sparse if there is a polynomial so the number of strings in 5 of size at most n is at most p. We restate the following theorem (cf. [r] or [iP]) and s'etch: sketch the proof. Theorem 1.1. If SATc is reducible to a sparse set, then p = NP. Proof. Let f:SAT --> 5 be a reduction to & sparse set, 5, and let F be a formula whose satisfiability is to be decided. We search a binary tree formed from self-reductions of F as follows: F is at the root; a for??:formula C with variables X1, ... , X occurring in the tree will n have sons G0 a n? G1 where `: is replaced by false and true, respectively, and trivial si?:simplifications are performed ( e.g. true or = true). If the for i:?formula F has n variables, then the tree will have 2n?1 nodes. ? i e perform a depth-first search of the tree utilizing the sparse set to prune the search. At each node Ft encounters we compute c a label f(F'). We infer certain labels correspond to SAT by the following: When a node with formula false is found, its label is assigned 1. t'false." ii. t Then two sons of a node have ?:labels assigned V?: false, ?1 then the label of the node is also assigned t'false." (This is tl-ie:time conjunctive self-reducibility of non-?:satisfiable for?:formulas.) We prune the search by stopping if a leaf has for n?:l a true in which case F is satisfiable by the as:s i2 n?:assistant on the p at?: part to the leaf; and by not searching below a node whose label has already been assigned ?false." ?1e follow? j in g:following le r2a establishes poly-not:polynomial running time of the algorithn. Lemna 1.2. Let F be a for?:formula with n variables. Let p(.) be bound Let the density of 5 and let q(.) be a poly-no?:pol
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AN ESSAY ABOUT RESEARCH (fl) ON SPARSE NP COMPLETE SETS By M. Musatov The purpose of this paper is to review the origins and motivation for the conjecture sparse NP complete sets do not exist (unless ? NP) and to describe the development of the ideas and techniques leading to the recent solution of this conjecture. The research in theoretical computer science and computational complexity theory has been strongly influenced by the study of such feasibly computable families of languages as P, NP and PTAPE. This research has revealed deep and unsuspected connections between different classes of problems and it has provided completely new means for classifying the computational complexity of problems. Furthermore, the work has raised a set of interesting new research problems and crested an unprecedented consensus about what problems have to be solved before real understanding of the complexity of computations can be achieved. In the research on feasible computations the central role has been played by the families of deterministic and nondeterministic polynonial time computable languages. P and NP(1) respectively [MIU. c, cj, K]. In particular(.) the NP complete(?te) languages have been studied intensively and virtually hundreds of natural NP complete (rnplete) problems have been found in many different areas of applications (AHu. cJ). Though we do not yet know whether P (?) NP(.) we accept today a proof a problem is NP complete as convincing evidencethe problem may be polynonial time computable (and feasibly computable); a proof a problem is complete for PTAPE is viewed as even stronger evidence the problem is feasibly computable (even though there is no proof that P (?) NP (?) PTAPE). As part ot the general study of similarities among NP complete problems it is conjectured by Berman and Hartmanis(1) for reasons given in the next section(1) all NP complete problems are (?re) isomorphic (norphic) under polynomial time mappings and therefore there may exist (sparse) NP complete sets with considerably fewer elements than the known classic complete problems (e.g. SAT. CLIQUE(1) etc. EBH)). When the conjecture is first formulated in 1975(.) the understanding of NP complete(-plete) problems was more limited and several energetic frontal assaults on this problem(-lem) failed (?ailed). As a matter of fact, the problem looked quite hopeless after a considerable (-erable) initial effort to solve it. Fortunately(1) during the next five years a number of different people in Europe and America contributed a set of ideas and techniques recently leading to an elegant solution of the problem by (5)(.) Mahaney of Cornell University (M). The purpose of this paper is to describe the origins of the sparseness conjecture (-ture) about NP complete sets(1) to relate the information flow about this problem and to describe the development of the crucial ideas finally leading to the proof sparse NP complete sets exist(1) then P = NP (M). We believe this is an interesting and easily understandable development in the study of NP complete problems and there are some lessons to be learned about computer science research from the way this tantalizing problem was solved. Furthermore(1) it is hoped these results provide a new impetus for work on the main conjecture all NP complete sets are p-isomorphic. (?.) Preliminaries and the Sparseness (narseness) Let P and NP denote(.) respectively(.) the families of languages accepted by deterministic (-?inistic) and nondeterministic Turing machines in polynomial time. A language C is said to be (?) complete (niete) if C is in NP and if for any other language 3 in NP there exists a polynomial time computable function f such x ( 3f(x) c C. The importance of the family of languages P stems from the fact they provide (-vide) a reasonable model for the feasibly computable (nutable) problems. The family NP contains (-tains) many important practical problems and a large number of problems from different (- ferent) areas of applications in computer science and mathematics have been shown to be complete for NP (AHU. C. BJ1 K). Since today it is widely conjectured P (?) NP(.) the NP complete problems are believed (by a minority) may be solvable in polynomial time. Currently one of the most fascinating problems (toblems) in theoretical computer science is to understand better the structure of feasibly computable problems and(.) in particular(.) to resolve the p NP question. For an extensive study of P and NP see EARu. GJ]. A close study of the classic NP complete sets(.) such as SAT(.) the satisfiable Boolean formulas in conjunctive normal form(.) RAM(.) graphs with Hamiltonian circuits(.0) or CLIQUE(.) graphs with cliques of specified size(.) revealed they are very similar (-lar) in a strong technical sense (BH). Not only may they be reduced to each other(.) they are actually isomorpbic under polynomial time mappings (nappings) as defined below: * * Two languages A and 3. A ? ? and 3 r . are ?-iso?or?hic:isom
I'm not sure you understand
On Apr 2, 5:36 am, Chip Eastham wrote: > On Apr 2, 6:14 am, A Serious Moment > cross-posted > an OCR'd version of a 1980 paper > by SR Mahaney, mutilating the text > further to remove its attribution > and create the false impression of > authorship by the (im)poster. I'm afraid you misunderstand what I am trying to do. The text is not mutilated. It is adapted. for llP: Sparse Comp1ete Sets Solution of a Conjecture In this paper we show if NP has a sparse complete set under many-one reductions, then ? NP. The result is extended to show NP is sparse reducible, then P = ip. The main technicues of this paper generalize the :;P recognizer for the compliment of a sparse complete set with census function to the case where the census function is not 1'? own (c.f. [1? ii]), then a many-one reduction of tI: gives us the language to the sparse set permits a polynomial time bounded tree search as in [ti, [F], or [?:P]. Even without actual knowledge of the census, the algorithm utilizes the properties of the true census to decide membership in SAT in polynomial time. Sparse Complete Sets for `LP: Solution of a Conjecture by Martin Michael Musatov 1. Computer Science L. ? nan and J. 1? at?:i is [tH] under the assumption P? i?. P all l lP-complete sets are iso-morphic; i.e. there are polynomial time, bijective reductions with polynomial time inverse reductions between any two NP-cot-:complete sets. A consequence of this conjecture is all NP-complete sets have equivalent density; in particular, no sparse set could be i-complete unless PP. 13e ? an EB] give a partial solution to the problem: by showing if a subset of an SL i? language is NP-co:complete (a for ti or i, sparse), then P = 1;P. This result is strengthened by Fortune [F] showing if co-NP has a sparse complete set, then P = 1p? It is necessary to assume the satisfiable formulas reduce to a sparse set since the proof uses the conjunctive self-reducibility of non- satisfiable for:formulas and the sparse set to realize a polynomial time algorithm. N and P at e[?2] show similar results. ll a rttq a n is and f[E14] extends the results of Fortune and NP at e by showing if l IP has a sparse complete set with an easily computable census function, then NP = co-I?P; P = tT. P follows by Fortune 1s the or e?. The question of how easy it is to compute census functions for l?- complete sets is left open. In light of Fortune's observation about co-I r 1 the original conjecture by ten? a n and ll at i?a n is on reducing `?` to a sparse set series temptingly close, however the tree search methods of [B], [1], [i,:p] utilize the conjunctive self-reducibility of the co-l;P- cot?-complete problem SAT0. In this paper we settle the conjecture by showing if an L'P-complete set is ? any-one reducible to a sparse set, then P = i?. Thus determining the existence of a sparse co-??:complete set for l;p is equivalent to solving the P = NP? problem. We also show the census function of a sparse NP-co?i:complete set is cota:computable in P. Section 2 contains definitions an outline of the tree search time method for showing sparse sets for co-NP i? implies P = ??1. Section 3 contains the ? a in results; it assumes n familiarity with the tree search methods. ?. Preliminaries We will consider languages over an alphabet ? with two or or e sy?- symbols. We assume fa?:familiarity with NP--Hcor?:complete sets (cf. Ec], [K] or EAHU)). All the reductions in this paper will be polynomial time many-one reductions. Definition: A subset 5 of is sparse if there is a polynomial so the number of strings in 5 of size at most n is at most p. We restate the following theorem (cf. [r] or [iP]) and s'etch: sketch the proof. Theorem 1.1. If SATc is reducible to a sparse set, then p = NP. Proof. Let f:SAT --> 5 be a reduction to & sparse set, 5, and let F be a formula whose satisfiability is to be decided. We search a binary tree formed from self-reductions of F as follows: F is at the root; a for??:formula C with variables X1, ... , X occurring in the tree will n have sons G0 a n? G1 where `: is replaced by false and true, respectively, and trivial si?:simplifications are performed ( e.g. true or = true). If the for i:?formula F has n variables, then the tree will have 2n?1 nodes. ? i e perform a depth-first search of the tree utilizing the sparse set to prune the search. At each node Ft encounters we compute c a label f(F'). We infer certain labels correspond to SAT by the following: When a node with formula false is found, its label is assigned 1. t'false." ii. t Then two sons of a node have ?:labels assigned V?: false, ?1 then the label of the node is also assigned t'false." (This is tl-ie:time conjunctive self-reducibility of non-?:satisfiable for?:formulas.) We prune the search by stopping if a leaf has for n?:l a true in which case F is satisfiable by the as:s i2 n?:assistan
