E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM J. Comput., 26 (1997) 93-109.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....because most researchers dealing with the complexity of the transversal hypergraph thought so far that these problems are completely unrelated to limited nondeterminism. 4. 2 Tractable cases A large number of tractable cases of TRANS HYP and TRANS ENUM are known in the literature, e.g. [7, 5, 4, 9, 11, 14, 12, 21, 35, 37], and references therein. Examples of tractable classes are instances (H, has the edge sizes bounded by a constant, or where is acyclic. Various degrees of hypergraph acyclicity have been defined in the literature [15] The most general notion of hypergraph acyclicity (applying to the ....

E. Boros, P. Hammer, T. Ibaraki, and K. Kawakami. Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle. SIAM J. Comput., 26(1):93--109, 1997.

....application areas, and given the long standing failure to settle the complexity of these problems, emphasis was put on finding tractable cases of Dual and corresponding polynomial total time cases of Dualization. In fact, several relevant tractable classes were found by various authors; see e.g. [6, 7, 8, 10, 12, 13, 31, 32, 35, 37] and references therein. Moreover, classes of formulas were identified on which Dualization is not just polynomial total time, but where the conjuncts of the dual formula can be enumerated with incremental polynomial delay, i.e. with delay polynomial in the size of the input plus the size of all ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami. Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle. SIAM J. Comput., 26:93-109, 1997.

....application areas, and given the long standing failure to settle the complexity of these problems, emphasis was put on finding tractable cases of DUAL and corresponding polynomial total time cases of DUALIZATION. In fact, several relevant tractable classes were found by various authors; see e.g. [4, 8, 9, 10, 12, 14, 15, 20, 35, 36, 39, 41] and references therein. Moreover, classes of formulas were identified on which INFSYS RR 1843 02 05 DUALIZATION is not just polynomial total time, but where the conjuncts of the dual formula can be enumerated with incremental polynomial delay, i.e. with delay polynomial in the size of the input ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing, 26 (1997) 93-109.

....Science B.V. All rights reserved. Keywords: Monotone Boolean function; Learning; Membership query; Computational complexity 1. Introduction An important open problem in machine learning is whether monotone Boolean functions are exactly learnable in polynomial time using only membership queries [2,5,4,3]. Given access to a membership query oracle, a function is considered to be identified if both its CNF and DNF representations are explicitly obtained. The complexity of this identification problem is measured in terms of both the input and output sizes. The input size n is the number of variables ....

E. Boros, P.L. Hammer, T. Ibaraki, K. Kawakami, Polynomialtime recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM J. Computing 26 (1) (1997) 93--109.

....de nition of g regularity in Section 6) as applications of the above transformation. In addition to the theory of coteries, the concepts of self duality and regularity play important roles in diverse areas such as computational learning theory (e.g. identi cation of positive Boolean functions [8, 10, 24, 25]) threshold logic [27] operations research [6, 11, 29, 31] clutters in set theory [7] minimal transversals in hypergraphs [16] and coherent systems of reliability theory [33] The results of this paper are relevant to all these problems. 2 De nitions and Basic Properties A Boolean function, ....

....A positive function f is called 2 monotonic if there exists a linear ordering on V , for which f is regular. The 2 monotonicity and related concepts have been studied in various contexts in elds such as threshold logic [6, 12, 27, 31] game theory [33] hypergraph theory [11] and learning theory [10, 24, 25]. The 2monotonicity was originally introduced in conjunction with threshold functions (e.g. 27] where a positive function f is a threshold function if there exist n nonnegative real numbers (weights) w 1 ; w 2 ; wn and a non negative real number (threshold) t such that: f(x) 1; ....

E. Boros, P.L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM J. Computing, 26 (1997) 93-109.

....DUAL(H,X ) can be solved by an NC algorithm for dim(H) 3 and by a randomized NC algorithm for dim(H) 4, 5. E#cient algorithms also exist for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs of unbounded dimension (see e.g. [5, 19, 22, 24, 26, 27, 46, 49, 50, 58, 59]) 2 Even though no incremental polynomial time algorithm for the dualization of arbitrary hypergraphs is known, the dualization problem for hypergraphs of unbounded dimension is very unlikely to be NP hard since it can be solved in incremental quasi polynomial time (see [30] or [36] ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM J. Comput., 26 (1997) 93-109.

....we can conclude that x satis es (21) and consequently, it must have been created in Step 2. 15 We mention in closing that all the results of the paper remain valid for systems of 2 monotonic inequalities in integer variables. The case of a single Boolean 2monotonic inequality is discussed in [2, 5, 6, 9, 12, 14, 15]. ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93-109.

....that run with polynomial delay, i.e. in poly( V , A ) time for a specific sequence # # B # B . see e.g. 20, 21, 33] E#cient algorithms exist also for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs (see e.g. [3, 10, 12, 13, 23, 24, 27, 28]) Even though no incremental polynomial time algorithm for the dualization of arbitrary hypergraphs is known, an incremental quasi polynomial time one exists (see [15] This algorithm solves the dualization problem in O(nm) m time, where n = and m = A B (see also [19] for more ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93-109.

.... poly( V , A ) time, where B is systematically enlarged from B = # during the generation process of A d (see e.g. 18, 20, 32] E#cient algorithms exist also for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs (see e.g. [3, 7, 9, 23, 26, 27]) Though no incremental polynomial time algorithm is known for the general case, an incremental quasi polynomial time one exists (see [11] This algorithm solves Dualization(A, B) in O(nm) m o(log m) time, where n = V and m = A B (see also [16] for more detail) We should ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93-109.

.... poly( V , A ) time, where B is systematically enlarged from B = # during the generation process of A d (see e.g. 20, 22, 34] E#cient algorithms exist also for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs (see e.g. [4, 8, 10, 25, 28, 29]) Even though no incremental polynomial time algorithm for the dualization of arbitrary hypergraphs is known, an incremental quasi polynomial time one exists (see [13] This algorithm solves the dualization problem in O(nm) m o(log m) time, where n = V and m = A B (see also ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93-109.

.... that run with polynomial delay, i.e. in poly(jV j; jAj) time for a speci c sequence ; B 1 B 2 : A d , see e.g. 21, 23, 36] Ecient algorithms exist also for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs (see e.g. [4, 10, 12, 13, 25, 26, 30, 31]) Even though no incremental polynomial time algorithm for the dualization of arbitrary hypergraphs is known, an incremental quasi polynomial time one exists (see [15] This algorithm solves the dualization problem in O(nm) m o(log m) time, where n = jV j and m = jAj jBj (see also [19] ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93-109.

.... i.e. in poly(jV j; jAj) time, where B is systematically enlarged from B = during the generation process of A d (see e.g. 18, 20, 32] Ecient algorithms exist also for the dualization of 2 monotonic, threshold, matroid, read bounded, acyclic and some other classes of hypergraphs (see e.g. [4, 7, 9, 23, 26, 27]) Even though no incremental polynomial time algorithm for the dualization of arbitrary hypergraphs is known, an incremental quasi polynomial time one exists (see [11] This 2 algorithm solves the dualization problem in O(nm) m o(log m) time, where n = jV j and m = jAj jBj (see also ....

E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, SIAM Journal on Computing 26 (1997) 93-109. { 17 {

....(with respect to a permutation oe) then jJ (f)j (n 1 Gamma log 2 jI(f)j)jI(f)j; 12) furthermore, J (f) can be generated in O(n 2 jI(f)j) time. Let us mention finally that if a regular function is given by an oracle, then its prime implicants can be generated efficiently, see e.g. [6, 7]. 7 4 Strategies and binary decision trees As we noted earlier, inspection strategies can naturally be represented as binary decision trees, i.e. as rooted trees, in which every node has two or zero successors. Nodes with zero successors are called the leaves, and are labeled by T or F ....

E. Boros, P. L. Hammer, T. Ibaraki, and K. Kawakami. Polynomial time recognition of 2-monotonic positive boolean functions given by an oracle. SIAM Journal on Computing, 26:93--109, 1997.

....(with respect to a permutation oe) then jJ (f)j (n 1 Gamma log 2 jI(f)j)jI(f)j; 12) furthermore, J (f) can be generated in O(n 2 jI(f)j) time. Let us mention finally that if a regular function is given by an oracle, then its prime implicants can be generated efficiently, see e.g. [5, 6]. 4 Strategies and binary decision trees As we noted earlier, inspection strategies can naturally be represented as binary decision trees, i.e. as rooted trees, in which every node has two or zero successors. Nodes with zero successors are called the leaves, and are labeled by T or F ....

E. Boros, P. L. Hammer, T. Ibaraki, and K. Kawakami. Polynomial time recognition of 2-monotonic positive boolean functions given by an oracle. SIAM Journal on Computing, 26:93--109, 1997.

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