We study the learnability of monotone term decision lists in the exact model of equivalence and m... more We study the learnability of monotone term decision lists in the exact model of equivalence and membership queries. We show that, for any constant k _> 0, k-term monotone decision lists are exactly and properly learnable with n ~ membership queries in O(n k3) time. We aJso show n n(k) membership queries are necessary for exact learning. In contrast, both k-term monotone decision lists (k > 2) and general monotone decision lists are not learnable with equivalence queries alone.
Decision trees are popular representations of Boolean functions. We show that, given an alternati... more Decision trees are popular representations of Boolean functions. We show that, given an alternative representation of a Boolean function , say as a read-once branching program, one can find a decision tree Ì which approximates to any desired amount of accuracy. Moreover, the size of the decision tree is at most that of the smallest decision tree which can represent and this construction can be obtained in quasi-polynomial time. We also extend this result to the case where one has access only to a source of random evaluations of the Boolean function instead of a complete representation. In this case, we show that a similar approximation can be obtained with any specified amount of confidence (as opposed to the absolute certainty of the former case.) This latter result implies proper PAC-learnability of decision trees under the uniform distribution without using membership queries.
We investigate the query complexity of exact learning in the membership and (proper) equivalence ... more We investigate the query complexity of exact learning in the membership and (proper) equivalence query model. We give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model. We give applications of this characterization, including results on learning a natural subclass of DNF formulas, and on learning with membership queries alone. Query complexity has previously been used to prove lower bounds on the time complexity of exact learning. We show a new relationship between query complexity and time complexity in exact learning: If any “honest” class is exactly and properly learnable with polynomial query complexity, but not learnable in polynomial time, then P = NP. In particular, we show that an honest class is exactly polynomial-query learnable if and only if it is learnable using an oracle for Γ p 4 .
Aslam and Rivest considered the problem of inferring the smallest edge-colored graph of degree bo... more Aslam and Rivest considered the problem of inferring the smallest edge-colored graph of degree bound k consistent with the sequence of colors seen in a walk of the graph. Using Church-Rosser properties of certain sets of rewrite rules, they gave a polynomial time algorithm for the case of k = 2. The straightforward implementation of their ideas results in an O(n 5) algorithm, where n is the length of the walk. In this paper, we develop their ideas further and give an O(n log n) algorithm for the same problem. We also show that if the degree bound k is greater than two, then the decision version of the problem is NP-complete, thus settling a conjecture of Aslam and Rivest.
We prove the following results. Any Boolean function of O(log n) relevant variables can be exactl... more We prove the following results. Any Boolean function of O(log n) relevant variables can be exactly learned with a set of nonadaptive membership queries alone and a minimum sized decision tree representation of the function constructed, in polynomial time. In contrast, such a function cannot be exactly learned with equivalence queries alone using general decision trees and other representation classes as hypotheses. Our results imply others which may be of independent interest. We show that truth-table minimization of decision trees can be done in polynomial time, complementing the well-known result of Masek that truth-table minimization of DNF formulas is NP-hard. The proofs of our negative results show that general decision trees and related representations are not learnable in polynomial time using equivalence queries alone, confirming a folklore theorem.
We prove that decision trees exhibit the \approximate ngerprint" property, and therefore are not ... more We prove that decision trees exhibit the \approximate ngerprint" property, and therefore are not polynomially learnable using only equivalence queries. A slight modi cation of the proof extends this result to several other representation classes of boolean concepts which h a ve b e e n studied in computational learning theory.
We introduce a new representation class of Boolean functions -monotone term decision listswhich c... more We introduce a new representation class of Boolean functions -monotone term decision listswhich combines compact representation size with tractability of essential operations. We present many properties of the class which make it an attractive alternative to traditional universal representation classes such as DNF formulas or decision trees. We study the learnability of monotone term decision lists in the exact model of equivalence and membership queries. We show that, for any constant k¿0, k-term monotone decision lists are exactly and properly learnable with n O(k) membership queries in n O(k 3 ) time. We also show that n (k) membership queries are necessary for exact learning. In contrast, both k-term monotone decision lists (k¿2) and general monotone term decision lists are not learnable with equivalence queries alone. We also show that a subclass of monotone term decision lists (disj-MDL) is learnable with equivalence and membership queries, while neither type of query alone su ces.
We consider here scalar aggregation queries in databases that may violate a given set of function... more We consider here scalar aggregation queries in databases that may violate a given set of functional dependencies. We deÿne consistent answers to such queries to be greatest-lowest/leastupper bounds on the value of the scalar function across all (minimal) repairs of the database. We show how to compute such answers. We provide a complete characterization of the computational complexity of this problem. We also show how tractability can be improved in several special cases (one involves a novel application of Boyce-Codd Normal Form) and present a practical hybrid query evaluation method.
Bshouty, Goldman, Hancock and Matar have shown that up to %/logn term DNF formulas can be properl... more Bshouty, Goldman, Hancock and Matar have shown that up to %/logn term DNF formulas can be properly learned in the exact model with equivalence and membership queries. Given standard complexitytheoretical assumptions, we show that this positive result for proper learning cannot be significantly improved in the exact model or the PAC model extended to allow membership queries. Our negative results are derived from two general techniques for proving such results in the exact model and the extended PAC model. As a further application of these techniques, we consider read-thrice DNF formulas. Here we improve on Aizenstein, Hellerstein, and Pitt's negative result for proper learning in the exact model in two ways. First, we show that their assumption of NP ~ co-NP can be replaced with the weaker assumption of P ~ NP. Second, we show that read-thrice DNF formulas are not properly learnable in the extended PAC model, assuming RP 5& NP.
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Papers by Vijay Raghavan