We show that Inner Product in 2n variables, IP_n(x, y) = x₁y₁ ⊕ … ⊕ x_ny_n, can be computed by depth-3 bottom fan-in 2 circuits of size poly(n)⋅ (9/5)ⁿ, matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps. 1) We provide a general template for constructing optimal depth-3 circuits with bottom fan-in k for an arbitrary function f. We do this in two steps. First, we partition f^{-1}(1) into orbits of its automorphism group. Second, for each orbit, we construct one k-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by f. 2) We instantiate the template for IP_n and k = 2. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller 2-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. 3) We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.

Inductive program synthesis, from input/output examples, can provide an opportunity to automatically create programs from scratch without presupposing the algorithmic form of the solution. For induction of general programs with loops (as opposed to loop-free programs, or synthesis for domain-specific languages), the state of the art is at the level of introductory programming assignments. Most problems that require algorithmic subtlety, such as fast sorting, have remained out of reach without the benefit of significant problem-specific background knowledge. A key challenge is to identify cues that are available to guide search towards correct looping programs. We present MAKESPEARE, a simple delayed-acceptance hillclimbing method that synthesizes low-level looping programs from input/output examples. During search, delayed acceptance bypasses small gains to identify significantly-improved stepping stone programs that tend to generalize and enable further progress. The method performs well on a set of established benchmarks, and succeeds on the previously unsolved “Collatz Numbers” program synthesis problem. Additional benchmarks include the problem of rapidly sorting integer arrays, in which we observe the emergence of comb sort (a Shell sort variant that is empirically fast). MAKESPEARE has also synthesized a record-setting program on one of the puzzles from the TIS100 assembly language programming game.

Article Sample complexity of model-based search Share on Author: Christopher D. Rosin The Scripps Research Institute and University of California, San Diego, CSE Dept., and The Scripps Research Institute, Mail Drop MB5, 10550 North Torrey Pines Road, La Jolla, CA The Scripps Research Institute and University of California, San Diego, CSE Dept., and The Scripps Research Institute, Mail Drop MB5, 10550 North Torrey Pines Road, La Jolla, CAView Profile Authors Info & Claims COLT' 98: Proceedings of the eleventh annual conference on Computational learning theoryJuly 1998 Pages 259–267https://doi.org/10.1145/279943.279994Online:24 July 1998Publication History 0citation220DownloadsMetricsTotal Citations0Total Downloads220Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

An understanding of antiviral drug resistance is important in the design of effective drugs. Comprehensive features of the interaction between drug designs and resistance mutations are difficult to study experimentally because of the very large numbers of drugs and mutants involved. We describe a computational framework for studying antiviral drug resistance. Data on HIV-1 protease are used to derive an approximate model that predicts interaction of a wide range of mutant forms of the protease with a broad class of protease inhibitors. An algorithm based on competitive coevolution is used to find highly resistant mutant forms of the protease, and effective inhibitors against such mutants, in the context of the model. We use this method to characterize general features of inhibitors that are effective in overcoming resistance, and to study related issues of selection pathways, cross-resistance, and combination therapies.

We consider "competitive coevolution," in which fitness is based on direct competition among individuals selected from two independently evolving populations of "hosts" and "parasites." Competitive coevolution can lead to an "arms race," in which the two populations reciprocally drive one another to increasing levels of performance and complexity. We use the games of Nim and 3-D Tic-Tac-Toe as test problems to explore three new techniques in competitive coevolution. "Competitive fitness sharing" changes the way fitness is measured; "shared sampling" provides a method for selecting a strong, diverse set of parasites; and the "hall of fame" encourages arms races by saving good individuals from prior generations. We provide several different motivations for these methods and mathematical insights into their use. Experimental comparisons are done, and a detailed analysis of these experiments is presented in terms of testing issues, diversity, extinction, arms race progress measurements, and drift.

Machine learning of game strategies has often depended on competitive methods that continually develop new strategies capable of defeating previous ones. We use a very inclusive definition of game and consider a framework within which a competitive algorithm makes repeated use of a strategy learning component that can learn strategies which defeat a given set of opponents. We describe game learning in terms of sets H and X of first and second player strategies, and connect the model with more familiar models of concept learning. We show the importance of the ideas of teaching set [20] and specification number [19] k in this new context. The performance of several competitive algorithms is investigated, using both worst-case and randomized strategy learning algorithms. Our central result (Theorem 4) is a competitive algorithm that solves games in a total number of strategies polynomial in lg(jHj), lg(jX j), and k. Its use is demonstrated, including an application in concept learning ...