Boolean function

Abstract: Dynamical systems theory and complexity science provide powerful tools for analysing artificial agents and robots. Furthermore, they have been recently proposed also as a source of design principles and guidelines. Boolean networks are a prominent example of complex dynamical systems and they have been shown to effectively capture important phenomena in gene regulation. From an engineering perspective, these models are very compelling, because they can exhibit rich and complex behaviours, in spite of the compactness of their description. In this paper, we propose the use of Boolean networks for controlling robots' behaviour. The network is designed by means of an automatic procedure based on stochastic local search techniques. We show that this approach makes it possible to design a network which enables the robot to accomplish a task that requires the capability of navigating the space using a light stimulus, as well as the formation and use of an internal memory.

Abstract: The common feature of nearly all logic and memory devices is that they make use of stable units to represent 0’s and 1’s. A completely different paradigm is based on three-terminal stochastic units which could be called “p-bits”, where the output is a random telegraphic signal continuously fluctuating between 0 and 1 with a tunable mean. p-bits can be interconnected to receive weighted contributions from others in a network, and these weighted contributions can be chosen to not only solve problems of optimization and inference but also to implement precise Boolean functions in an inverted mode. This inverted operation of Boolean gates is particularly striking: They provide inputs consistent to a given output along with unique outputs to a given set of inputs. The existing demonstrations of accurate invertible logic are intriguing, but will these striking properties observed in computer simulations carry over to hardware implementations? This paper uses individual micro controllers to emulate p-bits, and we present results for a 4-bit ripple carry adder with 48 p-bits and a 4-bit multiplier with 46 p-bits working in inverted mode as a factorizer. Our results constitute a first step towards implementing p-bits with nano devices, like stochastic Magnetic Tunnel Junctions.

Abstract: We present a two-level Boolean minimization tool (BOOM) based on a new implicant generation paradigm. In contrast to all previous minimization methods, where the implicants are generated bottom-up, the proposed method uses a top-down approach. Thus instead of increasing the dimensionality of implicants by omitting literals from their terms, the dimension of a term is gradually decreased by adding new literals. Unlike most other minimization tools like ESPRESSO, BOOM does not use the definition of the function to be minimized as a basis for the solution, and thus the original coverage influences the solution only indirectly through the number of literals used. Most minimization methods use two basic phases introduced by Quine-McCluskey, known as prime implicant (PI) generation and the covering problem solution. Some more modern methods, like ESPRESSO, combine these two phases, reducing the number of PIs to be processed. This approach is also used in BOOM, where the search for new literals to be included into a term aims at maximum coverage of the output function. The function to be minimized is defined by its on-set and off-set, listed in a truth table. Thus the don't care set, often representing the dominant part of the truth table, need not be specified explicitly. The proposed minimization method is efficient above all for functions with a large number of input variables while only few care terms are defined. The minimization procedure is very fast, hence if the first solution does not meet the requirements, it can be improved in an iterative manner. The method has been tested on several different kinds of problems, like the MCNC standard benchmarks or larger problems generated randomly.

Abstract: This paper connects hard-core set construction, a type of hardness amplification from computational complexity, and boosting, a technique from computational learning theory. Using this connection we give fruitful applications of complexity-theoretic techniques to learning theory and vice versa. We show that the hard-core set construction of Impagliazzo (1995), which establishes the existence of distributions under which boolean functions are highly inapproximable, may be viewed as a boosting algorithm. Using alternate boosting methods we give an improved bound for hard-core set construction which matches known lower bounds from boosting and thus is optimal within this class of techniques. We then show how to apply techniques from Impagliazzo (1995) to give a new version of Jackson's celebrated Harmonic Sieve algorithm for learning DNF formulae under the uniform distribution using membership queries. Our new version has a significant asymptotic improvement in running time. Critical to our arguments is a careful analysis of the distributions which are employed in both boosting and hard-core set constructions.