A Sound and Complete Axiomatization of Majority-n Logic

Manipulating logic functions via majority operators recently drew the attention of researchers in computer science. For example, circuit optimization based on majority operators enables superior results as compared to traditional logic systems. Also, the Boolean satisfiability problem finds new solving approaches when described in terms of majority decisions. To support computer logic applications based on majority a sound and complete set of axioms is required. Most of the recent advances in majority logic deal only with ternary majority (MAJ- 3) operators because the axiomatization with solely MAJ-3 and complementation operators is well understood. However, it is of interest extending such axiomatization to n-ary majority operators (MAJ-n) from both the theoretical and practical perspective. In this work, we address this issue by introducing a sound and complete axiomatization of MAJ-n logic. Our axiomatization naturally includes existing majority logic systems. Based on this general set of axioms, computer applications can now fully exploit the expressive power of majority logic.Comment: Accepted by the IEEE Transactions on Computer

Massively parallel computing on an organic molecular layer

Current computers operate at enormous speeds of ~10^13 bits/s, but their principle of sequential logic operation has remained unchanged since the 1950s. Though our brain is much slower on a per-neuron base (~10^3 firings/s), it is capable of remarkable decision-making based on the collective operations of millions of neurons at a time in ever-evolving neural circuitry. Here we use molecular switches to build an assembly where each molecule communicates-like neurons-with many neighbors simultaneously. The assembly's ability to reconfigure itself spontaneously for a new problem allows us to realize conventional computing constructs like logic gates and Voronoi decompositions, as well as to reproduce two natural phenomena: heat diffusion and the mutation of normal cells to cancer cells. This is a shift from the current static computing paradigm of serial bit-processing to a regime in which a large number of bits are processed in parallel in dynamically changing hardware.Comment: 25 pages, 6 figure

Making Classical Ground State Spin Computing Fault-Tolerant

We examine a model of classical deterministic computing in which the ground state of the classical system is a spatial history of the computation. This model is relevant to quantum dot cellular automata as well as to recent universal adiabatic quantum computing constructions. In its most primitive form, systems constructed in this model cannot compute in an error free manner when working at non-zero temperature. However, by exploiting a mapping between the partition function for this model and probabilistic classical circuits we are able to show that it is possible to make this model effectively error free. We achieve this by using techniques in fault-tolerant classical computing and the result is that the system can compute effectively error free if the temperature is below a critical temperature. We further link this model to computational complexity and show that a certain problem concerning finite temperature classical spin systems is complete for the complexity class Merlin-Arthur. This provides an interesting connection between the physical behavior of certain many-body spin systems and computational complexity.Comment: 24 pages, 1 figur