We introduce a class of digital machines, we name Digital Memcomputing Machines, (DMMs) able to solve a wide range of problems including Non-deterministic Polynomial (NP) ones with polynomial resources (in time, space, and energy). An abstract DMM with this power must satisfy a set of compatible mathematical constraints underlying its practical realization. We prove this by making a connection with the dynamical systems theory. This leads us to a set of physical constraints for poly-resource resolvability. Once the mathematical requirements have been assessed, we propose a practical scheme to solve the above class of problems based on the novel concept of self-organizing logic gates and circuits (SOLCs). These are logic gates and circuits able to accept input signals from any terminal, without distinction between conventional input and output terminals. They can solve boolean problems by self-organizing into their solution. They can be fabricated either with circuit elements with memory (such as memristors) and/or standard MOS technology. Using tools of functional analysis, we prove mathematically the following constraints for the poly-resource resolvability: (i) SOLCs possess a global attractor; (ii) their only equilibrium points are the solutions of the problems to solve; (iii) the system converges exponentially fast to the solutions; (iv) the equilibrium convergence rate scales at most polynomially with input size. We finally provide arguments that periodic orbits and strange attractors cannot coexist with equilibria. As examples, we show how to solve the prime factorization and the search version of the NP-complete subset-sum problem. Since DMMs map integers into integers, they are robust against noise and hence scalable. We finally discuss the implications of the DMM realization through SOLCs to the NP = P question related to constraints of poly-resources resolvability.
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A semigroup T(t), t > 0, is said to be conditionally completely continuous for t ≥ t1 if, for each t ≥ t1 and each bounded set B in X for which is bounded, we have T(t)B precompact. A semigroup T(t), t ≥ 0, is completely continuous if it is conditionally completely continuous and, for each t ≥ 0, the set is bounded if B is bounded.4
If the integer n requires three or more primes to factorize, either the circuit needs to be extended to include extra output numbers, or we let the circuit with only p and q to break n into two integers (depending on the initial conditions). We can then proceed by reducing even further these numbers till primes are found.
