I’ll present an approach from mathematical logic which shows how sub-symbolic dynamics may give rise to higher-level cognitive representations of structures, systems of knowledge, and algorithmic processes. This approach posits that learners posses a system for expressing isomorphisms with which they create mental models with arbitrary dynamics. The theory formalizes one account of how novel conceptual content may arise, allowing us to explain how even elementary logical and computational operations may be learned. I provide an implementation that learns to represent a variety of structures, including logic, number, kinship trees, regular languages, context-free languages, domains of theories like magnetism, dominance hierarchies, list structures, quantification, and computational primitives like repetition, reversal, and recursion. Moreover, the account is based on simple discrete dynamical processes that could be implemented in a variety of different physical or biological systems. In particular, I describe how the required dynamics can be directly implemented in an existing connectionist framework. The resulting theory provides an “assembly language” for cognition, where high-level theories cognition and computation can be translated into simple and neurally plausible underlying dynamics.

If biology is the study of self-replicating entities, and we want to understand the role of information, it makes sense to see how information theory is connected to the ‘replicator equation’ — a simple model of population dynamics for self-replicating entities. The relevant concept of information turns out to be the information of one probability distribution relative to another, also known as the Kullback–Liebler divergence. Using this we can get a new outlook on free energy, see evolution as a learning process, and give a clearer, more general formulation of Fisher’s fundamental theorem of natural selection.

Which features make the brain such a powerful and energy-efficient computing machine? Can we reproduce them in the solid state, and if so, what type of computing paradigm would we obtain? I will show that a machine that uses memory (time non-locality) to both process and store information, like our brain, and is endowed with intrinsic parallelism and information overhead – namely takes advantage, via its collective state, of the network topology related to the problem – has a computational power far beyond our standard digital computers [1, 2]. We have named this novel computing paradigm “*memcomputing*” [1, 2, 3, 4]. As examples, I will show the polynomial-time solution of prime factorization, the search version of the subset-sum problem [5], and approximations to the Max-SAT beyond the inapproximability gap [6] using polynomial resources and self-organizing logic gates, namely gates that self-organize to satisfy their logical proposition [5]. I will also show that these machines are described by a topological field theory, and they compute via an instantonic phase, implying that they are robust against noise and disorder [7]. The digital memcomputing machines we propose can be efficiently simulated, are *scalable*and can be easily realized with available nanotechnology components. Work supported in part by CMRR and MemComputing, Inc. (http://memcpu.com/).

[1] M. Di Ventra and Y.V. Pershin, Computing: the Parallel Approach, *Nature Physics *9, 200 (2013).

[2]F. L. Traversa and M. Di Ventra, Universal Memcomputing Machines, *IEEE Transactions on Neural Networks and Learning Systems *26, 2702 (2015).

[3] M. Di Ventra and Y.V. Pershin, Just add memory, *Scientific American ***312**, 56 (2015).

[4] M. Di Ventra and F.L. Traversa, Memcomputing: leveraging memory and physics to compute efficiently, *J. Appl. Phys. ***123**, 180901 (2018).

[5] F. L. Traversa and M. Di Ventra, Polynomial-time solution of prime factorization and NP-complete problems with digital memcomputing machines, *Chaos: An Interdisciplinary Journal of Nonlinear Science ***27**, 023107 (2017). *
*[6] F. L. Traversa, P. Cicotti, F. Sheldon, and M. Di Ventra, Evidence of an exponential speed-up in the solution of hard optimization problems

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*Complexity*

**2018**, 7982851 (2018).

[7] M. Di Ventra, F. L. Traversa and I.V. Ovchinnikov, Topological field theory and computing with instantons,

*Annalen der Physik*1700123 (2017).