Dynamical systems via domains:Toward a unified foundation of symbolic and non-symbolic computation

Non-symbolic computation (as, e.g., in biological and artificial neural networks) is astonishingly good at learning and processing noisy real-world data. However, it lacks the kind of understanding we have of symbolic computation (as, e.g., specified by programming languages). Just like symbolic computation, also non-symbolic computation needs a semantics—or behavior description—to achieve structural understanding. Domain theory has provided this for symbolic computation, and this thesis is about extending it to non-symbolic computation. Symbolic and non-symbolic computation can be described in a unified framework as state-discrete and state-continuous dynamical systems, respectively. So we need a semantics for dynamical systems: assigning to a dynamical system a domain—i.e., a certain mathematical structure—describing the system’s behavior. In part 1 of the thesis, we provide this domain-theoretic semantics for the ‘symbolic’ state-discrete systems (i.e., labeled transition systems). And in part 2, we do this for the ‘non-symbolic’ state-continuous systems (known from ergodic theory). This is a proper semantics in that the constructions form functors (in the sense of category theory) and, once appropriately formulated, even adjunctions and, stronger yet, equivalences. In part 3, we explore how this semantics relates the two types of computation. It suggests that non-symbolic computation is the limit of symbolic computation (in the ‘profinite’ sense). Conversely, if the system’s behavior is fairly stable, it may be described as realizing symbolic computation (here the concepts of ergodicity and algorithmic randomness are useful). However, the underlying concept of stability is limited by a no-go result due to a novel interpretation of Fitch’s paradox. This also has implications for AI-safety and, more generally, suggests fruitful applications of philosophical tools in the non-symbolic computation of modern AI