How should a machine intelligence perform unsupervised structure discovery over streams of sensory input? One approach to this problem is to cast it as an apperception task [1]. Here, the task is to construct an explicit interpretable theory that both explains the sensory sequence and also satisfies a set of unity conditions, designed to ensure that the constituents of the theory are connected in a relational structure. However, the original formulation of the apperception task had one fundamental limitation: it assumed (...) the raw sensory input had already been parsed using a set of discrete categories, so that all the system had to do was receive this already-digested symbolic input, and make sense of it. But what if we don't have access to pre-parsed input? What if our sensory sequence is raw unprocessed information? The central contribution of this paper is a neuro-symbolic framework for distilling interpretable theories out of streams of raw, unprocessed sensory experience. First, we extend the definition of the apperception task to include ambiguous (but still symbolic) input: sequences of sets of disjunctions. Next, we use a neural network to map raw sensory input to disjunctive input. Our binary neural network is encoded as a logic program, so the weights of the network and the rules of the theory can be solved jointly as a single SAT problem. This way, we are able to jointly learn how to perceive (mapping raw sensory information to concepts) and apperceive (combining concepts into declarative rules). (shrink)

This paper attempts to answer a central question in unsupervised learning: what does it mean to “make sense” of a sensory sequence? In our formalization, making sense involves constructing a symbolic causal theory that both explains the sensory sequence and also satisfies a set of unity conditions. The unity conditions insist that the constituents of the causal theory – objects, properties, and laws – must be integrated into a coherent whole. On our account, making sense of sensory input is a (...) type of program synthesis, but it is unsupervised program synthesis. Our second contribution is a computer implementation, the Apperception Engine, that was designed to satisfy the above requirements. Our system is able to produce interpretable human-readable causal theories from very small amounts of data, because of the strong inductive bias provided by the unity conditions. A causal theory produced by our system is able to predict future sensor readings, as well as retrodict earlier readings, and impute (fill in the blanks of) missing sensory readings, in any combination. In fact, it is able to do all three tasks simultaneously. We tested the engine in a diverse variety of domains, including cellular automata, rhythms and simple nursery tunes, multi-modal binding problems, occlusion tasks, and sequence induction intelligence tests. In each domain, we test our engine's ability to predict future sensor values, retrodict earlier sensor values, and impute missing sensory data. The Apperception Engine performs well in all these domains, significantly out-performing neural net baselines. We note in particular that in the sequence induction intelligence tests, our system achieved human-level performance. This is notable because our system is not a bespoke system designed specifically to solve intelligence tests, but a general-purpose system that was designed to make sense of any sensory sequence. (shrink)

This paper formalizes part of the cognitive architecture that Kant develops in the Critique of Pure Reason. The central Kantian notion that we formalize is the rule. As we interpret Kant, a rule is not a declarative conditional stating what would be true if such and such conditions hold. Rather, a Kantian rule is a general procedure, represented by a conditional imperative or permissive, indicating which acts must or may be performed, given certain acts that are already being performed. These (...) acts are not propositions; they do not have truth-values. Our formalization is related to the input/ output logics, a family of logics designed to capture relations between elements that need not have truth-values. In this paper, we introduce KL3 as a formalization of Kant’s conception of rules as conditional imperatives and permissives. We explain how it differs from standard input/output logics, geometric logic, and first-order logic, as well as how it translates natural language sentences not well captured by first-order logic. Finally, we show how the various distinctions in Kant’s much-maligned Table of Judgements emerge as the most natural way of dividing up the various types and sub-types of rule in KL3. Our analysis sheds new light on the way in which normative notions play a fundamental role in the conception of logic at the heart of Kant’s theoretical philosophy. (shrink)