In spite of the rapidly increasing number of applications of machine learning in various domains, a principled and systematic approach to the incorporation of domain knowledge in the engineering process is still lacking and ad hoc solutions that are difficult to validate are still the norm in practice, which is of growing concern not only in mission-critical applications. In this note, we introduce Probabilistic Approximate Logic (PALO) as a logic based on the notion of mean approximate probability to overcome conceptual and computational difficulties inherent to strictly probabilistic logics. The logic is approximate in several dimensions. Logical independence assumptions are used to obtain approximate probabilities, but by averaging over many instances of formulas a useful estimate of mean probability with known confidence can usually be obtained. To enable efficient computational inference, the logic has a continuous semantics that reflects only a subset of the structural properties of classical logic, but this imprecision can be partly compensated by richer theories obtained by classical inference or other means. Computational inference, which refers to the construction of models and validation of logical properties, is based on Stochastic Gradient Descent (SGD) and Markov Chain Monte Carlo (MCMC) techniques and hence another dimension where approximations are involved. We also present the Logical Imagination Engine (LIME), a prototypical implementation of PALO based on TensorFlow. Albeit not limited to the biological domain, we illustrate its operation in a quite substantial bioinformatics machine learning application concerned with network synthesis and analysis in a recent DARPA project.
In this paper we investigate a propositional fuzzy logical system L? which contains the well-known Lukasiewicz, Product and Godel fuzzy logics as sublogics. We define the corresponding algebraic structures, called L?-algebras and prove the following completeness result: a formula f is provable in the L? logic iff it is a tautology for all linear L?-algebras. Moreover, linear L?-algebras are shown to be embeddable in linearly ordered abelian rings with a strong unit and cancellation law. Probability theory and fuzzy logic have been presented as quite distinct theoretical foundations for reasoning and decision making in situations of uncertainty. This paper establishes a common basis for both forms of logic of uncertainty in which a basic uncertainty logic is defined in terms of a valuation on a lattice of propositions. The (non-truth-functional) connectives for conjunction, disjunction, equivalence, implication, and negation are defined in terms which closely resemble those of probability theory. Addition of the axiom of the excluded middle to the basic logic gives a standard probability logic. Alternatively, addition of a requirement for strong truth-functionality (truth-value of connective determined by truth-value of constituents) gives a fuzzy logic with connectives, including implication, as in Lukasiewicz' infinitely valued logic. A common semantics for all such variants is given in terms of binary responses from a population. The type of population, e.g., physical events, people, or neurons, determines whether the model is of physical probability, subjective belief, or human decision-making. The formal theory and the semantics together illustrate clearly the precise similarities and differences between fuzzy and probability logics.