Progress in computing thrives from progress in a diversity of fields. Matrices and Linear Algebra, an old stalwart of computing, are nowadays at the core of much work in Artificial Intelligence. Formalisms based on Mathematical Logic, such as propositional logic, datalog, or logic programming enable principled programming where one can distinguish meaning from execution. Graphs arise naturally in programming, and are often used to understand properties of data.

The presentation starts by quickly mentioning some important properties connecting these areas. Graphs can encode programs, and they can be encoded as matrices, which can be encoded by programs themselves. This connection has led to interest in combining the approaches into a single system, and has led to developments such as Statistical Relational Learning (SRL) and the neuro-symbolic systems.

The SRL approach is about dealing with uncertainty on structured domains. In a simplest form, it can be seen as extending a graphical network with repeated information: say, students that perform well in Algebra I will also perform well in Algebra II. Logical rules can be used to express this common information, as exemplified in CLP(BN), an extension of Prolog that builds directed acyclic graphs with probabilistic annotations, that is, Bayesian networks. A different approach is used in Problog. This language is also based in Prolog, but it softens the language by introducing Probabilistic facts. These facts can be used to abduce on different values, but they can also be used to give probability to rules. One should also mention Markov Logic Networks. In these models, logic atoms are connected in clauses. These connections generate an undirected graphical model.

One problem with these models is that they may require heavy temporal and memory resources and can quickly become expensive. Deep neural networks have become successful by implementing a compound function f = f₀ • f₁ ... f_n, where each f_i is a layer. Layers can be implemented through matrices, and parameter learning can use the derivative of f to optimize parameters through gradient based search.

This development has led to rapid growth in the landscape of neuro-symbolic models. TensorLog is an influential example that constructs a deep neural network out of logical rules. This is achieved through restriction to chain-rules, where each goal has a single input variable, shared with the previous goal, and a single output variable, connected to the next goal. The model is suited for edge-detection problems, where it performs well, but is not ideal for pure classification. Several extensions have been proposed; we discuss how one, NeuralLog, tackles this problem. Notice these are but a few examples; very different approaches exist, such as Deep-Problog.

Meanwhile, there has been further progress in neural networks. Graph Neural Networks are close in many ways to logic based proposals, but are based on graph theory, so they approach learning in a different way to logic based systems. But focus has settled on Large Language Models. Although most developments are empirical, there exist already proposals for logic reasoning in LLMs, such as LOGIC-LLM and LOGICLM. We expect to see others soon.

Program generation is a powerful and expressive approach to eliminating abstraction overhead and improving program performance, which has been studied and implemented in a variety of languages with different forms.

In this talk we overview MacoCaml, a new design and implementation of compile-time computation for OCaml. MacoCaml features a unifying and novel combination of phase separation and quotation-based staging. We review MacoCaml's recent developments, including a comprehensive formalism of a feature-rich macro calculus with key meta-theoretic properties, and an extension to module functors that leads to explicit phase distinction. We describe how the meta-theoretical results offer practical benefits for programmers, and conclude the talk with a few directions for future exploration.