Semiring programming: A semantic framework for generalized sum product problems

Abstract

To solve hard problems, AI relies on a variety of disciplines such as logic, probabilistic reasoning, machine learning and mathematical programming. Although it is widely accepted that solving real-world problems requires an integration amongst these, contemporary representation methodologies offer little support for this.In an attempt to alleviate this situation, we position and motivate a new declarative programming framework in this paper. We focus on the semantical foundations in service of providing abstractions of well-known problems such as SAT, Bayesian inference, generative models, learning and convex optimization. Programs are understood in terms of first-order logic structures with semiring labels, which allows us to freely combine and integrate problems from different AI disciplines and represent non-standard problems over unbounded domains. Thus, the main thrust of this paper is to view such well-known problems through a unified lens in the hope that appropriate solver strategies (exact, approximate, portfolio or hybrid) may emerge that tackle real-world problems in a principled way.

Tractable Boolean and Arithmetic Circuits

“…Recent theoretical results have shown that full recovery from some modeling errors is possible under certain conditions [60]. We finally note that some of the key properties of tractable circuits, including decomposability and determinism, have been exploited for tractable reasoning and learning in a more general, semiring setting; see, e.g., [65,66,55,9].…”

“…This can always be done and the size of resulting circuit is linear in the size of multiplied factors, not the size of their product which can be exponential. 9 The interest, however, is in tractable arithmetic circuits that can reason about a factor, not ones that can only look up its values. One fundamental reasoning task is that of computing the value of a partial variable instantiation, known as the marginals (MAR) problem [82].…”

GPU Semiring Primitives for Sparse Neighborhood Methods

“…The use of semirings in graph theory dates back to the early 1970s [42], when "good old-fashioned artificial intelligence", or Symbolic AI, was a dominant paradigm in research. Semirings have also been used for some time to implement more modern machine learning methods [11], with the more recent invention of semiring programming attempting to further consolidate these concepts under a single framework and set of symbolic routines. Semirings can be a useful building-block for probabilistic models, such as Bayesian networks [49] and the use of Tropical semiring in Markov networks [26].…”

Symbolic Logic Meets Machine Learning: A Brief Survey in Infinite Domains

“…Following works such as [14,33], an interesting observation made in [52] is that by appealing to a sum of products computation over different semiring structures, we can realize a large number of tasks such as satisfiability, unweighted model counting, sensitivity analysis, gradient computations, in addition to WMC. It was then shown in [9] that the idea could be generalized further for infinite domains: by defining a measure on first-order models, WMI and convex optimization can also be captured. As the underlying language is a logical one, composition can already be defined using logical connectives.…”