Related papers: BDDs Naturally Represent Boolean Functions, and ZD…
This paper treats the variation of sets. We attempt to formulate convergence and continuity of set-valued functions in a different way from the theories on sequences of sets and correspondence. In the final section, we also attempt to…
Recent work has shown that the input-output behavior of some machine learning systems can be captured symbolically using Boolean expressions or tractable Boolean circuits, which facilitates reasoning about the behavior of these systems.…
Brain-wide recordings of large-scale networks of neurons now provide an unprecedented view into how the brain drives behavior. However, brain activity contains both information directly related to behavior as well as the potential for many…
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other…
We develop and investigate a general theory of representations of second-order functionals, based on a notion of a right comodule for a monad on the category of containers. We show how the notion of comodule representability naturally…
Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical…
Score functions for learning the structure of Bayesian networks in the literature assume that data are a homogeneous set of observations; whereas it is often the case that they comprise different related, but not homogeneous, data sets…
Boolean networks model finite discrete dynamical systems with complex behaviours. The state of each component is determined by a Boolean function of the state of (a subset of) the components of the network. This paper addresses the…
In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially…
We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if…
Decision diagrams (DDs) are powerful tools to represent effectively propositional formulas, which are largely used in many domains, in particular in formal verification and in knowledge compilation. Some forms of DDs (e.g., OBDDs, SDDs) are…
Dialectical frameworks are a unifying model of formal argumentation, where argumentative relations between arguments are represented by assigning acceptance conditions to atomic arguments. Their generality allow them to cover a number of…
Boolean circuit is a computational graph that consists of the dynamic directed graph structure and static functionality. The commonly used logic optimization and Boolean matching-based transformation can change the behavior of the Boolean…
In this paper, we consider decision trees that use both queries based on one attribute each and queries based on hypotheses about values of all attributes. Such decision trees are similar to ones studied in exact learning, where not only…
In this thesis, we develop various techniques for working with sets in machine learning. Each input or output is not an image or a sequence, but a set: an unordered collection of multiple objects, each object described by a feature vector.…
Hilary Putnam once suggested that "the actual existence of sets as 'intangible objects' suffers... from a generalization of a problem first pointed out by Paul Benacerraf... are sets a kind of function or are functions a sort of set?"…
Binary relations are an important abstraction arising in many data representation problems. The data structures proposed so far to represent them support just a few basic operations required to fit one particular application. We identify…
We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…
This paper provides some counterexamples to Cantor's contributions to the foundations of Set Theory. The first counterexample forces Cantor's Diagonal Method (DM) to yield one of the numbers in the target list. To study this anomaly, and…
GADTs can be represented either as their Church encodings a la Atkey, or as fixpoints a la Johann and Polonsky. While a GADT represented as its Church encoding need not support a map function satisfying the functor laws, the fixpoint…