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Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we…
Diagrammatic expansions are a paradigmatic and powerful tool of quantum many-body theory. Their evaluation to high order, e.g., by the Diagrammatic Monte Carlo technique, can provide unbiased results in strongly correlated and challenging…
We consider anti-unification for simply typed lambda terms in associative, commutative, and associative-commutative theories and develop a sound and complete algorithm which takes two lambda terms and computes their generalizations in the…
We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…
This book explores an alternative to the current dominant paradigm where a discrete computer model is constructed as an attempt to approximate some continuum theory. We focus on a class of discrete computer models that are based on simple…
A compiler approach for generating low-level computer code from high-level input for discontinuous Galerkin finite element forms is presented. The input language mirrors conventional mathematical notation, and the compiler generates…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We consider the power of Boolean circuits with MOD$_{6}$ gates. First, we introduce a few basic notions of computational complexity, and describe the standard models with which we study the complexity of problems. We then define the model…
Programs with control are usually modeled using lambda calculus extended with control operators. Instead of modifying lambda calculus, we consider a different model of computation. We introduce continuation calculus, or CC, a deterministic…
The ability to model, analyze, and predict execution time of computations is an important building block supporting numerous efforts, such as load balancing, performance optimization, and automated performance tuning for high performance,…
Dominant areas of computer science and computation systems are intensively linked to the hypercube-related studies and interpretations. This article presents some transformations and analytics for some example algorithms and Boolean domain…
We describe how dagger-Frobenius monoids give the correct categorical description of certain kinds of finite-dimensional 'quantum algebras'. We develop the concept of an involution monoid, and use it to construct a correspondence between…
We use the notion of Borel generators to give alternative methods for computing standard invariants, such as associated primes, Hilbert series, and Betti numbers, of Borel ideals. Because there are generally few Borel generators relative to…
Static verification techniques leverage Boolean formula satisfiability solvers such as SAT and SMT solvers that operate on conjunctive normal form and first order logic formulae, respectively, to validate programs. They force bounds on…
When writing high-performance code for numerical computation in a scripting language like MATLAB, it is crucial to have the operations in a large for-loop vectorized. If not, the code becomes too slow to use, even for a moderately large…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
In this article we show the rough outline of a computer algorithm to generate lower bounds on the exponential function of (in principle) arbitrary precision. We implemented this to generate all necessary analytic terms for the Boltzmann…
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
Representation theory provides a suitable framework to count and classify invariants in tensor models. We show that there are two natural ways of counting invariants, one for arbitrary rank of the gauge group and a second, which is only…