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William DeMeo

Publications and source records attributed to William DeMeo.

12 recordsLinked to original sources
arXiv2022

Universal Algebraic Methods for Constraint Satisfaction Problems

After substantial progress over the last 15 years, the "algebraic CSP-dichotomy conjecture" reduces to the following: every local constraint satisfaction problem (CSP) associated with a finite idempotent algebra is tractable if and only if the algebra has a Taylor term operation. Despite the tremendous achievements in this area (including recently announce proofs of the general conjecture), there remain examples of small algebras with just a single binary operation whose CSP resists direct classification as either tractable or NP-complete using known methods. In this paper we present some new methods for approaching such problems, with particular focus on those techniques that help us attack the class of finite algebras known as "commutative idempotent binars" (CIBs). We demonstrate the utility of these methods by using them to prove that every CIB of cardinality at most 4 yields a tractable CSP.

cs.LO↗
arXiv2021

A Machine-checked proof of Birkhoff's Variety Theorem in Martin-Löf Type Theory

The Agda Universal Algebra Library (agda-algebras) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and proof assistant. In this paper we draw on and explain many components of the agda-algebras library, which we extract into a single Agda module in order to present a self-contained formal and constructive proof of Birkhoff's HSP theorem in Martin-Löf dependent type theory. In the course of our presentation, we highlight some of the more challenging aspects of formalizing the basic definitions and theorems of universal algebra in type theory. Nonetheless, we hope this paper and the agda-algebras library serve as further evidence in support of the claim that dependent type theory and the Agda language, despite the technical demands they place on the user, are accessible to working mathematicians (such as ourselves) who possess sufficient patience and resolve to formally verify their results with a proof assistant. Indeed, the agda-algebras library now includes a substantial collection of definitions, theorems, and proofs from universal algebra, illustrating the expressive power of inductive and dependent types for representing and reasoning about general algebraic and relational structures.

cs.LO↗
arXiv2021

The Agda Universal Algebra Library, Part 1: Foundation

The Agda Universal Algebra Library (UALib) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and proof assistant. The UALib includes a substantial collection of definitions, theorems, and proofs from general algebra and equational logic, including many examples that exhibit the power of inductive and dependent types for representing and reasoning about relations, algebraic structures, and equational theories. In this paper we discuss the logical foundations on which the library is built, and describe the types defined in the first 13 modules of the library. Special attention is given to aspects of the library that seem most interesting or challenging from a type theory or mathematical foundations perspective.

cs.LO↗
arXiv2021

The Agda Universal Algebra Library, Part 2: Structure

The Agda Universal Algebra Library (UALib) is a library of types and programs (theorems and proofs) we developed to formalize the foundations of universal algebra in dependent type theory using the Agda programming language and proof assistant. The UALib includes a substantial collection of definitions, theorems, and proofs from universal algebra, equational logic, and model theory, and as such provides many examples that exhibit the power of inductive and dependent types for representing and reasoning about mathematical structures and equational theories. In this paper, we describe the the types and proofs of the UALib that concern homomorphisms, terms, and subalgebras.

cs.LO↗
arXiv2021

Constraint Satisfaction Problems over Finite Structures

We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a natural algebraic question: which finite algebras admit only polynomially many homomorphisms into them? We give some sufficient and some necessary conditions for a finite algebra to have this property. In particular, we show that every finite equationally nontrivial algebra has this property which gives us, as a simple consequence, a complete complexity classification of CSPs over two-element structures, thus extending the classification for two-element relational structures by Schaefer (STOC'78). We also present examples of two-element structures that have bounded width but do not have relational width (2,3), thus demonstrating that, from a descriptive complexity perspective, allowing operations leads to a richer theory.

cs.LO↗
arXiv2020

Polynomial-time Tests for Difference Terms in Idempotent Varieties

We consider the following practical question: given a finite algebra A in a finite language, can we efficiently decide whether the variety generated by A has a difference term? We answer this question (positively) in the idempotent case and then describe algorithms for constructing difference term operations.

math.LO↗
arXiv2019

Bounded homomorphisms and finitely generated fiber products of lattices

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms $g\colon A\to D$, $h\colon B\to D$, where $A$, $B$ are finitely generated and $D$ satisfies (D), we show the following: If $g$ and $h$ are bounded, then their fiber product (pullback) $C=\{(a,b)\in A\times B\ |\ g(a)=h(b)\}$ is finitely generated. While the converse is not true in general, it does hold when $A$ and $B$ are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

math.LO↗
arXiv2014

Interval enforceable properties of finite groups

We propose a classification of group properties according to whether they can be deduced from the assumption that a group's subgroup lattice contains an interval isomorphic to some lattice. We are able to classify a few group properties as being "interval enforceable" in this sense, and we establish that other properties satisfy a weaker notion of "core-free interval enforceable." We also show that if there exists a group property and its negation that are both core-free interval enforceable, this would settle an important open question in universal algebra.

math.GR↗
arXiv2013

Expansions of finite algebras and their congruence lattices

We present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra $(B_0, \dots)$, let $B_1, B_2, \dots, B_K$ be sets that either intersect $B_0$ or intersect each other at certain points. We construct an \emph{overalgebra} $(A, F_A)$, by which we mean an expansion of $(B_0, \dots)$ with universe $A = B_0 \cup B_1 \cup \cdots \cup B_K$, and a certain set $F_A$ of unary operations that includes mappings $e_i$ satisfying $e_i^2 = e_i$ and $e_i(A) = B_i$, for $0\leq i \leq K$. We explore two such constructions and prove results about the shape of the new congruence lattices $Con(A, F_A)$ that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore.

math.RA↗