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Lucy Ham

Publications and source records attributed to Lucy Ham.

6 recordsLinked to original sources
arXiv2024

Mapping, modeling, and reprogramming cell-fate decision making systems

Many cellular processes involve information processing and decision making. We can probe these processes at increasing molecular detail. The analysis of heterogeneous data remains a challenge that requires new ways of thinking about cells in quantitative, predictive, and mechanistic ways. We discuss the role of mathematical models in the context of cell-fate decision making systems across the tree of life. Complex multi-cellular organisms have been a particular focus, but single celled organisms also have to sense and respond to their environment. We center our discussion around the idea of design principles which we can learn from observations and modeling, and exploit in order to (re)-design or guide cellular behavior.

q-bio.CB↗
arXiv2022

Finite model theory for pseudovarieties and universal algebra: preservation, definability and complexity

We explore new interactions between finite model theory and classical streams of universal algebra and semigroup theory. A key result is an example of finite algebras whose variety is not finitely axiomatisable in first order logic, but where the class of finite members are finitely axiomatisable amongst finite algebras. These algebras present a negative solution to a first order formulation of the Eilenberg-Sch\"utzenberger problem, and witness the simultaneous failure of the {\L}os-Tarski Theorem, the SP-Preservation Theorem and Birkhoff's HSP-Preservation Theorem at the finite level. The examples also show that a pseudovariety without any finite pseudoequational basis may be finitely axiomatisable in first order logic amongst finite algebras. Other results include the undecidability of deciding first order definability of the pseudovariety of a finite algebra, and a mapping from any fixed finite template constraint satisfaction problem to a first order equivalent variety membership problem.

math.LO↗
arXiv2017

All or nothing: toward a promise problem dichotomy for constraint problems

A finite constraint language $\mathscr{R}$ is a finite set of relations over some finite domain $A$. We show that intractability of the constraint satisfaction problem $\operatorname{CSP}(\mathscr{R})$ can, in all known cases, be replaced by an infinite hierarchy of intractable promise problems of increasingly disparate promise conditions: where instances are guaranteed to either have no solutions at all, or to be $k$-robustly satisfiable (for any fixed $k$), meaning that every "reasonable" partial instantiation on~$k$ variables extends to a solution. For example, subject to the assumption $\texttt{P}\neq \texttt{NP}$, then for any~$k$, we show that there is no polynomial time algorithm that can distinguish non-$3$-colourable graphs, from those for which any reasonable $3$-colouring of any $k$ of the vertices can extend to a full $3$-colouring. Our main result shows that an analogous statement holds for all known intractable constraint problems over fixed finite constraint languages.

cs.CC↗
arXiv2017

Gap theorems for robust satisfiability: Boolean CSPs and beyond

A computational problem exhibits a "gap property" when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the complexity of possible ${\bf NP}$-hard gaps in the case of Boolean domains. As a consequence, we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem: all are either polynomial-time tractable or $\mathbf{NP}$-complete. Schaefer's original dichotomy for $\textsf{SAT}$ variants is a notable particular case. Universal algebraic methods have been central to recent efforts in classifying the complexity of constraint satisfaction problems. A second contribution of the article is to develop aspects of the algebraic approach in the context of a number of variants of the constraint satisfaction problem. In particular, this allows us to lift our results on Boolean domains to many templates on non-Boolean domains.

cs.CC↗
arXiv2015

Relativised homomorphism preservation at the finite level

In this article, we investigate the status of the homomorphism preservation property amongst restricted classes of finite relational structures and algebraic structures. We show that there are many homomorphism-closed classes of finite lattices that are definable by a first-order sentence but not by existential positive sentences, demonstrating the failure of the homomorphism preservation property for lattices at the finite level. In contrast to the negative results for algebras, we establish a finite-level relativised homomorphism preservation theorem in the relational case. More specifically, we give a complete finite-level characterisation of first-order definable finitely generated anti-varieties relative to classes of relational structures definable by sentences of some general forms. When relativisation is dropped, this gives a fresh proof of Atserias's characterisation of first-order definable constraint satisfaction problems over a fixed template, a well known special case of Rossman's Finite Homomorphism Preservation Theorem.

math.LO↗