Gregory L. McColm

A computational model for self-assembling flexible tiles

We present a theoretical model for self-assembling tiles with flexible branches motivated by DNA branched junction molecules. We encode an instance of a “problem” as a pot of such tiles, and a “solution” as an assembled complete complex without any free sticky ends (called ports), whose number of tiles is within predefined bounds. We develop an algebraic representation of this self-assembly process and use it to prove that this model of self-assembly precisely captures NP-computability when the number of tiles in the minimal complete complexes is bounded by a polynomial.

Hierarchies in transitive closure logic, stratified Datalog and infinitary logic☆

Abstract We establish a general hierarchy theorem for quantifier classes in the infinitary logic L∞ωωon finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman. We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog.

When is arithmetic possible

Abstract When a structure or class of structures admits an unbounded induction, we can do arithmetic on the stages of that induction: if only bounded inductions are admitted, then clearly each inductively definable relation can be defined using a finite explicit expression. Is the converse true? We examine evidence that the converse is true, in positive elementary induction (where explicit = elementary). We present a stronger conjecture involving the language L consisting of all L ∞ω formulas with a finite number of variables, and examine a combinatorial property equivalent to “all L -definable relations are elementary”.

On the power of deterministic transitive closures

Abstract We show that transitive closure logic (FO + TC) is strictly more powerful than deterministic transitive closure logic (FO + DTC) on finite (unordered) structures. In fact, on certain classes of graphs, such as hypercubes or regular graphs of large degree and girth, every DTC-query is bounded and therefore first order expressible. On the other hand, there are simple (FO + pos TC) queries on these classes that cannot be defined by first order formulae.

Parametrization over inductive relations of a bounded number of variables

Abstract We present a Parametrization Theorem for (positive elementary) inductions that use a bounded number of variables. We investigate associated halting problem(s) on classes of finite structures and on solitary ‘unreasonable’ structures. These results involve the complexity of the inductive relations—and the complexity of the structure or class of structures on which these relations live. We also apply this Parametrization Theorem to Moschovakis closure ordinals, to determine when the closure ordinal is greater than ω, and to investigate the closure ordinals of unreasonable structures.

Deterministic vs. nondeterministic transitive closure logic

It is shown that transitive closure logic (FO+TC) is strictly more powerful than deterministic transitive closure logic (FO+DTC) on unordered structures. In fact, on certain classes of graphs, such as hypercubes or regular graphs of large degree and girth, every query in (FO+DTC) is first-order expressible. On the other hand, there are simple (FO+pos TC) queries on these classes that cannot be defined by first-order formulas.<<ETX>>

On stoichiometry for the assembly of flexible tile DNA complexes

Given a set of flexible branched junction DNA molecules with sticky-ends (building blocks), called here “tiles”, we consider the problem of determining the proper stoichiometry such that all sticky-ends could end up connected. In general, the stoichiometry is not uniform, and the goal is to determine the proper proportion (spectrum) of each type of molecule within a test tube to allow for complete assembly. According to possible components that assemble in complete complexes we partition multisets of tiles, called here “pots”, into classes: unsatisfiable, weakly satisfiable, satisfiable and strongly satisfiable. This classification is characterized through the spectrum of the pot, and it can be computed in PTIME using the standard Gauss-Jordan elimination method. We also give a geometric description of the spectrum as a convex hull within the unit cube.

Spectrum of a pot for DNA complexes

Given a set of flexible branched junction DNA molecules (building blocks) with sticky ends we consider the question of determining the proper stoichiometry such that all sticky ends could end up connected. The idea is to determine the proper proportion (spectrum) of each type of molecules present, which in general is not uniform. We classify the pot in three classes: weakly satisfiable, satisfiable and strongly satisfiable according to possible components that assemble in complete complexes. This classification is characterized through the spectrum of the pot, which can be computed in PTIME using the standard Gauss-Jordan elimination method.

MSO zero-one laws on random labelled acyclic graphs☆

We use Ehrenfeucht-type games to prove that Monadic Second Order logic admits labelled zero-one laws for random free trees, generating the complete almost sure theory. Our method will be to dissect random trees to get a picture of what almost all random free trees look like. We will use elementary (second moment) methods to obtain probability results.

A Splitting Inequality

AbstractWe prove that if A is a finite set, and if