## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, March 17, 2016 — 2:30 PM EDT

**Matt Valeriote, McMaster University **

“Testability”

I will discuss a question dealing with the solutions of primitive positive (pp) formulas over finite structures. The question is concerned with the query complexity of algorithms for making decisions about the correctness of a proposed solution to a pp-formula over a finite structure. In general, in order to determine with high probability whether some hidden proposed solution to a pp-formula is close to being an actual solution, one will need to check a large fraction of its values. It turns out that for some (special) structures, there are algorithms that only need to check (or query) a constant number of values of a proposed solution to a pp-formula (no matter how many free variables the formula has) in order to conclude, with high probability, whether or not it is close to a solution to the formula over the structure.

With Chen and Yoshida, we characterize, for finite structures A, the query complexity of testing solutions of pp-formulas over A in terms of algebraic conditions on A. In particular, we characterize those structures A for which solution testing can be carried out by an algorithm that only needs to make a constant number of queries of a proposed solution to a pp-formula, independent of the number of free variables the formula has.

MC 5403

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.