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**11 - 14**of**14**### Testing Properties of Constraint-Graphs Extended Abstract + Appendix

"... We study a model of graph related formulae that we call the Constraint-Graph model. A constraintgraph is a labeled multi-graph (a graph where loops and parallel edges are allowed), where each edge e is labeled by a distinct Boolean variable and every vertex is associate with a Boolean function over ..."

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We study a model of graph related formulae that we call the Constraint-Graph model. A constraintgraph is a labeled multi-graph (a graph where loops and parallel edges are allowed), where each edge e is labeled by a distinct Boolean variable and every vertex is associate with a Boolean function over the variables that label its adjacent edges. A Boolean assignment to the variables satisfies the constraint graph if it satisfies every vertex function. We associate with a constraint-graph G the property of all assignments satisfying G, denoted SAT (G). We show that the above model is quite general. That is, for every property of strings P there exists a property of constraint-graphs PG such that P is testable using q queries iff PG is thus testable. In addition, we present a large family of constraint-graphs for which SAT (G) is testable with constant number of queries. As an implication of this, we infer the testability of some edge coloring problems (e.g. the property of two coloring of the edges in which every node is adjacent to at least one vertex of each color). Another implication is that every property of Boolean strings that can be represented by a Read-twice CNF formula is testable. We note that this is the best possible in terms of the number of occurrences of every variable in a formula. 1

### MODELS OF QUERY COMPLEXITY FOR BOOLEAN FUNCTIONS

, 2008

"... In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm ..."

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In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm that computes f. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to ”approximately ” compute f) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries). The main results in this thesis are: • We study the relation between deterministic decision tree complexity and other combinatorial measures of complexity measures like sensitivity and block sensitivity. We prove that for minterm-transitive functions the sensitivity is quadratically related to block sensitivity which is polynomially

### Property Testing of Massively Parametrized problems- A survey ∗

"... We survey here property testing results for the so called ’massively parametrized ’ model (or problems). This paper is based on a survey talk gave at the workshop on property testing, Beijing, ..."

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We survey here property testing results for the so called ’massively parametrized ’ model (or problems). This paper is based on a survey talk gave at the workshop on property testing, Beijing,