### 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 ..."

Abstract
- Add to MetaCart

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

### Testing Membership in Formal Languages Implicitly Represented by Boolean Functions

"... Abstract: Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron in [Goldreich et al. (1998)] and inspired by Rubinfeld and Sudan in [Rubinfeld and Sudan 1996], deals with the relaxation of decision problems. Given a property P the aim is to decide whether a given input ..."

Abstract
- Add to MetaCart

Abstract: Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron in [Goldreich et al. (1998)] and inspired by Rubinfeld and Sudan in [Rubinfeld and Sudan 1996], deals with the relaxation of decision problems. Given a property P the aim is to decide whether a given input satisfies the property P or is far from having the property. A property P can be described as a language, i.e., a nonempty family of binary words. The associated property to a family of boolean functions f =(fn) is the set of 1-inputs of f. By an attempt to correlate the notion of testing to other notions of low complexity property testing has been considered in the context of formal languages. Here, a brief summary of results on testing properties defined by formal languages and by languages implicitly represented by small restricted branching programs is provided. Key Words: binary decision diagrams (BDDs), boolean functions, branching programs (BPs), computational complexity, formal languages, property testing, randomness, sublinear algorithms Category: F.1.3, F.2.2, G.3.1

### 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 ..."

Abstract
- Add to MetaCart

(Show Context)
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

### Approximations in Model-Checking and Testing

, 2008

"... We describe different approximations in the context of Model-Checking and Testing. ..."

Abstract
- Add to MetaCart

We describe different approximations in the context of Model-Checking and Testing.

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

, 2010

"... 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, Jan 2010. ..."

Abstract
- Add to MetaCart

(Show Context)
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, Jan 2010.

### Testing graphs for colorability properties \Lambda

"... Abstract Let P be a property of graphs. An ffl-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the ca ..."

Abstract
- Add to MetaCart

(Show Context)
Abstract Let P be a property of graphs. An ffl-test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than ffl \Gamma

### unknown title

"... We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include a couple of observations which are of independent interest. 1. In the context of linear property testing, adaptive 2-sided error tests have no more power than non-adaptive 1-sided error ..."

Abstract
- Add to MetaCart

We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include a couple of observations which are of independent interest. 1. In the context of linear property testing, adaptive 2-sided error tests have no more power than non-adaptive 1-sided error tests. 2. Random linear LDPC codes with linear distance and constant rate are very far from being locally testable.

### Electronic Colloquium on Computational Complexity, Report No. 54 (2007) Testing Properties of Constraint-Graphs

"... 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 ..."

Abstract
- Add to MetaCart

(Show Context)
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

### Contents

, 2010

"... Model Checking and Testing are two areas with a similar goal: to verify that a system satisfies a property. They start with different hypothesis on the systems and develop many techniques with different notions of approximation, as an exact verification may be computationally too hard. We present so ..."

Abstract
- Add to MetaCart

(Show Context)
Model Checking and Testing are two areas with a similar goal: to verify that a system satisfies a property. They start with different hypothesis on the systems and develop many techniques with different notions of approximation, as an exact verification may be computationally too hard. We present some of notions of approximation with their Logic and Statistics backgrounds, which yield several techniques