# Some Studies on Circuit Complexity.

## Date of Submission

December 1996

## Date of Award

Winter 12-12-1997

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

## Subject Name

Computer Science

## Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

## Supervisor

Barua, Rana (TSMU-Kolkata; ISI)

## Abstract (Summary of the Work)

The Circuit Complexity of Boolean functions is a topic of long-standing interest in Computer Science. The subject originated form the requirement of minimising hard-ware circuit costs for computing Boolean functions. The pioneering contribution in this direction was made by Shannon in a paper[47] in which he proposed the size of smallest circuit computing a function to be the measure of its complexity. He proved an upper bound on the complexity of all n-input functions and used a counting argument to show that for most of the Boolean functions this bound is not too far off. The problem of finding lower bounds on the complexity of circuits computing Boolean functions has motivated considerable research over the past four decades.The other motivation to study this subject came from the need of developing suitable mathematical models to study the theory of Computational Complexity. The classification of problems from the point of view of computation can be divided into two major subdisciplines. One of these is the theory of Algorithms which gives the upper bounds on the amount of computational resources (time, space etc.) needed to solve particular problems. The other one consists of techniques for analysing a problem in some computational model irrespective of any algorithm and tries to get some lower bound on the amount of resources required to solve it. The most common model of computation is Turing Machine model. Hartmanis and Stearns(20] first formalised the measure of complexity of functions as time on Turing machine and Edmonds[18] felt the need of avoiding brute force search method and foresaw the issue of polynomial-vs-exponential complexity. Cook, in his 1971 paper[15], precisely formulated the P vs. NP conjecture. Savage(46] first showed the connection between circuit complexity and Turing machine time. Over the years, many researchers have felt that the combinatorial and static nature of Boolean circuits renders them more suitable for proving lower bounds on complexity of problems and allows for natural variations and restrictions. Thus, gradually it became apparent that devising new frameworks for studying complexity in circuit model and applying them to explicit problems may lead towards solving the interesting unsolved questions of Complexity Theory.Outline of this documentThis document consists of two parts. In the first part, described in Chapter 2 and Chapter 3, a brief survey on circuit complexity is presented. Details of major portion of this review are contained in the survey work by Boppana and Sipser(13), in the book by Wegener(55], in the Ph.D thesis of Mauricio Karchmer(23] and some recent papers on the fusion method(24],[9). We have tried to touch upon the different techniques available to prove circuit lower bounds and have attempted to list down those results that are considered as breakthroughs achieved in this subject. In Chapter 4, we describe some of our results. This is followed by Conclusion and then a list of references.

## Control Number

ISI-DISS-1996-135

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/6305

## Recommended Citation

Chakraborty, Pranab, "Some Studies on Circuit Complexity." (1997). *Master’s Dissertations*. 227.

https://digitalcommons.isical.ac.in/masters-dissertations/227

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843250