Theory of Boolean Circuits

Lecturer: Prof. Dr. Heribert Vollmer

Frequency: Every two years (even years) in the summer semester

Course type: Lecture, Exercise, and Seminar (2V + 1Ü + 2S, 7 ECTS)

Examination: Oral exam

The module provides advanced knowledge about the theoretical circuit model. After successfully completing the course, students will be able to analyze algorithmic problems with respect to their circuit complexity. They will assess consequences of upper and lower bounds in the circuit model. They develop Boolean circuits for new algorithmic problems.

Content:

In this lecture, we will study the computational model of Boolean circuits. Boolean circuits are directed acyclic graphs whose nodes (gates) evaluate Boolean functions (such as AND, OR, NOT). We will examine various fundamental functions (addition, multiplication, sorting, etc.) and construct circuits that realize these with as few gates as possible or with the shortest possible path lengths between input and output.

Outline:

• Boolean circuits and their complexity measures
• Circuits for fundamental functions (addition, multiplication, threshold)
• Reductions
• Reductions between fundamental functions (iterated addition, multiplication, sorting, iterated multiplication)
• TC0 vs. NC1
• Lower bounds for general circuits (parity, threshold)
• Probabilistic circuits
• Circuits with MOD gates
• Lower bounds for AC0(p)
• Circuits and polynomials
• The Smolensky theorem