Chapter 7. Proof Complexity and SAT Solving

This chapter gives an overview of proof complexity and connections to SAT solving, focusing on proof systems such as resolution, Nullstellensatz, polynomial calculus, and cutting planes (corresponding to conflict-driven clause learning, algebraic approaches using linear algebra or Gröbner bases, and pseudo-Boolean solving, respectively). There is also a discussion of extended resolution (which is closely related to DRAT proof logging) and Frege and extended Frege systems more generally. An ample supply of references for further reading is provided, including for some topics omitted in this chapter.

Example of polynomial controller synthesis for a non-square object with one input and two outputs

Polynomial methods for synthesizing controllers for automatic control systems with linear objects are becoming increasingly common. The synthesis of multichannel controllers is particularly difficult, which is caused by the need to use matrix polynomial calculus. However, this approach mainly considers objects with the number of inputs equal to the number of outputs. This is due to the convenience of solving a system of linear algebraic equations in matrix polynomial calculus. In this paper, we consider a polynomial method for synthesizing regulators for a non-square object, that is, one whose number of inputs is not equal to the number of outputs. The selected system contains not only a non-square object, but also a non-square controller.