## Final Report Summary - FEALORA (Feasibility, logic and randomness in computational complexity)

This was an interdisciplinary project focusing on connections between mathematical logic, mainly proof theory, and the branch of theoretical computer science called computational complexity. One of the main goals in mathematical logic is to fully understand which mathematical principles can be formalized and proved in which formal systems. The formal systems studied in the project were theories formalized in first-order logic and proof systems for propositional logic. The most important property of formal systems is their strength measured by the amount of mathematical principles that can be formalized in them. It is a well-known fact (a corollary of the Gödel incompleteness theorem) that first-order theories form an infinite hierarchy from the point of view of their strength. Computational complexity theory studies the amount of resources (such as time and memory) needed to solve some problem (e.g. to find a solution of an equation). Also in this theory there are infinite hierarchies of problems.

Connections between proofs and computations is a classical subject in proof theory. In the mid 1970s Stephen Cook initiated the study of “feasible” proofs and efficient computations. His ideas and results developed into a new discipline called proof complexity. The picture that gradually emerged from the results proved in this area suggested that there is a connection between unprovability in weak formal systems and the computational complexity of problems associated with the unprovable sentences. Approximately 10 years ago, the PI proposed the Feasible Incompleteness Thesis, which, roughly speaking, says that this connection is not restricted to weak theories, but is a general principle stating that incompleteness can be caused by computational complexity.

The objective of project FEALORA was to study these connections between provability and computational complexity. The plan was to approach this problem from three direction. First, we need to fully understand the nature of these connections. Specifically, we must state precisely which computational problems correspond to which sentences and how to measure the computational complexity of these problems. Second, we need to demonstrate this phenomenon on specific instances of proof systems, sentences and computational tasks. Third, in order to study specific instances, we need to develop methods of proving lower bounds on the complexity of proofs, and understand the limits of these methods. Additionally, our goal was also to study the role of randomness in this context.

We should point out that it is impossible to justify the feasible incompleteness thesis unconditionally, because it would imply that P = NP, which is considered to be impossible to prove with the currently available means. What is only possible is to develop a system of plausible conjectures and study relations between them. Furthermore, one can prove relativized versions (a.k.a. oracle versions) of the conjectures and prove separations of the relativized conjectures. In this direction we made considerable progress compared to the state of art at the beginning of the project. Specifically, we found a better classification of the conjectures, introduced a number of new ones, and found more connections with finite versions of the incompleteness theorem.

In the second direction the typical results are bounds on the lengths of proofs (upper and lower) that enable one to show that one proof system is weaker than another. We have proved several such results; one of the highlights of these results is our lower bound on the length of proofs of random formulas in the cutting plane proof system. This result represents a shift of the frontier in the area of random tautologies after 16 years of stagnation. Another highlight is our lower bound on the length of proofs in the random resolution proof system. These results also shed more light on the role of randomness in proof complexity.

Concerning the third direction, our research focused on two main methods of proving lower bounds on the length of proofs: the switching lemma and feasible interpolation. Our investigation of the switching lemma has not been completed, but already the results obtained so far are interesting. We studied feasible interpolation in connection with the cutting plane proof system and the related Lovasz-Schrijver system. Our results confirm the generally accepted belief that feasible interpolation should be applicable to a wider range of proof systems than it has been before.

Furthermore, we have proved a number of results on circuit complexity, non-classical logics, models of arithmetic, and introduced a set-theoretical approach to computational complexity.

Connections between proofs and computations is a classical subject in proof theory. In the mid 1970s Stephen Cook initiated the study of “feasible” proofs and efficient computations. His ideas and results developed into a new discipline called proof complexity. The picture that gradually emerged from the results proved in this area suggested that there is a connection between unprovability in weak formal systems and the computational complexity of problems associated with the unprovable sentences. Approximately 10 years ago, the PI proposed the Feasible Incompleteness Thesis, which, roughly speaking, says that this connection is not restricted to weak theories, but is a general principle stating that incompleteness can be caused by computational complexity.

The objective of project FEALORA was to study these connections between provability and computational complexity. The plan was to approach this problem from three direction. First, we need to fully understand the nature of these connections. Specifically, we must state precisely which computational problems correspond to which sentences and how to measure the computational complexity of these problems. Second, we need to demonstrate this phenomenon on specific instances of proof systems, sentences and computational tasks. Third, in order to study specific instances, we need to develop methods of proving lower bounds on the complexity of proofs, and understand the limits of these methods. Additionally, our goal was also to study the role of randomness in this context.

We should point out that it is impossible to justify the feasible incompleteness thesis unconditionally, because it would imply that P = NP, which is considered to be impossible to prove with the currently available means. What is only possible is to develop a system of plausible conjectures and study relations between them. Furthermore, one can prove relativized versions (a.k.a. oracle versions) of the conjectures and prove separations of the relativized conjectures. In this direction we made considerable progress compared to the state of art at the beginning of the project. Specifically, we found a better classification of the conjectures, introduced a number of new ones, and found more connections with finite versions of the incompleteness theorem.

In the second direction the typical results are bounds on the lengths of proofs (upper and lower) that enable one to show that one proof system is weaker than another. We have proved several such results; one of the highlights of these results is our lower bound on the length of proofs of random formulas in the cutting plane proof system. This result represents a shift of the frontier in the area of random tautologies after 16 years of stagnation. Another highlight is our lower bound on the length of proofs in the random resolution proof system. These results also shed more light on the role of randomness in proof complexity.

Concerning the third direction, our research focused on two main methods of proving lower bounds on the length of proofs: the switching lemma and feasible interpolation. Our investigation of the switching lemma has not been completed, but already the results obtained so far are interesting. We studied feasible interpolation in connection with the cutting plane proof system and the related Lovasz-Schrijver system. Our results confirm the generally accepted belief that feasible interpolation should be applicable to a wider range of proof systems than it has been before.

Furthermore, we have proved a number of results on circuit complexity, non-classical logics, models of arithmetic, and introduced a set-theoretical approach to computational complexity.