### The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs [chapter]

Albert Atserias
2013 Lecture Notes in Computer Science
In recent years there has been some progress in our understanding of the proof-search problem for very low-depth proof systems, e.g. proof systems that manipulate formulas of very low complexity such as clauses (i.e. resolution), DNF-formulas (i.e. R(k) systems), or polynomial inequalities (i.e. semi-algebraic proof systems). In this talk I will overview this progress. I will start with bounded-width resolution, whose specialized proof-search algorithm is as easy as uninteresting, but whose
more » ... f-search problem is unintentionally solved by certain versions of conflict-driven clause-learning algorithms with restarts. I will continue with R(k) systems, whose proof-search problem turned out to hide the complexity of certain two-player games of interest in the area of systems synthesis and verification. And I will close with bounded-degree semialgebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems. Research partially supported by project TIN2010-20967-C04-05 (TASSAT). makes it trivially automatizable in polynomial time, but for a silly reason. For contrast, interesting proof systems in this sense do include propositional resolution, for example, whose reasoning power is able to produce short proofs of non-trivial tautologies arising in multiple application contexts. For example, resolution admits polynomial-size proofs of the least-number principle (every finite linear order has a least element)  , which underlies many inductive proofs. The purpose of this paper is to discuss the status of the proof-search problem for inference-based proof systems that work with formulas of very low complexity. These include resolution or DNF-resolution, which work with clauses and DNFformulas, respectively, and semi-algebraic proofs, which work with polynomial inequalities over the reals. We also take the opportunity to discuss the connection to some of the lift-and-project methods in mathematical programming. Inference-based proof systems Most classical proof systems are inference-based: starting with a set of given hypotheses, some conclusions are produced syntactically by means of one or more inference rules, which are then added to the set of hypotheses to proceed. In producing proofs for a tautology, an inference-based proof system starts with the empty set of hypotheses and the goal is to produce the tautology. Of course this will mean that the set of inference rules includes some axioms, i.e. inference rules that can be fired without any hypotheses. In producing refutations for a contradiction, an inference-based proof system starts with the given contradiction and the goal is to produce some blatant inconsistency. All typical inference-based proof systems manipulate some particular type of formulas, be them clauses, DNF or CNF-formulas, propositional formulas of some higher but fixed depth of alternations between disjunctions and conjunctions, general propositional formulas, polynomial equations over some ring, polynomial inequalities over some ordered ring, disjunctions of those, decision trees branching on variables or more complicated formulas, binary decision diagrams of various sorts, Boolean circuits, etc. The inference rules are typically some more or less obvious, non-interesting, and polynomially checkable ways of producing some logical consequence of the hypotheses. In this sense, what makes an inference-based proof system more or less powerful is the expressive power of the type of formulas it manipulates. Systems that manipulate propositional formulas In resolution, the formulas are clauses, disjunctions of variables or negated variables, and the only inference rule is the resolution rule: