In previous work, an attempt was made to apply the schematic CERES method [8] to a formal proof with an arbitrary number of Π2 cuts (a recursive proof encapsulating the infinitary pigeonhole principle) [5]. However the derived schematic refutation for the characteristic clause set of the proof could not be expressed in the formal language provided in [8]. Without this formalization a Herbrand s...

We construct quasipolynomial-size proofs of the propositional pigeonhole principle for the fragment of the sequent calculus with no cuts between ancestors of left and right negation, weakening and contraction rules. The main construction of our argument, inspired by previous work on the monotone calculus by Atserias et al., provides formal proofs that permute the inputs of formulae computing th...

We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system beyond Resolution for which such lower bounds are ...

A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st -connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere. We prove that the propositional tautologies which encode the st -connect...

In this paper we prove general exact unprovability results that show how a threshold between provability and unprovability of a finite well-quasi-orderedness assertion of a combinatorial class is transformed by the sequence-construction, multisetconstruction, cycle-construction and labeled-tree-construction. Provability proofs use asymptotic pigeonhole principle, unprovability proofs use Weierm...

Our main result shows that a shortest proof size of tree-like resolution for the pigeonhole principle is superpolynomially larger than that of DAG-like resolution. In the proof of a lower bound, we exploit a relationship between tree-like resolution and backtracking, which has long been recognized in this eld but not been used before to give explicit results.

The schematic CERES method is a method of cut elimination for proof schemata, that is a sequence of proofs with a recursive construction. Proof schemata can be thought of as a way to circumvent the addition of an induction rule to the LK-calculus. In this work, we formalize a schematic version of the Infinitary Pigeonhole Principle (IPP), in the LKS-calculus [9], and analyse the extracted claus...

At first sight, the argument which F. P. Ramsey gave for (the infinite case of) his famous theorem from 1927, is hopelessly unconstructive. If suitably reformulated, the theorem is true intuitionistically as well as classically: we offer a proof which should convince both the classical and the intuitionistic reader.