Proof Complexity of Modal Resolution

We investigate the proof complexity of modal resolution systems developed by Nalon and Dixon (J Algorithms 62(3–4):117–134, 2007) and Nalon et al. (in: Automated reasoning with analytic Tableaux and related methods—24th international conference, (TABLEAUX’15), pp 185–200, 2015), which form the basis of modal theorem proving (Nalon et al., in: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI’17), pp 4919–4923, 2017). We complement these calculi by a new tighter variant and show that proofs can be efficiently translated between all these variants, meaning that the calculi are equivalent from a proof complexity perspective. We then develop the first lower bound technique for modal resolution using Prover–Delayer games, which can be used to establish “genuine” modal lower bounds for size of dag-like modal resolution proofs. We illustrate the technique by devising a new modal pigeonhole principle, which we demonstrate to require exponential-size proofs in modal resolution. Finally, we compare modal resolution to the modal Frege systems of Hrubeš (Ann Pure Appl Log 157(2–3):194–205, 2009) and obtain a “genuinely” modal separation.

Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size .

Combinatorial proofs of two theorems of Lutz and Stull

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if \[K \subset {\mathbb{R}^n}\] is any set with equal Hausdorff and packing dimensions, then \begin{equation} \[{\dim _{\text{H}}}{\pi _e}(K) = \min \{ {\dim _{\text{H}}}{\text{ }}K{\text{, 1}}\} \] \end{equation} for almost every \[e \in {S^{n - 1}}\] . Here \[{\pi _e}\] stands for orthogonal projection to span ( \[e\] ). The primary purpose of this paper is to present proofs for Lutz and Stull’s projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman’s “potential theoretic” method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections to m-planes in \[{\mathbb{R}^n}\] , for all \[0 < m < n\] .

The weakness of the pigeonhole principle under hyperarithmetical reductions

The infinite pigeonhole principle for 2-partitions ([Formula: see text]) asserts the existence, for every set [Formula: see text], of an infinite subset of [Formula: see text] or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that [Formula: see text] admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every [Formula: see text] set, of an infinite low[Formula: see text] subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak et al.

Pursue an easier approach, considering the Pigeonhole Principle

“Pursue an easier approach, considering the Pigeonhole Principle” offers an introduction to a mathematical principle by way of answering a version of a popular mathematics question: “are there are two non-bald people in London with the same number of hairs on their heads?” Formally, the Pigeonhole Principle is stated: If n items are put into m containers and n>m, then at least one container contains more than one item. The discussion is illustrated with numerous hand-drawn sketches. The Pigeonhole Principle allows readers to solve problems that seem to require counting, without ever having to count. Mathematics students and enthusiasts are encouraged to pursue engaging, if unconventional, paths as they work toward solutions in mathematics and life. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.