Polynomial time operations in explicit mathematics

In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on = {0, 1}* are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theor

Some observations on the logical foundations of inductive theorem proving

In this paper we study the logical foundations of automated inductive theorem proving. To that aim we first develop a theoretical model that is centered around the difficulty of finding induction axioms which are sufficient for proving a goal. Based on this model, we then analyze the following aspects: the choice of a proof shape, the choice of an induction rule and the language of the induction formula. In particular, using model-theoretic techniques, we clarify the relationship between notions of inductiveness that have been considered in the literature on automated inductive theorem proving. This is a corrected version of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017

The Complexity of Distributed Edge Coloring with Small Palettes

The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree $\Delta$. In this paper we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. 1. We simplify the \emph{round elimination} technique of Brandt et al. and prove that $(2\Delta-2)$-edge coloring requires $\Omega(\log_\Delta \log n)$ time w.h.p. and $\Omega(\log_\Delta n)$ time deterministically, even on trees. The simplified technique is based on two ideas: the notion of an irregular running time and some general observations that transform weak lower bounds into stronger ones. 2. We give a randomized edge coloring algorithm that can use palette sizes as small as $\Delta + \tilde{O}(\sqrt{\Delta})$, which is a natural barrier for randomized approaches. The running time of the algorithm is at most $O(\log\Delta \cdot T_{LLL})$, where $T_{LLL}$ is the complexity of a permissive version of the constructive Lovasz local lemma. 3. We develop a new distributed Lovasz local lemma algorithm for tree-structured dependency graphs, which leads to a $(1+\epsilon)\Delta$-edge coloring algorithm for trees running in $O(\log\log n)$ time. This algorithm arises from two new results: a deterministic $O(\log n)$-time LLL algorithm for tree-structured instances, and a randomized $O(\log\log n)$-time graph shattering method for breaking the dependency graph into independent $O(\log n)$-size LLL instances. 4. A natural approach to computing $(\Delta+1)$-edge colorings (Vizing's theorem) is to extend partial colorings by iteratively re-coloring parts of the graph. We prove that this approach may be viable, but in the worst case requires recoloring subgraphs of diameter $\Omega(\Delta\log n)$. This stands in contrast to distributed algorithms for Brooks' theorem, which exploit the existence of $O(\log_\Delta n)$-length augmenting paths

Unprovability of strong complexity lower bounds in bounded arithmetic

While there has been progress in establishing the unprovability of complexity statements in lower fragments of bounded arithmetic, understanding the limits of Jeˇr ́abek’s theory APC1 [Jeˇr07a] and of higher levels of Buss’s hierarchy Si 2 [Bus86] has been a more elusive task. Even in the more restricted setting of Cook’s theory PV [Coo75], known results often rely on a less natural formalization that encodes a complexity statement using a collection of sentences instead of a single sentence. This is done to reduce the quantifier complexity of the resulting sentences so that standard witnessing results can be invoked. In this work, we establish unprovability results for stronger theories and for sentences of higher quantifier complexity. In particular, we unconditionally show that APC1 cannot prove strong complexity lower bounds separating the third level of the polynomial hierarchy. In more detail, we consider non-uniform average-case separations, and establish that APC1 cannot prove a sentence stating that ∀n ≥ n0 ∃ fn ∈ Π3-SIZE[nd] that is (1/n)-far from every Σ3-SIZE[2nδ] circuit. This is a consequence of a much more general result showing that, for every i ≥ 1, strong separations for Πi-SIZE[poly(n)] versus Σi-SIZE[2nΩ(1)] cannot be proved in the theory Ti PV consisting of all true ∀Σb i−1- sentences in the language of Cook’s theory PV. Our argument employs a convenient game-theoretic witnessing result that can be applied to sentences of arbitrary quantifier complexity. We combine it with extensions of a technique introduced by Kraj ́ıˇcek [Kra11] that was recently employed by Pich and Santhanam [PS21] to establish the unprovability of lower bounds in PV (i.e., the case i = 1 above, but under a weaker formalization) and in a fragment of APC1

Nondeterminism and an abstract formulation of Ne\v{c}iporuk's lower bound method

A formulation of "Ne\v{c}iporuk's lower bound method" slightly more inclusive than the usual complexity-measure-specific formulation is presented. Using this general formulation, limitations to lower bounds achievable by the method are obtained for several computation models, such as branching programs and Boolean formulas having access to a sublinear number of nondeterministic bits. In particular, it is shown that any lower bound achievable by the method of Ne\v{c}iporuk for the size of nondeterministic and parity branching programs is at most $O(n^{3/2}/\log n)$