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We survey recent progress in the proof complexity of strong proof systems and its connection to algebraic circuit complexity, showing how the synergy between the two gives rise to new approaches to fundamental open questions, solutions to…
Building on ideas of Gurevich and Shelah for the G\"odel Class, we present a new probabilistic proof of the finite model property for the Guarded Fragment of First-Order Logic. Our proof is conceptually simple and yields the optimal…
While Kolmogorov complexity is the accepted absolute measure of information content of an individual finite object, a similarly absolute notion is needed for the relation between an individual data sample and an individual model summarizing…
We consider the problem of inferring the probability distribution associated with a language, given data consisting of an infinite sequence of elements of the languge. We do this under two assumptions on the algorithms concerned: (i) like a…
We provide here a proof theoretic account of constraint programming that attempts to capture the essential ingredients of this programming style. We exemplify it by presenting proof rules for linear constraints over interval domains, and…
A well-known technique in estimating probabilities of rare events in general and in information theory in particular (used, e.g., in the sphere-packing bound), is that of finding a reference probability measure under which the event of…
Current discrete randomness and information conservation inequalities are over total recursive functions, i.e. restricted to deterministic processing. This restriction implies that an algorithm can break algorithmic randomness conservation…
Several logics for expressing coalitional ability under resource bounds have been proposed and studied in the literature. Previous work has shown that if only consumption of resources is considered or the total amount of resources produced…
Although many investigators affirm a desire to build reasoning systems that behave consistently with the axiomatic basis defined by probability theory and utility theory, limited resources for engineering and computation can make a complete…
We carry out a proof theoretic analysis of the wellfoundedness of recursive path orders in an abstract setting. We outline a very general termination principle and extract from its wellfoundedness proof subrecursive bounds on the size of…
Some established and also novel techniques in the field of applications of algorithmic (Kolmogorov) complexity currently co-exist for the first time and are here reviewed, ranging from dominant ones such as statistical lossless compression…
We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees…
A strong direct product theorem states that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then the overall success probability will be exponentially small in…
Suppose a language $L$ can be decided by a bounded-error randomized algorithm that runs in space $S$ and time $n \cdot \text{poly}(S)$. We give a randomized algorithm for $L$ that still runs in space $O(S)$ and time $n \cdot \text{poly}(S)$…
We develop the technique of reduced word manipulation to give a range of results concerning reduced words and permutations more generally. We prove a broad connection between pattern containment and reduced words, which specializes to our…
Muchnik's theorem about simple conditional descriprion states that for all words $a$ and $b$ there exists a short program $p$ transforming $a$ to $b$ that has the least possible length and is simple conditional on $b$. This paper presents a…
The Ku\v{c}era--G\'{a}cs theorem is a fundamental result in algorithmic randomness. It states that every infinite sequence $X$ is Turing reducible to a Martin-L\"of random $R$. This paper studies resource-bounded analogues of the…
The incompressibility method is a counting argument in the framework of algorithmic complexity that permits discovering properties that are satisfied by most objects of a class. This paper gives a preliminary insight into Kolmogorov's…
Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and…
We propose a measure based upon the fundamental theoretical concept in algorithmic information theory that provides a natural approach to the problem of evaluating $n$-dimensional complexity by using an $n$-dimensional deterministic Turing…