Related papers: Probable Counterexamples of the ABC Conjecture
Does combining a finite collection of objects infinitely many times guarantee the construction of a particular object? Here we use recursive function theory to examine the popular scenario of an infinite collection of typing monkeys…
Proof systems for the Relativized Propositional Calculus are defined and compared.
We elaborate on using importance sampling for causal reasoning, in particular for counterfactual inference. We show how this can be implemented natively in probabilistic programming. By considering the structure of the counterfactual query,…
We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method…
We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many…
The incompressibility method is an elementary yet powerful proof technique. It has been used successfully in many areas. To further demonstrate its power and elegance we exhibit new simple proofs using the incompressibility method.
We prove that for any positive integer c and any s > 0 there are representations of c as a sum a+b of two coprime positive integers a, b, such that the respective radicals are all greater than K(s)R(c)^(1-s)c^2. For the reprasentations in…
In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or…
We propose the Transcendental Encoding Conjecture for decision problems, which asserts that every language in complexity class P encodes to an algebraic real (possibly rational or algebraic irrational) under its binary characteristic…
This Chapter, "Overview of Approximate Bayesian Computation", is to appear as the first chapter in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and concepts behind ABC methods with many…
One well motivated explanation method for classifiers leverages counterfactuals which are hypothetical events identical to real observations in all aspects except for one feature. Constructing such counterfactual poses specific challenges…
Abstract argumentation offers an appealing way of representing and evaluating arguments and counterarguments. This approach can be enhanced by a probability assignment to each argument. There are various interpretations that can be ascribed…
Covering space theory is used to construct new examples of buildings.
When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
Counterfactual explanations are a popular type of explanation for making the outcomes of a decision making system transparent to the user. Counterfactual explanations tell the user what to do in order to change the outcome of the system in…
A conjecture concerning some pairs of interfering estimates for some integrals is formulated in three equivalent versions. Its importance for the the Paley problem for plurisubharmonic functions and for certain classes of extremal problems…
In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.
Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…