Related papers: Recursive Prime Factorizations: Dyck Words as Numb…
We identify a recursive structure among factorizations of polynomial values into two integer factors. Polynomials for which this recursive structure characterizes all non-trivial representations of integer factorizations of the polynomial…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…
We introduce an algebraic analogue of dynamical systems, based on term rewriting. We show that a recursive function applied to the output of an iterated rewriting system defines a formal class of models into which all the main architectures…
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such…
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal…
Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful.…
An inductive logic can be formulated in which the elements are not propositions or probability distributions, but information systems. The logic is complete for information systems with binary hypotheses, i.e., it applies to all such…
Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common…
Human language understanding operates at multiple levels of granularity (e.g., words, phrases, and sentences) with increasing levels of abstraction that can be hierarchically combined. However, existing deep models with stacked layers do…
We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have…
The problem of representing the atomic elements of language in modern neural learning systems is one of the central challenges of the field of natural language processing. I present a survey of the distributional, compositional, and…
We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function. To do so we show that any such class is…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
We study some essential arithmetic properties of a new tree-based number representation, {\em hereditarily binary numbers}, defined by applying recursively run-length encoding of bijective base-2 digits. Our representation expresses giant…
The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite…
We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits…
This work investigates the algorithmic complexity of non-classical logics, focusing on superintuitionistic and modal systems. It is shown that propositional logics are usually polynomial-time reducible to their fragments with at most two…