Related papers: Non-algorithmic theory of randomness
This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a…
Some asymptotic notions for random variables are discussed. In particular, different versions of O and o for sequences of random variables are studied. The results are elementary and more or less well-known, but collected here for future…
The Random Language Model, proposed as a simple model of human languages, is defined by the averaged model of a probabilistic context-free grammar. This grammar expresses the process of sentence generation as a tree graph with nodes having…
Many complex generative systems use languages to create structured objects. We consider a model of random languages, defined by weighted context-free grammars. As the distribution of grammar weights broadens, a transition is found from a…
Non-asymptotic theory of random matrices strives to investigate the spectral properties of random matrices, which are valid with high probability for matrices of a large fixed size. Results obtained in this framework find their applications…
Kolmogorov complexity and algorithmic probability are defined only up to an additive resp. multiplicative constant, since their actual values depend on the choice of the universal reference computer. In this paper, we analyze a natural…
We shall show in this paper that there are experiments which are Bernoulli trials with success probability p > 0.5, and which have the curious feature that it is possible to correctly predict the outcome with probability > p.
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan…
Probabilistic context-free grammars have a long-term record of use as generative models in machine learning and symbolic regression. When used for symbolic regression, they generate algebraic expressions. We define the latter as equivalence…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
We study regular expressions that use variables, or parameters, which are interpreted as alphabet letters. We consider two classes of languages denoted by such expressions: under the possibility semantics, a word belongs to the language if…
We present a simple new method for proving that languages are not regular. We prove the correctness of the method, illustrate the ease of using the method on well-known examples of nonregular languages, and prove two additional theorems on…
In this paper we introduce RankPL, a modeling language that can be thought of as a qualitative variant of a probabilistic programming language with a semantics based on Spohn's ranking theory. Broadly speaking, RankPL can be used to…
This paper is an attempt to bring together two approaches to language analysis. The possible use of probabilistic information in principle-based grammars and parsers is considered, including discussion on some theoretical and computational…
Kolmogorov's setting for probability theory is given an original generalization to account for probabilities arising from Quantum Mechanics. The sample space has a central role in this presentation and random variables, i.e., observables,…
We study random words in a weighted regular language that achieve the maximal free energy using thermodynamics formalism. In particular, typical words in the language are algorithmically generated which have applications in computer…
Nonmonotonic reasoning is a pattern of reasoning that allows an agent to make and retract (tentative) conclusions from inconclusive evidence. This paper gives a possible-worlds interpretation of the nonmonotonic reasoning problem based on…
Regular expressions in an Automata Theory and Formal Languages course are mostly treated as a theoretical topic. That is, to some degree their mathematical properties and their role to describe languages is discussed. This approach fails to…
We adapt the Deutsch-Josza algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is…