Related papers: A Dichotomy in Machine Knowledge
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed…
A folk theorem says higher order arithmetic has the proof theoretic strength of set theory with limited power set. This paper makes the theorem precise in terms of several axiom system based on ZF.
In inductive learning of a broad concept, an algorithm should be able to distinguish concept examples from exceptions and noisy data. An approach through recursively finding patterns in exceptions turns out to correspond to the problem of…
Computational mechanics, an approach to structural complexity, defines a process's causal states and gives a procedure for finding them. We show that the causal-state representation--an $\epsilon$-machine--is the minimal one consistent with…
The existence of a coalition strategy to achieve a goal does not necessarily mean that the coalition has enough information to know how to follow the strategy. Neither does it mean that the coalition knows that such a strategy exists. The…
We offer a view of mathematics as an experimental science where axioms play the role of foundational theories like general relativity and quantum mechanics in physics. Under this view, axioms are provisional and inferred from experience…
What would you do if you were asked to "add" knowledge? Would you say that "one plus one knowledge" is two "knowledges"? Less than that? More? Or something in between? Adding knowledge sounds strange, but it brings to the forefront…
Argumentation is a non-monotonic process. This reflects the fact that argumentation involves uncertain information, and so new information can cause a change in the conclusions drawn. However, the base logic does not need to be…
As it is well known, classical mechanics consists of several basic features like determinism, reductionism, completeness of knowledge and mechanicism. In this article the basic assumptions are discussed which underlie those features. It is…
A model of knowledge representation is described in which propositional facts and the relationships among them can be supported by other facts. The set of knowledge which can be supported is called the set of cognitive units, each having…
This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set…
We introduce and discuss, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true and false over some relevant collections of instances. A rigorous, algorithmic…
The decidability of a logical system refers to the existence of an algorithm that can determine whether any given formula in that system is a theorem. In this paper, Harrop's lemma is used to prove the decidability of quantum modal logic.
This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a…
The paper investigates whether and how AI systems can realize states of uncertainty. By adopting a functionalist and behavioral perspective, it examines how symbolic, connectionist and hybrid architectures make room for uncertainty. The…
We present a seven-pronged no-go result for quantum mechanics: a "heptalemma". It shows that seven initially plausible theses about physical reality are jointly inconsistent with the predictions of quantum mechanics, while any six are…
The definition is a common form of human expert knowledge, a building block of formal science and mathematics, a foundation for database theory and is supported in various forms in many knowledge representation and formal specification…
First-order linear temporal logic (FOLTL) is a flexible and expressive formalism capable of naturally describing complex behaviors and properties. Although the logic is in general highly undecidable, the idea of using it as a specification…
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms…
The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can…