Related papers: Some Mathematicians Are Not Turing Machines
In various models of one-way pushdown automata, the explicit use of two designated endmarkers on a read-once input tape has proven to be extremely useful for making a conscious, final decision on the acceptance/rejection of each input word…
This paper reviews the Church-Turing Thesis (or rather, theses) with reference to their origin and application and considers some models of "hypercomputation", concentrating on perhaps the most straight-forward option: Zeno machines (Turing…
We consider computations of a Turing machine subjected to noise. In every step, the action (the new state and the new content of the observed cell, the direction of the head movement) can differ from that prescribed by the transition…
A fundamental question is whether Turing machines can model all reasoning processes. We introduce an existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the…
The current state-of-the-art in artificial intelligence is impressive, especially in terms of mastery of language, but not so much in terms of mathematical reasoning. What could be missing? Can we learn something useful about that gap from…
The ``impossibility theorem'' -- which is considered foundational in algorithmic fairness literature -- asserts that there must be trade-offs between common notions of fairness and performance when fitting statistical models, except in two…
We recently described a formalism for reasoning with if-then rules that re expressed with different levels of firmness [18]. The formalism interprets these rules as extreme conditional probability statements, specifying orders of magnitude…
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is…
It is shown that an equiprobability hypothesis leads to a scenario in which it is possible to predict the outcome of a single toss of a fair coin with a success probability greater than 50%. We discuss whether this hypothesis might be…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
We formulate the Hauptvermutung of Causal Set Theory in two mathematically well-defined but different ways one of which turns out to be wrong and the other one turns out to be true. A further result is that the Hauptvermutung is true if we…
Mermin states in a recent paper that his nontechnical version of Bell's theorem stands and is not invalidated by time and setting dependent instrument parameters as claimed in one of our previous papers. We identify a number of…
While there has been some discussion on how Symbolic Computation could be used for AI there is little literature on applications in the other direction. However, recent results for quantifier elimination suggest that, given enough example…
(l) I have enough evidence to render the sentence S probable. (la) So, relative to what I know, it is rational of me to believe S. (2) Now that I have more evidence, S may no longer be probable. (2a) So now, relative to what I know, it is…
The present work proves that P=NP. The proof, presented in this work, is a constructive one: the program of a polynomial time deterministic multi-tape Turing machine M_ExistsAcceptingPath, that determines if there exists an accepting…
The generally accepted wisdom in computational circles is that pure proof verification is a solved problem and that the computationally hard elements and fertile areas of study lie in proof discovery. This wisdom presumably does hold for…
Markov's principle is a statement that originated in the Russian school of Constructive Mathematics and stated originally that "if it is impossible that an algorithm does not terminate, then it will terminate". This principle has been…
We show that the question whether a term is typable is decidable for type systems combining inclusion polymorphism with parametric polymorphism provided the type constructors are at most unary. To prove this result we first reduce the…
The CMI Millennium "P vs NP Problem" can be resolved e.g. if one shows at least one counterexample to the conjecture "P is equal to NP". A certain class of problems being such counterexamples is formulated. This implies the rejection of the…
Semi-unification is the combination of first-order unification and first-order matching. The undecidability of semi-unification has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s by Turing reduction from Turing machine immortality…