Related papers: Five Conferences on Undecidability
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
According to Chaitin, G\"odel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable…
We present a propositional logic %which can be used to reason about the uncertainty of events, where the uncertainty is modeled by a set of probability measures assigning an interval of probability to each event. We give a sound and…
In light of G\"{o}del's undecidability results (incomplete theorems) for math, quantum indeterminism indicates that physics and the Universe may be indeterministic, incomplete, and open in nature, and therefore demand no single unification…
Determining whether a program terminates is a central problem in computer science. Turing's Halting Problem established termination as undecidable, showing that no algorithm can universally determine termination for all programs and inputs.…
This paper presents a theory of systemic undecidability, reframing incomputability as a structural property of systems rather than a localized feature of specific functions or problems. We define a notion of causal embedding and prove a…
The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the…
We commonly think of mathematics as bringing precision to application domains, but its relationship with computer science is more complex. This experience report on the use of Racket and Haskell to teach a required first university CS…
We show that many classical decision problems about 1-counter omega-languages, context free omega-languages, or infinitary rational relations, are $\Pi_2^1$-complete, hence located at the second level of the analytical hierarchy, and…
Hilbert's Entscheidungsproblem has given rise to a broad and productive line of research in mathematical logic, where the classification process of decidable classes of first-order sentences represent only one of the remarkable results.…
While modal extensions of decidable fragments of first-order logic are usually undecidable, their monodic counterparts, in which formulas in the scope of modal operators have at most one free variable, are typically decidable. This only…
Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…
Those lectures revolve around the following problem: given a system of n real polynomials in n variables, count the number of real roots. The first lecture is a course on Newton iteration and alpha-theory. The second describes an…
Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form $\exists y. \forall x. (\Box_x P(x,y) \supset\Diamond_y P(y,x))$. Like First order…
Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (\beta) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order…
We prove the decidability for a class of languages which extend BST and NP-completeness for a subclass of them. The languages BST extended with unordered cartesian product, BST extended with ordered cartesian product and BST extended with…
G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing…
We study the problem of completely automatically verifying uninterpreted programs---programs that work over arbitrary data models that provide an interpretation for the constants, functions and relations the program uses. The verification…
For any first order theory T we construct a Boolean valued model M, in which precisely the T--provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a first order…
Some common fallacies about fundamental themes of Logic are exposed: the First and Second incompleteness Theorem interpretations, Chaitin's various superficialities and the usual classification of the axiomatic Theories in function of its…