Related papers: Turing-Church thesis, constructve mathematics and …
Life is confronted with computation problems in a variety of domains including animal behavior, single-cell behavior, and embryonic development. Yet we currently do not know of a naturally existing biological system that is capable of…
While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool…
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically…
Computability logic is a formal theory of computational tasks and resources. Its formulas represent interactive computational problems, logical operators stand for operations on computational problems, and validity of a formula is…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
Classical probability theory is formulated using sets. In this paper, we extend classical probability theory with propositional computability logic. Unlike other formalisms, computability logic is built on the notion of events/games, which…
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…
Incompleteness theorems of Godel, Turing, Chaitin, and Algorithmic Information Theory have profound epistemological implications. Incompleteness limits our ability to ever understand every observable phenomenon in the universe.…
Turing Machines (TMs) are the canonical model of computation in computer science and physics. We combine techniques from algorithmic information theory and stochastic thermodynamics to analyze the thermodynamic costs of TMs. We consider two…
Evidence-based reasoning is at the core of many problem-solving and decision-making tasks in a wide variety of domains. Generalizing from the research and development of cognitive agents in several such domains, this paper presents progress…
Given a represented space (in the sense of TTE theory), an appropriate representation is constructed for the Moschovakis extension of its carrier (with paying attention to the cases of effective topological spaces and effective metric…
The recently initiated approach called computability logic is a formal theory of interactive computation. See a comprehensive online source on the subject at http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a soundness…
We show that Church's thesis, the axiom stating that all functions on the naturals are computable, does not hold in the cubical assemblies model of cubical type theory. We show that nevertheless Church's thesis is consistent with univalent…
Pure type systems arise as a generalisation of simply typed lambda calculus. The contemporary development of mathematics has renewed the interest in type theories, as they are not just the object of mere historical research, but have an…
Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented…
Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an…
This paper presents a plausible reasoning system to illustrate some broad issues in knowledge representation: dualities between different reasoning forms, the difficulty of unifying complementary reasoning styles, and the approximate nature…
Computational Logic is the use of computers to establish facts in a logical formalism. Originating in 19th-century attempts to understand the nature of mathematical reasoning, the subject now comprises a wide variety of formalisms,…
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the…
We describe a method to axiomatize computations in deterministic Turing machines. When applied to computations in non-deterministic Turing machines, this method may produce contradictory (and therefore trivial) theories, considering…