52 research outputs found
The origins of the halting problem
[EN] The halting problem is a prominent example of undecidable problem and its formulation and undecidability proof is usually attributed to Turing's 1936 landmark paper. Copeland noticed in 2004, though, that it was so named and, apparently, first stated in a 1958 book by Martin Davis. We provide additional arguments partially supporting this claim as follows: (i) with a focus on computable (real) numbers with infinitely many digits (e.g., pi), in his paper Turing was not concerned with halting machines; (ii) the two decision problems considered by Turing concern the ability of his machines to produce specific kinds of outputs, rather than reaching a halting state, something which was missing from Turing's notion of computation; and (iii) from 1936 to 1958, when considering the literature of the field no paper refers to any "halting problem" of Turing Machines until Davis' book. However, there were important preliminary contributions by (iv) Church, for whom termination was part of his notion of computation (for the lambda-calculus), and (v) Kleene, who essentially formulated, in his 1952 book, what we know as the halting problem now.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098.Lucas Alba, S. (2021). The origins of the halting problem. Journal of Logical and Algebraic Methods in Programming. 121:1-9. https://doi.org/10.1016/j.jlamp.2021.1006871912
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
The Un(solv)able Problem
After a years-long intellectual journey, three mathematicians have discovered that a problem of central importance in physics is impossible to solveâand that means other big questions may be undecidable, too.
In Brief:
Kurt GĂśdel famously discovered in the 1930s that some statements are impossible to prove true or falseâthey will always be âundecidable.â
Mathematicians recently set out to discover whether a certain fundamental problem in quantum physicsâthe so-called spectral gap questionâfalls into this category. The spectral gap refers to the energy difference between the lowest energy state a material can occupy and the next state up.
After three years of blackboard brainstorming, midnight calculating and much theorizing over coffee, the mathematicians produced a 146-page proof that the spectral gap problem is, in fact, undecidable. The result raises the possibility that other important questions may likewise be unanswerable
Is thinking computable?
Strong artificial intelligence claims that conscious thought can arise in computers containing the right algorithms even though none of the programs or components of those computers understand which is going on. As proof, it asserts that brains are finite webs of neurons, each with a definite function governed by the laws of physics; this web has a set of equations that can be solved (or simulated) by a sufficiently powerful computer. Strong AI claims the Turing test as a criterion of success. A recent debate in Scientific American concludes that the Turing test is not sufficient, but leaves intact the underlying premise that thought is a computable process. The recent book by Roger Penrose, however, offers a sharp challenge, arguing that the laws of quantum physics may govern mental processes and that these laws may not be computable. In every area of mathematics and physics, Penrose finds evidence of nonalgorithmic human activity and concludes that mental processes are inherently more powerful than computational processes
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