1,147 research outputs found
Notes on degrees of relative computable categoricity
We are studying the degrees in which a computable structure is relatively
computably categoricity, i.e., computably categorcial among all non-computable
copies of the structure. Unlike the degrees of computable categoricity we can
bound the possible degrees of relative computable categoricity by the oracle
0". In the case of rigid structures the bound is in fact 0'. These estimations
are precise, in particular we can build a computable structure which is
relatively computably categorical only in the degrees above 0"
A logical basis for constructive systems
The work is devoted to Computability Logic (CoL) -- the
philosophical/mathematical platform and long-term project for redeveloping
classical logic after replacing truth} by computability in its underlying
semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article
elaborates some basic complexity theory for the CoL framework. Then it proves
soundness and completeness for the deductive system CL12 with respect to the
semantics of CoL, including the version of the latter based on polynomial time
computability instead of computability-in-principle. CL12 is a sequent calculus
system, where the meaning of a sequent intuitively can be characterized as "the
succedent is algorithmically reducible to the antecedent", and where formulas
are built from predicate letters, function letters, variables, constants,
identity, negation, parallel and choice connectives, and blind and choice
quantifiers. A case is made that CL12 is an adequate logical basis for
constructive applied theories, including complexity-oriented ones
Uncomputability and Undecidability in Economic Theory
Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.General Equilibrium Theory, Game Theory, Recursive Macro-economics, (Un)computability, (Un)decidability, Constructivity
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Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
Fuzzy stability analysis of regenerative chatter in milling
During machining, unstable self-excited vibrations known as regenerative chatter can occur, causing excessive tool wear or failure, and a poor surface finish on the machined workpiece. Consequently it is desirable to predict, and hence avoid the onset of this instability. Regenerative chatter is a function of empirical cutting coefficients, and the structural dynamics of the machine-tool system. There can be significant uncertainties in the underlying parameters, so the predicted stability limits do not necessarily agree with those found in practice. In the present study, fuzzy arithmetic techniques are applied to the chatter stability problem. It is first shown that techniques based upon interval arithmetic are not suitable for this problem due to the issue of recursiveness. An implementation of fuzzy arithmetic is then developed based upon the work of Hanss and Klimke. The arithmetic is then applied to two techniques for predicting milling chatter stability: the classical approach of Altintas, and the time-finite element method of Mann. It is shown that for some cases careful programming can reduce the computational effort to acceptable levels. The problem of milling chatter uncertainty is then considered within the framework of Ben-Haim's information-gap theory. It is shown that the presented approach can be used to solve process design problems with robustness to the uncertain parameters. The fuzzy stability bounds are then compared to previously published data, to investigate how uncertainty propagation techniques can offer more insight into the accuracy of chatter predictions
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