Results 1 
4 of
4
Informationtheoretic Limitations of Formal Systems
 JOURNAL OF THE ACM
, 1974
"... An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these ..."
Abstract

Cited by 47 (8 self)
 Add to MetaCart
An attempt is made to apply informationtheoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms.
On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers
 J. ASSOC. COMPUT. MACH
, 1969
"... It is suggested that there are infinite computable sets of natural numbers with the property that no infinite subset can be computed more simply or more quickly than the whole set. Attempts to establish this without restricting in any way the computer involved in the calculations are not entirely su ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
It is suggested that there are infinite computable sets of natural numbers with the property that no infinite subset can be computed more simply or more quickly than the whole set. Attempts to establish this without restricting in any way the computer involved in the calculations are not entirely successful. A hypothesis concerning the computer makes it possible to exhibit sets without simpler subsets. A second and analogous hypothesis then makes it possible to prove that these sets are also without subsets which can be computed more rapidly than the whole set. It is then demonstrated that there are computers which satisfy both hypotheses. The general theory is momentarily set aside and a particular Turing machine is studied. Lastly, it is shown that the second hypothesis is more restrictive then requiring the computer to be capable of calculating all infinite computable sets of natural numbers.
Measure Independent G6del SpeedUps and the Relative Difficulty of Recognizing Sets
"... We provide and interpret a new measure independent characterization of the G6del speedup phenomenon. In particular, we prove a theorem that demonstrates the indifference of the concept of a measure independent G6del speedup to an apparent weakening of its definition that is obtained by requiring ..."
Abstract
 Add to MetaCart
We provide and interpret a new measure independent characterization of the G6del speedup phenomenon. In particular, we prove a theorem that demonstrates the indifference of the concept of a measure independent G6del speedup to an apparent weakening of its definition that is obtained by requiring only those measures appearing in some fixed Blum complexity measure to participate in the speedup, and by deleting the &quot;for all r &quot; condition from the definition so as to relax the required amount of speedup. We interpret our results as correlating the relative difficulty of mechanically recognizing theories with the relative power and the relative abstractness of the theories. We conclude by providing two open problems concerning possible similarities and relationships between the Blum speedability and G6del speedup phenomena.
A CONNECTION BETWEEN BLUM SPEEDABLE SETS AND GODELâ€™S SPEEDUP THEOREM
"... In the midsixties BLUM [2] announced his speedup theorem, which affirmed the existence of recursive sets having, in some sense, no optimal recognizers. It was observed ..."
Abstract
 Add to MetaCart
In the midsixties BLUM [2] announced his speedup theorem, which affirmed the existence of recursive sets having, in some sense, no optimal recognizers. It was observed