3,028 research outputs found
Consequences of nonclassical measurement for the algorithmic description of continuous dynamical systems
Continuous dynamical systems intuitively seem capable of more complex behavior than discrete systems. If analyzed in the framework of the traditional theory of computation, a continuous dynamical system with countably many quasistable states has at least the computational power of a universal Turing machine. Such an analysis assumes, however, the classical notion of measurement. If measurement is viewed nonclassically, a continuous dynamical system cannot, even in principle, exhibit behavior that cannot be simulated by a universal Turing machine
Proof of Church's Thesis
We prove that if our calculating capability is that of a universal Turing
machine with a finite tape, then Church's thesis is true. This way we
accomplish Post (1936) program.Comment: 6 page
Wang's B machines are efficiently universal, as is Hasenjaeger's small universal electromechanical toy
In the 1960's Gisbert Hasenjaeger built Turing Machines from
electromechanical relays and uniselectors. Recently, Glaschick reverse
engineered the program of one of these machines and found that it is a
universal Turing machine. In fact, its program uses only four states and two
symbols, making it a very small universal Turing machine. (The machine has
three tapes and a number of other features that are important to keep in mind
when comparing it to other small universal machines.) Hasenjaeger's machine
simulates Hao Wang's B machines, which were proved universal by Wang.
Unfortunately, Wang's original simulation algorithm suffers from an exponential
slowdown when simulating Turing machines. Hence, via this simulation,
Hasenjaeger's machine also has an exponential slowdown when simulating Turing
machines. In this work, we give a new efficient simulation algorithm for Wang's
B machines by showing that they simulate Turing machines with only a polynomial
slowdown. As a second result, we find that Hasenjaeger's machine also
efficiently simulates Turing machines in polynomial time. Thus, Hasenjaeger's
machine is both small and fast. In another application of our result, we show
that Hooper's small universal Turing machine simulates Turing machines in
polynomial time, an exponential improvement.Comment: 18 pages, 1 figure, 1 table, Conference: Turing in context II -
History and Philosophy of Computing, 201
P is not equal to NP
SAT is not in P, is true and provable in a simply consistent extension B' of
a first order theory B of computing, with a single finite axiom characterizing
a universal Turing machine. Therefore, P is not equal to NP, is true and
provable in a simply consistent extension B" of B.Comment: In the 2nd printing the proof, in the 1st printing, of theorem 1 is
divided into three parts a new lemma 4, a new corollary 8, and the remaining
part of the original proof. The 2nd printing contains some simplifications,
more explanations, but no error has been correcte
Probing quantum-classical boundary with compression software
We experimentally demonstrate that it is impossible to simulate quantum
bipartite correlations with a deterministic universal Turing machine. Our
approach is based on the Normalized Information Distance (NID) that allows the
comparison of two pieces of data without detailed knowledge about their origin.
Using NID, we derive an inequality for output of two local deterministic
universal Turing machines with correlated inputs. This inequality is violated
by correlations generated by a maximally entangled polarization state of two
photons. The violation is shown using a freely available lossless compression
program. The presented technique may allow to complement the common statistical
interpretation of quantum physics by an algorithmic one.Comment: 7 pages, 6 figure
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