2,912 research outputs found
nested PLS
In this note we will introduce a class of search problems, called nested
Polynomial Local Search (nPLS) problems, and show that definable NP search
problems, i.e., -definable functions in are characterized
in terms of the nested PLS
A proof of P!=NP
We show that it is provable in PA that there is an arithmetically definable
sequence of -sentences, such that
- PRA+ is -sound and
-complete
- the length of is bounded above by a polynomial function of
with positive leading coefficient
- PRA+ always proves 1-consistency of PRA+.
One has that the growth in logical strength is in some sense "as fast as
possible", manifested in the fact that the total general recursive functions
whose totality is asserted by the true -sentences in the sequence
are cofinal growth-rate-wise in the set of all total general recursive
functions. We then develop an argument which makes use of a sequence of
sentences constructed by an application of the diagonal lemma, which are
generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction
as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth
in Mathematics". The argument establishes the result that it is provable in PA
that . We indicate how to pull the argument all the way down into
EFA
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
Theories for TC0 and Other Small Complexity Classes
We present a general method for introducing finitely axiomatizable "minimal"
two-sorted theories for various subclasses of P (problems solvable in
polynomial time). The two sorts are natural numbers and finite sets of natural
numbers. The latter are essentially the finite binary strings, which provide a
natural domain for defining the functions and sets in small complexity classes.
We concentrate on the complexity class TC^0, whose problems are defined by
uniform polynomial-size families of bounded-depth Boolean circuits with
majority gates. We present an elegant theory VTC^0 in which the provably-total
functions are those associated with TC^0, and then prove that VTC^0 is
"isomorphic" to a different-looking single-sorted theory introduced by
Johannsen and Pollet. The most technical part of the isomorphism proof is
defining binary number multiplication in terms a bit-counting function, and
showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc
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