1,664 research outputs found
Relativized Propositional Calculus
Proof systems for the Relativized Propositional Calculus are defined and
compared.Comment: 8 page
Bounded Relativization
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ?, we say that a statement is ?-relativizing if the statement holds relative to every oracle ? ? ?. It is easy to see that every result that relativizes also ?-relativizes for every complexity class ?. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing.
First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ? > 0, BPE^{MCSP}/2^{?n} ? SIZE[2?/n].
We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021).
Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ? L. For example:
- Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ? BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ? L.
- Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ? L.
- Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ? L.
In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ? EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible
Algebraic Methods in Computational Complexity
From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 “Algebraic Methods in Computational Complexity “ was held in Schloss Dagstuhl-Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
09421 Abstracts Collection -- Algebraic Methods in Computational Complexity
From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 ``Algebraic Methods in Computational Complexity \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Quantum Measure Theory and its Interpretation
We propose a realistic, spacetime interpretation of quantum theory in which
reality constitutes a *single* history obeying a "law of motion" that makes
definite, but incomplete, predictions about its behavior. We associate a
"quantum measure" |S| to the set S of histories, and point out that |S|
fulfills a sum rule generalizing that of classical probability theory. We
interpret |S| as a "propensity", making this precise by stating a criterion for
|S|=0 to imply "preclusion" (meaning that the true history will not lie in S).
The criterion involves triads of correlated events, and in application to
electron-electron scattering, for example, it yields definite predictions about
the electron trajectories themselves, independently of any measuring devices
which might or might not be present. (So we can give an objective account of
measurements.) Two unfinished aspects of the interpretation involve
*conditonal* preclusion (which apparently requires a notion of coarse-graining
for its formulation) and the need to "locate spacetime regions in advance"
without the aid of a fixed background metric (which can be achieved in the
context of conditional preclusion via a construction which makes sense both in
continuum gravity and in the discrete setting of causal set theory).Comment: Changes to original version: correction to the description of the
quantum measure in the non relativistic case; some rewording in other places;
a few typos corrected. 23 pages, plaintex with 7 eps figure
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