132,866 research outputs found
Two-message quantum interactive proofs and the quantum separability problem
Suppose that a polynomial-time mixed-state quantum circuit, described as a
sequence of local unitary interactions followed by a partial trace, generates a
quantum state shared between two parties. One might then wonder, does this
quantum circuit produce a state that is separable or entangled? Here, we give
evidence that it is computationally hard to decide the answer to this question,
even if one has access to the power of quantum computation. We begin by
exhibiting a two-message quantum interactive proof system that can decide the
answer to a promise version of the question. We then prove that the promise
problem is hard for the class of promise problems with "quantum statistical
zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp
reduction from the QSZK-complete promise problem "quantum state
distinguishability" to our quantum separability problem. By exploiting Knill's
efficient encoding of a matrix description of a state into a description of a
circuit to generate the state, we can show that our promise problem is NP-hard
with respect to Cook reductions. Thus, the quantum separability problem (as
phrased above) constitutes the first nontrivial promise problem decidable by a
two-message quantum interactive proof system while being hard for both NP and
QSZK. We also consider a variant of the problem, in which a given
polynomial-time mixed-state quantum circuit accepts a quantum state as input,
and the question is to decide if there is an input to this circuit which makes
its output separable across some bipartite cut. We prove that this problem is a
complete promise problem for the class QIP of problems decidable by quantum
interactive proof systems. Finally, we show that a two-message quantum
interactive proof system can also decide a multipartite generalization of the
quantum separability problem.Comment: 34 pages, 6 figures; v2: technical improvements and new result for
the multipartite quantum separability problem; v3: minor changes to address
referee comments, accepted for presentation at the 2013 IEEE Conference on
Computational Complexity; v4: changed problem names; v5: updated references
and added a paragraph to the conclusion to connect with prior work on
separability testin
Automated Reasoning and Presentation Support for Formalizing Mathematics in Mizar
This paper presents a combination of several automated reasoning and proof
presentation tools with the Mizar system for formalization of mathematics. The
combination forms an online service called MizAR, similar to the SystemOnTPTP
service for first-order automated reasoning. The main differences to
SystemOnTPTP are the use of the Mizar language that is oriented towards human
mathematicians (rather than the pure first-order logic used in SystemOnTPTP),
and setting the service in the context of the large Mizar Mathematical Library
of previous theorems,definitions, and proofs (rather than the isolated problems
that are solved in SystemOnTPTP). These differences poses new challenges and
new opportunities for automated reasoning and for proof presentation tools.
This paper describes the overall structure of MizAR, and presents the automated
reasoning systems and proof presentation tools that are combined to make MizAR
a useful mathematical service.Comment: To appear in 10th International Conference on. Artificial
Intelligence and Symbolic Computation AISC 201
Carnap: an Open Framework for Formal Reasoning in the Browser
This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap’s design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap’s implementation, along with its current and projected pedagogical applications
ATP and Presentation Service for Mizar Formalizations
This paper describes the Automated Reasoning for Mizar (MizAR) service, which
integrates several automated reasoning, artificial intelligence, and
presentation tools with Mizar and its authoring environment. The service
provides ATP assistance to Mizar authors in finding and explaining proofs, and
offers generation of Mizar problems as challenges to ATP systems. The service
is based on a sound translation from the Mizar language to that of first-order
ATP systems, and relies on the recent progress in application of ATP systems in
large theories containing tens of thousands of available facts. We present the
main features of MizAR services, followed by an account of initial experiments
in finding proofs with the ATP assistance. Our initial experience indicates
that the tool offers substantial help in exploring the Mizar library and in
preparing new Mizar articles
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