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In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree. The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
In computational complexity theory, EXPSPACE is the set of all decision problems solvable by a deterministic Turing machine in exponential space, i.e., in space, where is a polynomial function of . Some authors restrict to be a linear function, but most authors instead call the resulting class ESPACE. If we use a nondeterministic machine instead, we get the class NEXPSPACE, which is equal to EXPSPACE by Savitch's theorem.
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In computer science, concurrency is the ability of different parts or units of a program, algorithm, or problem to be executed out-of-order or in partial order, without affecting the outcome. This allows for parallel execution of the concurrent units, which can significantly improve overall speed of the execution in multi-processor and multi-core systems. In more technical terms, concurrency refers to the decomposability of a program, algorithm, or problem into order-independent or partially-ordered components or units of computation.
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In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second-order formulas over graphs of bounded treewidth. It is also of fundamental importance in automata theory, where the Büchi–Elgot–Trakhtenbrot theorem gives a logical characterization of the regular languages.
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A vector addition system (VAS) is one of several mathematical modeling languages for the description of distributed systems. Vector addition systems were introduced by Richard M. Karp and Raymond E. Miller in 1969, and generalized to vector addition systems with states (VASS) by John E. Hopcroft and Jean-Jacques Pansiot in 1979. Both VAS and VASS are equivalent in many ways to Petri nets introduced earlier by Carl Adam Petri.
The ZX-calculus is a rigorous graphical language for reasoning about linear maps between qubits, which are represented as string diagrams called ZX-diagrams. A ZX-diagram consists of a set of generators called spiders that represent specific tensors. These are connected together to form a tensor network similar to Penrose graphical notation. Due to the symmetries of the spiders and the properties of the underlying category, topologically deforming a ZX-diagram does not affect the linear map it represents. In addition to the equalities between ZX-diagrams that are generated by topological deformations, the calculus also has a set of graphical rewrite rules for transforming diagrams into one another. The ZX-calculus is universal in the sense that any linear map between qubits can be represented as a diagram, and different sets of graphical rewrite rules are complete for different families of linear maps. ZX-diagrams can be seen as a generalisation of quantum circuit notation.
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References
- ↑ Vorobyov, Sergei; Voronkov, Andrei (1998), "Complexity of Nonrecursive Logic Programs with Complex Values", Proceedings of the Seventeenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS '98), New York, NY, USA: ACM, pp. 244–253, CiteSeerX 10.1.1.39.8822 , doi:10.1145/275487.275515, ISBN 978-0-89791-996-8 .
- ↑ Stockmeyer, Larry J. (1974), The Complexity of Decision Problems in Automata Theory and Logic (PDF), Ph.D. dissertation, Massachusetts Institute of Technology
- ↑ Libkin, Leonid (2006), "Logics for unranked trees: an overview", Logical Methods in Computer Science, 2 (3): 3:2, 31, arXiv: cs.LO/0606062 , doi:10.2168/LMCS-2(3:2)2006, MR 2295773 .
- ↑ Vorobyov, Sergei (1996), "An improved lower bound for the elementary theories of trees", Automated Deduction — CADE-13: 13th International Conference on Automated Deduction New Brunswick, NJ, USA, July 30 – August 3, 1996, Proceedings, Lecture Notes in Computer Science, vol. 1104, Springer, pp. 275–287, CiteSeerX 10.1.1.39.1499 , doi:10.1007/3-540-61511-3_91, ISBN 978-3-540-61511-8 .
- ↑ Pratt-Hartmann, Ian; Szwast, Wieslaw; Tendera, Lidia (2016), "Quine's fluted fragment is non-elementary", in Talbot, Jean-Marc; Regnier, Laurent (eds.), 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France, LIPIcs, vol. 62, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 39:1–39:21, doi: 10.4230/LIPIcs.CSL.2016.39
- ↑ Statman, Richard (1979), "The typed λ-calculus is not elementary recursive", Theoretical Computer Science , 9: 73–81, doi:10.1016/0304-3975(79)90007-0, hdl: 2027.42/23535 .
- ↑ Czerwiński, Wojciech; Orlikowski, Łukasz (2021). Reachability in Vector Addition Systems is Ackermann-complete. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). arXiv: 2104.13866 .
- 1 2 Brubaker, Ben (4 December 2023). "An Easy-Sounding Problem Yields Numbers Too Big for Our Universe". Quanta Magazine .
- ↑ Leroux, Jerome (February 2022). "The Reachability Problem for Petri Nets is Not Primitive Recursive". IEEE Xplore. IEEE: 1241–1252. arXiv: 2104.12695 . doi:10.1109/FOCS52979.2021.00121. ISBN 978-1-6654-2055-6.