John Baez

Logic In Computer Science (LICS 2009), UCLA

August 13, 2009

Computation and the Periodic Table

In physics, Feynman diagrams are used to reason about quantum processes. Similar diagrams can also be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of topological quantum field theory and quantum computation, it became clear that diagrammatic reasoning takes advantage of an extensive network of interlocking analogies between physics, topology, logic and computation. These analogies can be made precise using the formalism of symmetric monoidal closed categories. But symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized "periodic table" of k-tuply monoidal n-categories. This raises the question of how these analogies extend. An important clue comes from the way symmetric monoidal closed 2-categories describe rewrite rules in the lambda calculus and multiplicative intuitionistic linear logic. This talk is based on work in progress with Paul-André Melliès and Mike Stay.

Click on this to see the transparencies of the talk:

• Computation and the Periodic Table - in PDF and Postscript

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, to appear in New Structures in Physics, ed. Bob Coecke.

• John Baez and Aaron Lauda, A prehistory of n-categorical physics, in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson.

• Albert Burroni, Higher-dimensional word problems with applications to equational logic, Theor. Comp. Sci. 115 (1993), 43-62.

• Yves Guiraud, The three dimensions of proofs, Ann. Pure Appl. Logic 141 (2006), 266-295.

• Barnaby P. Hilken, Towards a proof theory of rewriting: the simply-typed 2λ-calculus, Theor. Comp. Sci. 170 (1996), 407-444.

• C. Barry Jay and Neil Ghani, The virtues of eta-expansion, J. Functional Programming 1 (1993), 1-19.

• R. A. G. Seely, Weak adjointness in proof theory in Proc. Durham Conf. on Applications of Sheaves, Springer Lecture Notes in Mathematics 753, Springer, Berlin, 1979, pp. 697-701.

• R. A. G. Seely, Modeling computations: a 2-categorical framework, in Proc. Symposium on Logic in Computer Science 1987, Computer Society of the IEEE, pp. 65-71.

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