nLab Geometry of Interaction

Idea

What has been called Geometry of Interaction (Girard 89) is a kind of semantics for linear logic/linear type theory that is however different in method from the usual categorical semantics in monoidal categories. Instead of interpreting a proof of a linear entailment $A\vdash B$ as a morphism between objects $A$ and $B$ in a monoidal category as in categorical semantics, the Geometry of Interaction interprets it as an endomorphism on the object $A\multimap B$. This has been named operational semantics to contrast with the traditional denotational semantics.

That also the “operational semantics” of GoI has an interpretation in category theory, though, namely in compact closed categories induced from traced monoidal categories was first suggested in (Joyal-Street-Verity 96) and then developed in (Hines 97,Haghverdi 00, Abramsky-Haghverdi-Scott 02, Haghverdi-Scott 05). See (Shirahata) for a good review.

Properties

Relation to superoperators and quantum operations

We discuss a relation of the GoI to superoperators/quantum operations in quantum physics.

According to (Abramsky-Haghverdi-Scott 02, remark 5.8, Haghverdi 00b, section 6) all the standard and intended interpretations of the GoI take place in those compact closed categories $Int(\mathcal{C})$ free on a traced monoidal category $\mathcal{C}$ (as discussed there). In these references therefore the notation $Int(-)$ from (Joyal-Street-Verity 96) is changed to $\mathcal{G}(-)$, for “Geometry of Interaction construction” (see Abramsky-Haghverdi-Scott 02, p. 11).

The objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism in $\mathcal{C}$ of the form

$$\array{ A^+\otimes B^- \\ \downarrow \\ A^- \otimes B^+ }$$

in $\mathcal{C}$, and composition of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by tracing out $B^+$ and $B^-$ in the evident way.

We observe now that subcategories of such $Int(\mathcal{C})$ are famous in quantum physics as “categories of superoperators” or “categories of quantum operations”, formalizing linear maps on spaces of operators on a Hilbert spaces that takes density matrices (hence mixed quantum states) to density matrices (“completely positive maps”).

First notice that if the traced monoidal category structure on $\mathcal{C}$ happens to be that induced by a compact closed category structure, then by compact closure the morphisms $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ above are equivalently morphisms in $\mathcal{C}$ of the form

$$(A^-)^\ast \otimes A^+ \longrightarrow (B^-)^\ast \otimes B^+$$

and if moreover one concentrates on the case where $A \coloneqq A^- \simeq A^+$ etc. then these are morphisms of the form

$$End(A) \longrightarrow End(B) \,.$$

Under this identification the somewhat curious-looking composition in $Int(\mathcal{C})$ given by tracing in $\mathcal{C}$ becomes just plain composition in $\mathcal{C}$.

Subcategories (non-full) of $Int(\mathcal{C})$ consisting of morphisms of this form in some ambient $\mathcal{C}$ have been considered in (Selinger 05), there denoted $CPM(\mathcal{C})$, and shown to formalize the concept of quantum operations in quantum physics. For a quick review of Selinger’s construction see for instance (Coecke-Heunen 11, section 2). Comments related to the relation between GoI and such constructions appear in (Abramsky-Coecke 02).

Now the $CPM(-)$ construction requires and assumes that $\mathcal{C}$ has all dual objects (which in the standard model means to restrict to Hilbert spaces of quantum states with are of finite dimension). To generalize away from this requirement a more general definition of quantum operations in $\mathcal{C}$ is given in (Coecke-Heunen 11, def. 1), there denoted $CP(\mathcal{C})$. Direct inspection shows that $CP(\mathcal{C})$ is still a (non-full) subcategory of $Int(\mathcal{C})$.

Related concepts

• transcendental syntax

References

The original references are

• Jean-Yves Girard. Towards a geometry of interaction. In Categories in Computer Science and Logic, pages 69 – 108, Providence, 1989. American Mathematical Society. Proceedings of Symposia in Pure Mathematics n92.

• Jean-Yves Girard. Geometry of interaction I: interpretation of system F. In Ferro, Bonotto, Valentini, and Zanardo, editors, Logic Colloquium ‘88, pages 221 – 260, Amsterdam, 1989. North-Holland.

• Jean-Yves Girard. Geometry of interaction II : deadlock-free algorithms. In Martin-Löf and Mints, editors, Proceedings of COLOG 88, volume 417 of Lecture Notes in Computer Science, pages 76 – 93, Heidelberg, 1990. Springer-Verlag.

• Jean-Yves Girard. Geometry of interaction III : accommodating the additives. In Girard, Lafont, and Regnier, editors, Advances in Linear Logic, pages 329 – 389, Cambridge, 1995. Cambridge University Press.

• Jean-Yves Girard. Geometry of interaction IV : the feedback equation. In Stoltenberg-Hansen and Väänänen, editors, Logic Colloquium ‘03, pages 76 – 117. Association for Symbolic Logic, 2006.

• Jean-Yves Girard. Geometry of interaction V : logic in the hyperfinite factor, in memoriam Claire Delaleu (1991-2009). Fully revised version (October 2009). (pdf)

• Jean-Yves Girard. Geometry of Interaction VI: a blueprint for Transcendental Syntax, January 2013, revised August 28th 2013. (pdf)

Reviews include

• Jean-Yves Girard, Part VI of Lectures on Logic

• Masaru Shirahata, Geometry of Interaction explained, (pdf)

• Samson Abramsky, E. Haghverdi and P. Scott, “Geometry of Interaction and linear combinatory algebra,” Mathematical Structures in Computer Science (2002), vol. 12, pp. 625-665. (ps)

• Samson Abramsky and R. Jagadeesan, “New Foundations for the Geometry of Interaction,” Proceedings 7th Annual IEEE Symp. on Logic in Computer Science, LICS’92, Santa Cruz, CA, USA, 22–25 June 1992. (pdf)

• Harry G. Mairson, “From Hilbert Spaces to Dilbert Spaces: Context Semantics Made Simple”, Proceedings of FST TCS 2002. (pdf)

• Linear Logic Wiki, Geometry of Interaction

• Daniel Murfet, section 5 of Logic and linear algebra: an introduction (arXiv:1407.2650)

The operational categorical semantics in traced monoidal categories is due to

• André Joyal, Ross Street, Dominic Verity, Traced monoidal categories, Math. Proc. Camb. Phil. Soc. (1996), 119, 447 (pdf)

• Peter Hines?, The Algebra of Self-Similarity and its Applications, PhD Thesis, University of Wales, U.K. 1997 (pdf)

• Esfandir Haghverdi, A Categorical Approach to Linear Logic, Geometry of Proofs and Full Completeness, PhD Thesis, University of Ottawa, Canada 2000.

• Esfandir Haghverdi, Unique Decomposition Categories, Geometry of Interaction, and Combinatory Logic, Math. Struc. in Comp. Sci., 10, pp. 205-231.

• Samson Abramsky, Esfandir Haghverdi, Philip Scott, Geometry of Interaction and Linear Combinatory Algebras. MSCS, vol. 12(5), 2002, 625-665, CUP (2002) (citeseer)

• Esfandir Haghverdi, Philip Scott, A categorical model for the geometry of interactions ,Theoretical Computer Science, 2005 (pdf)

Discussion of quantum operations/completely positive maps mentioned above is in

• Peter Selinger, Dagger-compact closed categories and completely positive maps, Electronic Notes in Theoretical Computer

  Science (special issue: Proceedings of the 3rd International Workshop on Quantum Programming Languages). 2005 (pdf, ps)

• Samson Abramsky, Bob Coecke, Physical Traces: Quantum vs. Classical Information Processing, Electronic Notes in Theoretical Computer Science 69 (2003) (arXiv:0207057)

• Bob Coecke, Chris Heunen, Pictures of complete positivity in arbitrary dimension, EPTCS 95, 2012, pp. 27-35 (arXiv:1110.3055)