String diagrams for regular logic David I. Spivak (joint with Brendan Fong) Presented on 2018/10/27 Octoberfest David I. Spivak String diagrams for regular logic Presented on 2018/10/27 0 / 19
Introduction Outline 1 Introduction Application: playing with logic Implications for string diagrams String diagrams for regular logic 2 Regular categories and regular logic 3 Bringing it all together David I. Spivak String diagrams for regular logic Presented on 2018/10/27 0 / 19
Introduction Application: playing with logic Minority Report The 2002 movie Minority report showed detective Tom Cruise playing seamlessly with logic. A computer database held relevant information. Cruise could pull it up, and manipulate it, to solve crimes. David I. Spivak String diagrams for regular logic Presented on 2018/10/27 1 / 19
Introduction Application: playing with logic Minority Report The 2002 movie Minority report showed detective Tom Cruise playing seamlessly with logic. A computer database held relevant information. Cruise could pull it up, and manipulate it, to solve crimes. Let’s imagine such a detective scenario. The knowledge base says: Any two people who work in the same tiny company are acquainted. Categorical Informatics is a tiny company. David works at Categorical Informatics . Ryan works at Categorical Informatics . We of course want to conclude that David and Ryan are acquainted. David I. Spivak String diagrams for regular logic Presented on 2018/10/27 1 / 19
Introduction Application: playing with logic Sample scenario Assume: person company person works works acquainted ⊢ tiny David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19
Introduction Application: playing with logic Sample scenario Assume: company Ci person company person works works acquainted ⊢ = Ci Ci tiny David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19
Introduction Application: playing with logic Sample scenario Assume: company Ci person company person works works acquainted ⊢ = Ci Ci tiny = = Ryan = tiny David works Ci works Ci Ci true David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19
Introduction Application: playing with logic Sample scenario Assume: company Ci person company person works works acquainted ⊢ = Ci Ci tiny = = Ryan = tiny David works Ci works Ci Ci true Show: = David acquainted Ryan true David I. Spivak String diagrams for regular logic Presented on 2018/10/27 2 / 19
Introduction Application: playing with logic Picture proof = = = David works Ci Ryan works Ci Ci tiny true Combine! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Combined: David works Ci true = Ryan works Ci tiny Ci Group two Ci’s! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Ci’s grouped: David works Ci true = Ryan works Ci tiny Ci Substitute! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Substituted: David works true = tiny Ci Ryan works Ci Group two Ci’s! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Two Ci’s grouped: David works true = tiny Ci Ryan works Ci Substitute! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Substituted: David works Ci true = tiny Ryan works Group Ci! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Ci grouped: David works Ci true = tiny Ryan works Discard group! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Group discarded: David works true = tiny Ryan works Group! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Grouped: David works true = tiny Ryan works Substitute! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Application: playing with logic Picture proof Substituted: David true = acquainted Ryan Done! David I. Spivak String diagrams for regular logic Presented on 2018/10/27 3 / 19
Introduction Implications for string diagrams Two-dimensional manipulation of string diagrams In this talk we discuss a 2-dimensional language for wiring diagrams. It includes all the sorts of operations shown above. Together with operations like discarding and breaking wires: ⊢ ⊢ ⊢ etc... David I. Spivak String diagrams for regular logic Presented on 2018/10/27 4 / 19
Introduction Implications for string diagrams Comparing to other string diagram languages Let’s compare to string diagram calculus for traced SMCs and hypercats. David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19
Introduction Implications for string diagrams Comparing to other string diagram languages Let’s compare to string diagram calculus for traced SMCs and hypercats. In traced SMCs, you can compose, tensor, swap, and trace. You can do these anywhere in the diagram, with axioms. David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19
Introduction Implications for string diagrams Comparing to other string diagram languages Let’s compare to string diagram calculus for traced SMCs and hypercats. In traced SMCs, you can compose, tensor, swap, and trace. You can do these anywhere in the diagram, with axioms. These can be considered generators and relations for an operad. Traced categories are algebras on the operad 1-Cob. − X 1 a Y − Y a − X 1 b + Ya Yc X 1 c − Y b X 1 a X 1 X 2 X 2 b X 1 c X 2 a − X 2 a + Y c X 1 b X 2 c Yb Yd + X 2 b + Y d + X 2 c David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19
Introduction Implications for string diagrams Comparing to other string diagram languages Let’s compare to string diagram calculus for traced SMCs and hypercats. In traced SMCs, you can compose, tensor, swap, and trace. You can do these anywhere in the diagram, with axioms. These can be considered generators and relations for an operad. Traced categories are algebras on the operad 1-Cob. In hypergraph categories, add Frobenius maps, plus axioms. Hypergraph categories are algebras on the operad Cospan. a b c 4 v 1 2 3 4 1 2 1 2 3 w 2 u 3 a b 3 1 1 4 2 s y x y s t u v w x z c 1 2 t 3 2 z 5 1 6 1 2 3 4 5 6 outer David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19
Introduction Implications for string diagrams Comparing to other string diagram languages Let’s compare to string diagram calculus for traced SMCs and hypercats. In traced SMCs, you can compose, tensor, swap, and trace. You can do these anywhere in the diagram, with axioms. These can be considered generators and relations for an operad. Traced categories are algebras on the operad 1-Cob. In hypergraph categories, add Frobenius maps, plus axioms. Hypergraph categories are algebras on the operad Cospan. In our picture proof, we had more operations and relations. Order on elements of each arity, preserved by substitution. Meet-semilattice structures on elements of each arity. Top element ( true ) can be discarded; corresponding structure for ∧ . Removing dots, breaking wires. We will see that this is a 2-dimensional structure. David I. Spivak String diagrams for regular logic Presented on 2018/10/27 5 / 19
Introduction String diagrams for regular logic Formal presentation of the calculus I. The graphical calculus shown above can be understood as follows. Fix a set Λ (elements will be string labels). Consider the monoidal bicategory C ospan co Λ . David I. Spivak String diagrams for regular logic Presented on 2018/10/27 6 / 19
Introduction String diagrams for regular logic Formal presentation of the calculus I. The graphical calculus shown above can be understood as follows. Fix a set Λ (elements will be string labels). Consider the monoidal bicategory C ospan co Λ . v → Λ, i.e. lists ( v (1) , . . . , v ( n )) ∈ Λ n . Objects: arities n − n 1 n 12 n 2 1-morphisms: v 1 v 2 Λ 2-morphisms: opposite of usual direction (hence − co ) Monoidal structure: (0 , +). David I. Spivak String diagrams for regular logic Presented on 2018/10/27 6 / 19
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