What is computation?

Transcription

1 What is computation? Daniel Murfet October 2014 therisingsea.org Logic and linear algebra On Sweedler s cofree cocommutative coalgebra Computing with cut systems

2 Turing machines, Lambda calculus, logic Semantics : syntax :: representations : group Homotopy type theory (Awodey, Voevodsky 2012) Girard Towards a Geometry of Interaction (1989)

3 Sense & Denotation Frege On sense and denotation (1892) A sentence denotes or refers to some external object, and expresses its sense, which is the mode of presentation of its denotation. 2 2=4 same denotation, different sense

4 Sense as algorithm 2 2=4 mult 2 (2) = 4 Turing machine input output 2 4 mult 2 computation

5 Sense as topology 2 2=4 proof-nets = diagrammatics of linear logic int A 2 mult 2 (2) computation 4

6 Sense as algebra T = Z 2 -graded triangulated category [1] [1] = id End T (Y ) = Hom T (Y,Y ) Hom T (Y,Y [1]) C = Z 2 -graded algebra A C-module in T is a morphism C! End T (Y )

7 T = Z 2 -graded triangulated category [1] [1] = id Example ample. C 1 =End k (k k[1]) a = a = = kha, a i with a 2 =(a ) 2 =0,aa + a a = C n = kha 1,...,a n,a 1,...,a ni with Cli ord relations T = C n -modules in T for n 0 C 0 = k

8 Sense as algebra T = Z 2 -graded triangulated category A C 1 -module in T is (Y,a,a ) Y = X X[1] X =Im(aa ) (Y,a,a ) = X 2 2=4 same denotation, di erent sense

9 A bicategory has objects, 1-morphisms and 2- morphisms, and composition functors B(b, c) B(a, b)! B(a, c) (mult 2, 2) 7! 4=X 4 4 A cut system is similar, except it has cut functors int A B(b, c) B(a, b)! B(a, c) 2 (computable) mult 2 (2) (mult 2, 2) 7! (Y,a,a ) mult 2

10 Theorem There is a bicategorical semantics of intuitionistic propositional linear logic in the cocompletion of a cut system B defined on the bicategory of Landau- Ginzburg models (hypersurface singularities and matrix factorisations). Lambda calculus embeds in intuitionistic linear logic The Clifford actions are derived from Atiyah classes of matrix factorisations (homological perturbation lemma under the hood).

11 Universal examples of same denotation, different sense: Turing machines, proof-nets, Clifford representations in triangulated categories (?) (Y,a,a ) = X 2 2=4 Cut system B 1 2 int mult 2 int (Y,a,a ) X =4

12 The adjoint Y! X ( Z of a morphism : X Y! Z is depicted ( X ( Z X Z!V = universal coalgebra over V Y V!V!V!V!V!V

13 int A =!(A ( A) ( (A ( A) 7! 2 int A!(A ( A) A ( A A A A ( A A A ( A 2

14 2:!(A ( A)! (A ( A) e2 :!(A ( A)!!(A ( A)

15 input to the right leg. yields the The numeral. The This manipulation fromfirst (4.3)equality to (4.4) below. is to take the second left leg equality and feed itfollows as an We feed 2 as input to mult2 by cutting: input to a the right leg. This yields the first equality below. The second equality follows from the fact that promotion box represents a morphism of coalgebras, and thus can be that a promotion a morphism of coalgebras, and thus can be commuted from pastthe thefact coproduct wherebybox it represents is duplicated: commuted past the coproduct whereby it is duplicated: inta (4.5) = = (4.5) (4.5) = (4.6) (4.6) = thisthe point the promotions with the derelictions the identity (3.6), releasing At thisat point promotions cancel cancel with the derelictions by theby identity (3.6), releasing theofpair of Church numerals contained the promotion This yields the first the pair Church numerals contained in the inpromotion boxes. boxes. This yields the first equality while the second is an application the general form the identity equality below, below, while the second is an application of the ofgeneral form of the ofidentity represented the transformation of diagrams represented by the by transformation of diagrams in (4.3)in (4.3) (4.4): (4.4): oofs by mult2 2. Not surprisingly, the cut-free normalisaumeral 4. Each of the proof transformations generated by plied to mult2 2 (given in Appendix A) yields a new proof his sequence of proofs represents a particular sequence of ram. grammatic transformations. From (4.5) the first step is to= (4.6) = (4.6) = = r adjunction, as in the manipulation from (4.1) to (4.2). (4.3), with 2 a part of 2 and 1 the promoted Church m (4.3) to (4.4) is to take the left leg and feed it as an ds the first equality below. The second equality follows box represents a morphism of coalgebras, and thus can be hereby it is duplicated: This last diagram is the denotation 4, conclude so we conclude that (atat least theoflevel This last diagram is the denotation of 4, soofwe that (at least the at level the of the