Physics, Language, Maths & Music

Transcription

1 Physics, Language, Maths & Music (partly in arxiv: ) Bob Coecke, Oxford, CS-Quantum SyFest, Vienna, July 2013 ALICE f f = f f = f f = ALICE BOB BOB meaning vectors of words does not Alice not like pregroup grammar Bob = Alice like not Bob

2 ... via (some sort of) Logic (partly in arxiv: ) Bob Coecke, Oxford, CS-Quantum SyFest, Vienna, July 2013 ALICE f f = f f = f f = ALICE BOB BOB meaning vectors of words does not Alice not like pregroup grammar Bob = Alice like not Bob

3 PHYSICS Samson Abramsky & BC (2004) A categorical semantics for quantum protocols. In: IEEE-LiCS 04. quant-ph/ BC (2005) Kindergarten quantum mechanics. quant-ph/

4 genesis [von Neumann 1932] Formalized quantum mechanics in Mathematische Grundlagen der Quantenmechanik

5 genesis [von Neumann 1932] Formalized quantum mechanics in Mathematische Grundlagen der Quantenmechanik [von Neumann to Birkhoff 1935] I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more. (sic)

6 genesis [von Neumann 1932] Formalized quantum mechanics in Mathematische Grundlagen der Quantenmechanik [von Neumann to Birkhoff 1935] I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more. (sic) [Birkhoff and von Neumann 1936] The Logic of Quantum Mechanics in Annals of Mathematics.

7 genesis [von Neumann 1932] Formalized quantum mechanics in Mathematische Grundlagen der Quantenmechanik [von Neumann to Birkhoff 1935] I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more. (sic) [Birkhoff and von Neumann 1936] The Logic of Quantum Mechanics in Annals of Mathematics. [ ] many followed them,... and FAILED.

8 genesis [von Neumann 1932] Formalized quantum mechanics in Mathematische Grundlagen der Quantenmechanik [von Neumann to Birkhoff 1935] I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more. (sic) [Birkhoff and von Neumann 1936] The Logic of Quantum Mechanics in Annals of Mathematics. [ ] many followed them,... and FAILED.

9 the mathematics of it

10 the mathematics of it Hilbert space stuff: continuum, field structure of complex numbers, vector space over it, inner-product, etc.

11 the mathematics of it Hilbert space stuff: continuum, field structure of complex numbers, vector space over it, inner-product, etc. WHY?

12 the mathematics of it Hilbert space stuff: continuum, field structure of complex numbers, vector space over it, inner-product, etc. WHY? von Neumann: only used it since it was available.

13 the physics of it

14 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition.

15 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition. Schrödinger (1935): the stuff which is the true soul of quantum theory is how quantum systems compose.

16 the physics of it von Neumann crafted Birkhoff-von Neumann Quantum Logic to capture the concept of superposition. Schrödinger (1935): the stuff which is the true soul of quantum theory is how quantum systems compose. Quantum Computer Scientists: Schrödinger is right!

17 the game plan

18 the game plan Task 0. Solve: tensor product structure the other stuff =???

19 the game plan Task 0. Solve: tensor product structure the other stuff =??? i.e. axiomatize without reference to spaces.

20 the game plan Task 0. Solve: tensor product structure the other stuff =??? i.e. axiomatize without reference to spaces. Task 1. Investigate which assumptions (i.e. which structure) on is needed to deduce physical phenomena.

21 the game plan Task 0. Solve: tensor product structure the other stuff =??? i.e. axiomatize without reference to spaces. Task 1. Investigate which assumptions (i.e. which structure) on is needed to deduce physical phenomena. Task 2. Investigate wether such an interaction structure appear elsewhere in our classical reality.

22 wire and box language

23 wire and box language Box : = f output wire(s) input wire(s) Interpretation: wire := system ; box := process

24 wire and box language Box : = f output wire(s) input wire(s) Interpretation: wire := system ; box := process one system: n subsystems: no system:... }{{} 1 } {{ } n }{{} 0

25 wire and box games sequential or causal or connected composition: g f g f parallel or acausal or disconnected composition: f g f fg

26 merely a new notation? (g f ) (k h) = (g k) ( f h) g k g k f h = f h

27 quantitative metric f : A B B f A

28 quantitative metric f : B A A f B

29 = = asserting (pure) entanglement quantum classical =

30 = = asserting (pure) entanglement quantum classical = introduce parallel wire between systems: subject to: only topology matters!

31 quantum-like E.g. =

32 Transpose: f = f Conjugate: f = f

33 classical data flow? f f f f =

34 classical data flow? f f =

35 classical data flow? f f =

36 classical data flow? ALICE f ALICE f = BOB BOB quantum teleportation

37 symbolically: dagger compact categories Thm. [Kelly-Laplaza 80; Selinger 05] An equational statement between expressions in dagger compact categorical language holds if and only if it is derivable in the graphical notation via homotopy. Thm. [Hasegawa-Hofmann-Plotkin; Selinger 08] An equational statement between expressions in dagger compact categorical language holds if and only if it is derivable in the dagger compact category of finite dimensional Hilbert spaces, linear maps, tensor product and adjoints.

38 LANGUAGE BC, Mehrnoosh Sadrzadeh & Stephen Clark (2010) Mathematical foundations for a compositional distributional model of meaning. arxiv:

39 the logic of it WHAT IS LOGIC?

40 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language.

41 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Alice and Bob ate everything or nothing, then got sick. connectives (, ) : and, or negation ( ) : not (cf. nothing = not something) entailment ( ) : then quantifiers (, ) : every(thing), some(thing) constants (a, b) : thing variable (x) : Alice, Bob predicates (P(x), R(x, y)) : eating, getting sick truth valuation (0, 1) : true, false ( z : Eat(a, z) Eat(b, z)) ( z : Eat(a, z) Eat(b, z)) S ick(a), S ick(b)

42 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason.

43 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason. E.g. automated theory exploration,...

44 the logic of it WHAT IS LOGIC? Pragmatic option 1: Logic is structure in language. Pragmatic option 2: Logic lets machines reason. Our framework appeals to both senses of logic, and moreover induces important new applications: From truth to meaning in natural language processing: (December 2010) Automated theorem generation for graphical theories:

45 the from-words-to-a-sentence process Consider meanings of words, e.g. as vectors (cf. Google): word 1 word 2 word n...

46 the from-words-to-a-sentence process What is the meaning the sentence made up of these? word 1 word 2 word n...

47 the from-words-to-a-sentence process I.e. how do we/machines produce meanings of sentences?? word 1 word 2 word n...

48 the from-words-to-a-sentence process I.e. how do we/machines produce meanings of sentences? grammar word 1 word 2 word n...

49 the from-words-to-a-sentence process Information flow within a verb: object subject verb

50 = the from-words-to-a-sentence process Information flow within a verb: object subject verb Again we have:

51 pregoup grammar Lambek s residuated monoids (1950 s): b a c a b c a c b

52 pregoup grammar Lambek s residuated monoids (1950 s): b a c a b c a c b or equivalently, a (a c) c a (a c) (c b) b c (c b) b

53 pregoup grammar Lambek s residuated monoids (1950 s): b a c a b c a c b or equivalently, a (a c) c a (a c) (c b) b c (c b) b Lambek s pregroups (2000 s): a a 1 a a b b 1 b b

54 pregoup grammar Lambek s residuated monoids (1950 s): b a c a b c a c b or equivalently, a (a c) c a (a c) (c b) b c (c b) b Lambek s pregroups (2000 s): a 1 a 1 1 a a b 1 b 1 b b 1

55 pregoup grammar -1 A A A A A A-1 A-1 A A A A A -1 A -1 A = A -1 A A A = A A -1-1 A = A A = A -1

56 pregoup grammar For noun type n, verb type is 1 n s n 1, so:

57 pregoup grammar For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n 1 s 1 s

58 pregoup grammar For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n 1 s 1 s

59 pregoup grammar For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n 1 s 1 s

60 pregoup grammar For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n 1 s 1 s Diagrammatic type reduction: n -1n s n-1 n

61 pregoup grammar For noun type n, verb type is 1 n s n 1, so: n 1 n s n 1 n 1 s 1 s Diagrammatic meaning: flow flow n verb n

62 algorithm for meaning of sentences

63 algorithm for meaning of sentences 1. Perform type reduction: (word type 1)... (word type n) sentence type

64 algorithm for meaning of sentences 1. Perform type reduction: (word type 1)... (word type n) sentence type 2. Interpret diagrammatic type reduction as linear map: f :: ii id ii i i

65 algorithm for meaning of sentences 1. Perform type reduction: (word type 1)... (word type n) sentence type 2. Interpret diagrammatic type reduction as linear map: f :: ii id ii 3. Apply this map to tensor of word meaning vectors: f ( v 1... v n ) i i

66 Alice does not like Bob

67 Alice does not like Bob grammar Alice does not not like Bob meaning vectors of words

68 Alice does not like Bob grammar Alice like Bob not meaning vectors of words

69 Alice does not like Bob grammar Alice like Bob not meaning vectors of words

70 Alice does not like Bob grammar Alice like Bob not meaning vectors of words = Alice not like Bob

71 Alice does not like Bob grammar Alice like Bob not meaning vectors of words = Alice not like Bob = Alice not like Bob Using: like = like like = like

72 experiment: word disambiguation E.g. what is saw in: Alice saw Bob with a saw. Edward Grefenstette & Mehrnoosh Sadrzadeh (2011) Experimental support for a categorical compositional distributional model of meaning. Accepted for: Empirical Methods in Natural Language Processing (EMNLP 11).

73 Frobenius algebras

74 such that, for k > 0: Frobenius algebras m { }}... { spiders = } {{... } m+m k { }} { } {{ } n+n k = n BC & Dusko Pavlovic (2007) Quantum measurement without sums. In: Mathematics of Quantum Computing and Technology. quant-ph/ BC, Dusko Pavlovic & Jamie Vicary (2008) A new description of orthogonal bases. Mathematical Structures in Computer Science

75 such that, for k > 0: Frobenius algebras m { }}... { spiders = } {{... } m+m k { }} { } {{ } n+n k = n BC & Dusko Pavlovic (2007) Quantum measurement without sums. In: Mathematics of Quantum Computing and Technology. quant-ph/ BC, Dusko Pavlovic & Jamie Vicary (2008) A new description of orthogonal bases. Mathematical Structures in Computer Science

76 Language-meaning: Frobenius algebras Bob = (the) man who Alice hates = Bob Stephen Clark, BC and Mehrnoosh Sadrzadeh (2013) The Frobenius Anatomy of Relative Pronouns. MOL 13.

77 Language-meaning: Frobenius algebras Bob = (the) man who Alice hates = Bob Stephen Clark, BC and Mehrnoosh Sadrzadeh (2013) The Frobenius Anatomy of Relative Pronouns. MOL 13.

78 Language-meaning: Frobenius algebras Bob = (the) man who Alice hates = Bob Stephen Clark, BC and Mehrnoosh Sadrzadeh (2013) The Frobenius Anatomy of Relative Pronouns. MOL 13.

79 MATHS

80 MATHS Topological QFT (Atiyah 88): F :: f : V V V

81 MATHS Topological QFT (Atiyah 88): F :: f : V V V Grammatical QFT: F :: ii id ii i i

82 MUSIC

83 MUSIC