Post on 07-Jun-2020
transcript
Physics, Language, Maths & Music(partly in arXiv:1204.3458)
Bob Coecke, Oxford, CS-Quantum
SyFest, Vienna, July 2013
=
f
f =
f f
f
ALICE
BOB
=
ALICE
BOB
f=
not
likeBobAlice
does
Alice not like
not
Bob
meaning vectors of words
pregroup grammar
. . . via (some sort of) Logic(partly in arXiv:1204.3458)
Bob Coecke, Oxford, CS-Quantum
SyFest, Vienna, July 2013
=
f
f =
f f
f
ALICE
BOB
=
ALICE
BOB
f=
not
likeBobAlice
does
Alice not like
not
Bob
meaning vectors of words
pregroup grammar
— PHYSICS —
Samson Abramsky & BC (2004) A categorical semantics for quantum proto-cols. In: IEEE-LiCS’04. quant-ph/0402130
BC (2005) Kindergarten quantum mechanics. quant-ph/0510032
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)
[Birkhoff and von Neumann 1936] The Logic of Quan-tum Mechanics in Annals of Mathematics.
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)
[Birkhoff and von Neumann 1936] The Logic of Quan-tum Mechanics in Annals of Mathematics.
[1936 – 2000] many followed them, ... and FAILED.
— genesis —
[von Neumann 1932] Formalized quantum mechanicsin “Mathematische Grundlagen der Quantenmechanik”
[von Neumann to Birkhoff 1935] “I would like tomake a confession which may seem immoral: I do notbelieve absolutely in Hilbert space no more.” (sic)
[Birkhoff and von Neumann 1936] The Logic of Quan-tum Mechanics in Annals of Mathematics.
[1936 – 2000] many followed them, ... and FAILED.
— the mathematics of it —
— the mathematics of it —
Hilbert space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.
— the mathematics of it —
Hilbert space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.
WHY?
— the mathematics of it —
Hilbert space stuff: continuum, field structure of com-plex numbers, vector space over it, inner-product, etc.
WHY?
von Neumann: only used it since it was ‘available’.
— the physics of it —
— the physics of it —
von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.
— the physics of it —
von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.
Schrodinger (1935): the stuff which is the true soul ofquantum theory is how quantum systems compose.
— the physics of it —
von Neumann crafted Birkhoff-von Neumann Quan-tum ‘Logic’ to capture the concept of superposition.
Schrodinger (1935): the stuff which is the true soul ofquantum theory is how quantum systems compose.
Quantum Computer Scientists: Schrodinger is right!
— the game plan —
— the game plan —
Task 0. Solve:tensor product structure
the other stuff= ???
— the game plan —
Task 0. Solve:tensor product structure
the other stuff= ???
i.e. axiomatize “⊗” without reference to spaces.
— 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 struc-ture) on ⊗ is needed to deduce physical phenomena.
— 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 struc-ture) on ⊗ is needed to deduce physical phenomena.
Task 2. Investigate wether such an “interaction struc-ture” appear elsewhere in “our classical reality”.
— wire and box language —
— wire and box language —
foutput wire(s)
input wire(s)Box =:
Interpretation: wire := system ; box := process
— wire and box language —
foutput wire(s)
input wire(s)Box =:
Interpretation: wire := system ; box := process
one system: n subsystems: no system:
︸︷︷︸1
. . .︸ ︷︷ ︸
n︸︷︷︸
0
— wire and box games —
sequential or causal or connected composition:
g ◦ f ≡g
f
parallel or acausal or disconnected composition:
f ⊗ g ≡ f fg
— merely a new notation? —
(g ◦ f ) ⊗ (k ◦ h) = (g ⊗ k) ◦ ( f ⊗ h)
=f h
g k
f h
g k
— quantitative metric —
f : A→ B
f
A
B
— quantitative metric —
f † : B→ A
f
B
A
— asserting (pure) entanglement —
quantumclassical
=
==
— asserting (pure) entanglement —
quantumclassical
=
==
⇒ introduce ‘parallel wire’ between systems:
subject to: only topology matters!
— quantum-like —
E.g.
=
Transpose:
ff
=Conjugate:
ff
=
classical data flow?
f
=
fff
classical data flow?
f
=
f
classical data flow?
f
=
f
classical data flow?
f
ALICE
BOB
=
ALICE
BOB
f
⇒ quantum teleportation
— symbolically: dagger compact categories —
Thm. [Kelly-Laplaza ’80; Selinger ’05] An equa-tional statement between expressions in dagger com-pact categorical language holds if and only if it isderivable in the graphical notation via homotopy.
Thm. [Hasegawa-Hofmann-Plotkin; Selinger ’08]An equational statement between expressions in dag-ger compact categorical language holds if and onlyif it is derivable in the dagger compact category of fi-nite dimensional Hilbert spaces, linear maps, tensorproduct and adjoints.
— LANGUAGE—
BC, Mehrnoosh Sadrzadeh & Stephen Clark (2010) Mathematical foundationsfor a compositional distributional model of meaning. arXiv:1003.4394
— the logic of it —
WHAT IS “LOGIC”?
— the logic of it —
WHAT IS “LOGIC”?
Pragmatic option 1: Logic is structure in language.
— 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, ornegation (¬) : not (cf. nothing = not something)entailment (⇒) : thenquantifiers (∀,∃) : every(thing), some(thing)constants (a, b) : thingvariable (x) : Alice, Bobpredicates (P(x),R(x, y)) : eating, getting sicktruth 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)
— the logic of it —
WHAT IS “LOGIC”?
Pragmatic option 1: Logic is structure in language.
Pragmatic option 2: Logic lets machines reason.
— 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, ...
— 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, andmoreover induces important new applications:From truth to meaning in natural language processing:
— (December 2010)
Automated theorem generation for graphical theories:
— http://sites.google.com/site/quantomatic/
— the from-words-to-a-sentence process —
Consider meanings of words, e.g. as vectors (cf. Google):
word 1 word 2 word n...
— the from-words-to-a-sentence process —
What is the meaning the sentence made up of these?
word 1 word 2 word n...
— the from-words-to-a-sentence process —
I.e. how do we/machines produce meanings of sentences?
word 1 word 2 word n...?
— the from-words-to-a-sentence process —
I.e. how do we/machines produce meanings of sentences?
word 1 word 2 word n...grammar
— the from-words-to-a-sentence process —
Information flow within a verb:
verb
object subject
— the from-words-to-a-sentence process —
Information flow within a verb:
verb
object subject
Again we have:
=
— pregoup grammar —
Lambek’s residuated monoids (1950’s):b ≤ a( c⇔ a · b ≤ c⇔ a ≤ c � b
— 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
— 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 · ab∗ · b ≤ 1 ≤ b · b∗
— 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 · −1a ≤ 1 ≤ −1a · a
b−1 · b ≤ 1 ≤ b · b−1
— pregoup grammar —
A A A AA A A A
-1
-1
-1
-1
=
A
A
A
A=A
A A
A=A
A
A
A=A
A A
A-1
-1
-1
-1 -1
-1 -1
-1
— pregoup grammar —
For noun type n, verb type is −1n · s · n−1, so:
— pregoup grammar —
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
— pregoup grammar —
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
— pregoup grammar —
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
— pregoup grammar —
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
Diagrammatic type reduction:
n nsn n-1 -1
— pregoup grammar —
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
Diagrammatic meaning:
verbn n
flow flow
— algorithm for meaning of sentences —
— algorithm for meaning of sentences —
1. Perform type reduction:
(word type 1) . . . (word type n) { sentence type
— 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 :: 7→
∑i
〈ii|
⊗ id ⊗
∑i
〈ii|
— 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 :: 7→
∑i
〈ii|
⊗ id ⊗
∑i
〈ii|
3. Apply this map to tensor of word meaning vectors:
f(−→v 1 ⊗ . . . ⊗
−→v n
)
—−−−−→Alice ⊗
−−−→does ⊗ −−→not ⊗
−−→like ⊗
−−→Bob —
—−−−−→Alice ⊗
−−−→does ⊗ −−→not ⊗
−−→like ⊗
−−→Bob —
Alice not like Bob
meaning vectors of words
not
grammar
does
—−−−−→Alice ⊗
−−−→does ⊗ −−→not ⊗
−−→like ⊗
−−→Bob —
Alice like Bob
meaning vectors of words
grammar
not
—−−−−→Alice ⊗
−−−→does ⊗ −−→not ⊗
−−→like ⊗
−−→Bob —
Alice like Bob
meaning vectors of words
grammar
not
—−−−−→Alice ⊗
−−−→does ⊗ −−→not ⊗
−−→like ⊗
−−→Bob —
Alice like Bob
meaning vectors of words
grammar
not= not
like
BobAlice
—−−−−→Alice ⊗
−−−→does ⊗ −−→not ⊗
−−→like ⊗
−−→Bob —
Alice like Bob
meaning vectors of words
grammar
not= not
like
BobAlice = not
likeBobAlice
Using: =
likelike
=
likelike
— experiment: word disambiguation —E.g. what is “saw”’ in: “Alice saw Bob with a saw”.
Edward Grefenstette & Mehrnoosh Sadrzadeh (2011) Experimental supportfor a categorical compositional distributional model of meaning. Acceptedfor: Empirical Methods in Natural Language Processing (EMNLP’11).
— Frobenius algebras —
— Frobenius algebras —
‘spiders’ =
m︷ ︸︸ ︷....
....︸ ︷︷ ︸n
such that, for k > 0:
m+m′−k︷ ︸︸ ︷........
....
....
....
︸ ︷︷ ︸n+n′−k
=
....
....
BC & Dusko Pavlovic (2007) Quantum measurement without sums. In: Math-ematics of Quantum Computing and Technology. quant-ph/0608035
BC, Dusko Pavlovic & Jamie Vicary (2008) A new description of orthogonalbases. Mathematical Structures in Computer Science. 0810.0812
— Frobenius algebras —
‘spiders’ =
m︷ ︸︸ ︷....
....︸ ︷︷ ︸n
such that, for k > 0:
m+m′−k︷ ︸︸ ︷........
....
....
....
︸ ︷︷ ︸n+n′−k
=
....
....
BC & Dusko Pavlovic (2007) Quantum measurement without sums. In: Math-ematics of Quantum Computing and Technology. quant-ph/0608035
BC, Dusko Pavlovic & Jamie Vicary (2008) A new description of orthogonalbases. Mathematical Structures in Computer Science. 0810.0812
— Frobenius algebras —Language-meaning:
Bob = (the) man who Alice hates = Bob
Stephen Clark, BC and Mehrnoosh Sadrzadeh (2013) The Frobenius Anatomyof Relative Pronouns. MOL ’13.
— Frobenius algebras —Language-meaning:
Bob = (the) man who Alice hates = Bob
Stephen Clark, BC and Mehrnoosh Sadrzadeh (2013) The Frobenius Anatomyof Relative Pronouns. MOL ’13.
— Frobenius algebras —Language-meaning:
Bob = (the) man who Alice hates = Bob
Stephen Clark, BC and Mehrnoosh Sadrzadeh (2013) The Frobenius Anatomyof Relative Pronouns. MOL ’13.
— MATHS —
— MATHS —
“Topological” QFT (Atiyah ’88):
F :: 7→ f : V ⊗ V → V
— MATHS —
“Topological” QFT (Atiyah ’88):
F :: 7→ f : V ⊗ V → V
“Grammatical” QFT:
F :: 7→
∑i
〈ii|
⊗id⊗
∑i
〈ii|
— MUSIC —
— MUSIC —