Linear Algebra and Geometric Approches to Meaning
1a. General Introduction
ESSLLI Summer School 2011, Ljubljana
August 1 – August 7, 2011
Reinhard Blutner
Universiteit van Amsterdam
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1. Geometric models of meaning
2. Phenomena and puzzles
3. Quantum Probabilities
4. Historical notes
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Geometric models in Cognitive
Psychology
• Basic claim: An understanding of problem solving,
categorization,
memory
retrieval,
inductive
reasoning, and other cognitive processes requires
that we understand how humans assess similarity.
• W. S. Torgerson (1965): Multidimensional scaling of
similarity. Psychometrika 30: 379–393.
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Geometric models in Cognitice
Psychology II
• A.
Tversky
(1977):
Features
of
similarity.
Psychological Review 84: 327–352.
• P. Gärdenfors: The Geometry of Thought (2000)
Concepts as convex spaces
• D. Widdows: Geometry
Distributional semantics
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and
Meaning
(2004)
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Voronoi Tessellation
If the closeness to a
prototyp
determines
class boundaries, then
we get a partition of
the conceptual space
into convex subspaces
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Concept combination á la
Peter Gärdenfors
What is the color of a red nose
(red flag, red tomato)?
skin colors
What is the computational
mechanism of combination?
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Natural concepts do not form
Boolean algebras
|7
•
P. Gärdenfors: The Geometry of Thought (2000)
Concepts as convex spaces
•
The intersection of convex sets is convex again, but
union and complement are not.
•
Hans Primas:
lattices.
•
Orthomodularity: If A  B then A = (AB)+(AB )
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Convex
sets
form
orthomodular
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Possible worlds and vectors
• Possible worlds: Isolated entities which are used for
modeling propositions (sets of possible worlds)
• Vector states: abstract objects which form vector
spaces.
– The addition of two vectors is an
operation which describes the superposition of possibly conflicting states
– The scalar product is an operation
which describes the similarity of two
states
– Projections are operators that map
vector spaces onto certain subspaces
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s
u
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A
Superposition
Superposing colors
Superposing pictures
Superposing meanings
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Superposing colors
x
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x
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x
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x
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Superposition of faces
x
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Structure of the task
Left: superposition of female human faces
Right: superposition of male human faces
Middle: superposition of male and female faces
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Experiment by Conte et al. (2007)
Test
Test III
Are the two lines
circles
of the same size?
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Comparison
Symbolic models
Geometric models
Formal semantics (Montague
1978); Partee, Kamp
Mental spaces (Gärdenfors
2000); Lakoff, Fouconnier
Concepts as
constructions
set-theoretic
Natural concepts as convex
subspaces
Qualitative aspects of meaning, feature and tree structs.
Quantitative aspects of
meaning, similarity structs.
Boolean algebra
Orthomodular lattice
Compositional architecture
Compositionality
= big problem
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2
1. Geometric models of meaning
2. Phenomena and puzzles
3. Quantum Probabilities
4. Historical notes
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Puzzle 1: Vagueness
•
A concept is vague if it does not have precise,
sharp boundaries and does not describe a welldefined set.
•
Vagueness is the inevitable result of a knowledge
system that stores the centers rather than the
boundaries of conceptual categories
•
Vagueness is different from typicality (centrality):
-
both robins and penguins are clearly birds, but
-
robins are more typical than penguins as birds
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Why is language vague?
“ It is not that people have a precise view of the world but
communicate it vaguely; instead, they have a vague view of
the world. I know of no model which formalizes this. I think
this is the real challenge posed by the question of my title
[Why is language vague?]" [Barton L. Lipman, 2001]
The geometric approach provides a new theory of vagueness
in the spirit of Lipman. It is able to solve some hard problems
such as the disjunction and the conjunction puzzle.
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Fuzzy set theory as a compositional
theory of graded membership
1. mA (x) membership function for instances x
of category A: 0  mA (x)  1
2. mA (x) = 1–mA (x)
3. mAB (x) = min(mA (x), mB (x))
4. mAB (x) = max(mA (x), mB (x))
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Why fuzzy sets do not work
Chose instance x such that
x
mCIRCLE(x) = mSQUARE(x)
1. Compositionality:
mCIRCLE(x)=mSQUARE(x)  mROUND CIRCLE(x)=mROUND SQUARE(x)
 Compositionality is violated
2. Monotonicity:
mROUND SQUARE (x)  mSQUARE (x)
 Monotonicity is violated (conjunction puzzle)
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Why fuzzy sets are not too bad
x
3. Boundary Contradiction: mSQUARE  SQUARE (x) > 0
Empirical counter-evidence by Bonini (1999), Alxatib &
Pelletier (2011), Ripley (2011), Sauerland (2011).
•
Supervaluation & probability theory both fail
•
Is there a solution to all three puzzles of vagueness
based on a uniform theory?
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Borderline contradictions
Alxatib & Pelletier 2011
asked subjects to judge
sentences such as
(*) x is tall and not tall
• 44.7%
of
their
subjects judge (*) to
be
true
for
the
borderline case (2)
•
Only 40.8% judge (*) to be false in that case.
•
Borderline contradictions are generally found to be
quite acceptable
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Puzzle 2: Probability judgements
violate the Kolmogorov axioms
•
A Boolean algebra over W is a set ℱ of subsets of W
[events, possibilities] that contains W and is closed
under union and complementation (intersection)
•
Normalized additive measure function
– P(A  B) = P(A) + P(B) for disjoint sets A and B
– P(W) = 1
•
Consequences:
– P(A  B)  P(A), P(A  B)  P(A)
[monotonicity]
– P(A) + P(B)  P(A  B) ≤ 1
[additivity I]
– P(A) + P(B)  P(A  B) ≤ 1
[additivity II]
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Disjunction puzzle
• Tversky and Shafir (1992) show that significantly more students report
they would purchase a nonrefundable Hawaiian vacation if they were to
know that they have passed or failed an important exam than report
they would purchase if they were not to know the outcome of the exam
• P(A|C)
P(A|C)
P(A)
= 0.54
= 0 .57
= 0 .32
• P(A) = P(A|C) P(C) + P(A|C) P(C)
since CA  (C)A = A (distributivity)
The ‘sure thing principle’ is violated empirically!
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Puzzle 3: Complementarity
•
Two Boolean descriptions are said to be
complementary if they cannot be embedded into a
single Boolean description.
•
Unicity: In classical probability theory, a single
sample space is proposed which provides a
complete and exhaustive description of all events
that can happen in an experiment.
–
–
•
If unicity is valid, then complementarity does not exist
If complementarity exists then unicity cannot be valid
Examples: physical time/mental time; physical/
mental objectivity (mind/body); Jung’s rational/
irrational functions.
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Cartographic map projections
All cartographic maps are valid only in the small, i.e. locally.
Adapted from Primas (2007)
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Puzzle 4: Question order effects
for attitude questions
Is Clinton honest? (50%)
Is Gore honest? (68%)
Is Gore honest? (60%)
Is Clinton honest? (57%)
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Puzzle 5: Asymmetric similarities
•
Korea is similar to China vs. China is similar to Korea
•
Chicago's linebackers are like tigers vs. *Tigers are
like Chicago's linebackers
•
From a classical perspective this is puzzling:
– sim (X, Y) = f (distance (X, Y))
– sim (Y, X) = f (distance (Y, X))
But distance is a symmetric function
•
How to express the basic cognitive operations
– asymmetric similarity, asymmetric conjunction –
in the geometric framework?
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3
1. Geometric models of meaning
2. Phenomena and puzzles
3. Quantum Probabilities
4. Historical notes
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Birkhoff and von Neumann
1936
Hence we conclude that the propositional
calculus of quantum mechanics has the same
structure as an abstract projective geometry.
The logic of quantum mechanics.
Annals of Mathematics 37(4), 1936
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Geometric model of probabilities
• In the vector model, pairwise disjoint possibilities are
represented by pairwise orthogonal subspaces.
• In the simplest case 1-dimensional subspaces are represented by the axes of a Cartesian coordinate system
• A state is described by a vector s of unit length
• The projections of s onto the
different axes are called
probability amplitudes.
• The square of the amplitudes
are the relevant probabilities.
• They sum up to 1 – the length
of s : (Pa s)2 + (Pb s)2 = 1.
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s
b
Pb(s)
a
Pa(s)
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Projections & probabilities
s
u'

Projector to subspace A: PA

Projected vector: PA s = u
unique vector u such that
s = u + u‘ , uU, u‘U ┴.

The length of the projection
is written |PA s| = |u |.
A
u
 The probability that s is about A (that s collapes
onto A) is the square of the length of the
corresponding projection: |PA s| 2 (Born rule)
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Order-dependence of projections
a
s
b
|Pa Pb s |  |Pa Pb s |
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Pa Pb s
Pb Pa s
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Asymmetric conjunction
•
The sequence of projections (Pa ; Pb) corresponds to
a Hermitian operator Pa Pb Pa .
(Pa ; Pb) =def Pa Pb Pa (Gerd Niestegge’s asymmetric
conjunction)
•
The expected probability for the sequence (Pa ; Pb) is
 ( P a ; P b) = | P b P a s | 2
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Solving the puzzles
•
The asymmetric conjunctions account for interference
effects, which partly can explain the puzzles
•
Many possible applications:
– Vagueness and probability judgements (Puzzles 1 & 2)
It does not account for borderline contradictions!
– Complementarity and uncertainty principle (Puzzle 3)
– Order effects for questions (Puzzle 4)
– Asymmetric similarity (Puzzle 5)
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4
1. Geometric models of meaning
2. Phenomena and puzzles
3. Quantum Probabilities
4. Historical notes
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Quantum Mechanics &
Quantum Cognition
Heisenberg
Aerts
1994
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Einstein
Conte
1989
Bohr
Khrennikov
1998
Pauli
Atmanspacher
1994
Sommerfeld-Bohr atomic model
• Electrons move on discrete orbits
• Electrons emit photons when jumping from one orbit
to the next
• Problems: many stipulations and conceptual inconsistencies. Empirical problems with certain spectra.
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Heisenberg‘s matrics mechanics
• Heisenberg
1925:
New
solution to old puzzle of
spectral lines of hydrogen.
The electrons do not move on
orbits
• “It was about three o' clock at night when the final
result of the calculation lay before me. At first I
was deeply shaken. I was so excited that I could
not think of sleep. So I left the house and awaited
the sunrise on the top of a rock.”
• Max Born: Linear algebra. Eigenvalue problem
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The pioneers and applications
to the macroworld
• William James was the first who introduced the idea
of complementarity into psychology
“It must be admitted, therefore, that in certain persons, at least,
the total possible consciousness may be split into parts which
coexist but mutually ignore each other, and share the objects of
knowledge between them. More remarkable still, they are
complementary” (James, the principles of psychology 1890, p. 206)
• Nils Bohr introduced it into physics (Complementarity
of momentum and place) and proposed to apply it
beyond physics to human knowledge.
• Beim Graben & Atmanspacher gave a systematic
treatment of complementarity in the macroworld.
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Interference effects and
puzzles of bounded rationality
• Conjunction and disjunction fallacy: Aerts et al.
(2005), Khrennikov (2006), Franco (2007), Conte et
al. (2008), Blutner (2008), Busemeyer et al. (2011).
• Prisoner’s dilemma: Pothos and Busemeyer (2009).
• Order effects: Trueblood and Busemeyer (in press).
• Categorization: Aerts and Gabora (2005), Aerts
(2009), Busemeyer, Wang, and Lambert-Mogiliansky
(2009).
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The recent geometric turn
• Quantum Interaction workshop at Stanford in 2007 organized by
Bruza, Lawless, van Rijsbergen, and Sofge (as part of the AAAI
Spring Symposium)
• Workshops at Oxford (England) in 2008,
• Workshop at Vaxjo (Sweden) in 2008,
• Workshop at Saarbrücken (Germany) in 2009,
• AAAI meeting Stanford in 2010.
• Busemeyer (Indiana University) et al. organized a special issue on
Quantum Cognition, which was published in the October (2009),
issue of Journal of Mathematical Psychology
• Busemeyer & Bruza (forthcoming): Quantum cognition and
decision. Cambridge University Press.
 www.quantum-cognition.de
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Conclusions
•
The inclusion of the concept of probability into
traditional geometric models is the beginning of a
real break through: it resolves a series of serious
puzzles and long-standing problems
•
Concepts of natural language semantics such as
similarity and logical operations do not directly
correspond to standard operations in the orthomodular (vector) framework but rather indirectly
•
The problem of conceptual combination can be
solved in the new framework. Extensional holism
coexists with intensional compositionality.
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