Course Structure:
- Introduction
- Computing Machinery and Intelligence (Turing)
- Logic
- Rules
- Concepts
- Analogy
- Images
- Connectionism
- Neuroscience
- Emotions
- Consciousness
- Embodiment
- Dynamical Systems
- Intentionality
- Externalism
- Conclusion
Cue:
- Lecture 1. Introduction
- Lecture 2. Computing Machinery and Intelligence (Turing)
- Lecture 3. Logic
- Lecture 4. Rules
- Lecture 5. Concepts
- Lecture 6. Analogy
- Lecture 7. Images
- Lecture 8. Connectionism
- Lecture 9. Neuroscience
- Lecture 10. Emotions
- Lecture 11. Consciousness
- Lecture 12. Embodiment
- Lecture 13. Dynamical Systems
- Lecture 14. Intentionality
- Lecture 15. Externalism
- Lecture 16. Conclusion
- Cognitive Science (CogSci): Study of mind and intelligence
- Main concerns:
- Identify resources used
- Understand how they are deployed
- Classic view (1950s–1980s):
- Symbolic representations
- Symbol processing
- Recent challenges:
- Adequacy of symbol processing
- Brain studies
- Consciousness, emotions
- Aims of the course:
- Examine classic CogSci as an account of human thinking and intelligence
- Examine challenges to classic CogSci
- For now:
- The CogSci paradigm
- History of CogSci
- Paradigm: a framework for constructing theories
- Cognitive Scientists disagree on the nature of thinking and intelligence
- Central Thesis: Thinking is like computation (in a digital computer)
- Information is represented (data structures)
- Calculations are performed
- Note: The thesis is an analogy, not a claim of physical resemblance between brains and PCs
- The thesis is a paradigm (Kuhn) more than a theory; it tells researchers …
- What to investigate,
- What sorts of theories to test, and
- How to test and evaluate them.
- For CogSci:
- Investigate intelligent behaviours
- Theorize about mental representations and procedures
- Test using computational models, experiments, etc.
Q: What activities require intelligence?
- Typical answers include
recreational challenges,argumentation,technological work - In classic CogSci: intelligence is any activity in which
knowledgeandexpertiseplays a major role - Intelligence is knowledge-intensive
- Mental representation - statements:
- Block A is on block B.
- Block B is on the ground.
- Block C is on the ground.
- Block C is right of block B.
- Mental procedures - rules:
- To have block x on block y, place block x on top of block y.
- To place block x on top of block y, remove other blocks from on top of y, pick up block x, move it on block y and let go of block x.
- Make a plan to spell "CAB"
- Theory:
modelorexplanationof how an intelligent activity occurs- Claim about mental representations and procedures, e.g., statements and rules
- Model confirmed if performance matches human behaviour (disconfirmed otherwise)
- This approach is referred as cognitive modeling - the operation of the computer program models or imitates the course of human thinking
- CogSci is highly interdisciplinary
- Different disciplines employ different testing methods, e.g., brain scans in neuroscience
- Central thesis: Thinking is like computing
- CRUM: Computational-Representational Understanding of Mind
- CRUM is a paradigm rather than a theory
- Intelligence is knowledge-intensive
- Produced by mental representations and procedures
- Theories are testable through simulation, experiment, etc.
- Evaluation of CRUM depends on
- Record of success or failure of CRUM theories
- Performance relative to other paradigms
- Prospects for future success
- Basic questions:
- What do you know and how do you know it? (epistemology)
- What kind of thing is a mind? (metaphysics)
- How does a mind give rise to thinking? (psychology)
- Some responses:
- Plato (ca. 400 BC): grasp of ideas, hydraulic analogy
- Locke (ca. 1700): possession of stattements, blank paper analogy
- Watson (ca. 1920): S-R arcs, switchboard anaology
- Weiner (ca. 1940): control configurations, rangefinder analogy
- 1940s: Turing, electromechanical computers, computer analogy
- 1950s:
- Miller: short term memory (7±2 chunks)
- Newell & Simon: General Problem Solver
- Chomsky: syntax as mental representation
- Some general historical trends:
- Thinking and intelligence have often been associated with information processing
- Information processing technology has often been used as a source of inspiration for theories of cognition
- Textbook
- Reading Materials
- Refer to "Grade Breakdown"
- Discussion in forum is strongly recommended, but not weighted in grade
...
- CRUM
- Central thesis: thinking is like computing
- Expertise and knowledge central to intelligence
- Computing machinery and intelligence (Turing 1950)
- Can machines think? (1953)
- Drew attention to the topic
- Laid out the classic paradigm
- Cambridge (1936), Princeton (1938)
- Developed general theory of computation
- Instrumental in breaking the Enigma code (WWII)
- Noted that computers can do intelligent work
- Committed suicide in 1954
- The imitation game: Why not just use a dictionary definition?
- Subject to prejudice:
- "The question of whether a computer can think is no more interesting than the question of whether a submarine can swim." (Dijkstra)
- Had history been different, our definitions would be different
- Imitation Game
- Imitate someone of the opposite gender:
- (a) man, (b) woman, (c) interrogator
- Goal: for (c) to distinguish (a) and (b)
- Teletype interface prevents superficial information from being used by (c)
- A computer might imitate a human:
- (a) computer, (b) person, (c) interrogator
- Goal: for (c) to distinguish (a) and (b)
- Teletype interface ensures that only some profound difference, e.g., intelligence, matters
- Imitate someone of the opposite gender:
- To suppose that a (human) brain is required for intelligence is to beg the question
- Why is conversing a good test of intelligence?
- It provides the computer an opportunity to avoid prejudice, superficial judgement
- We often judge a person's intelligence through conversation
- We would still need to avoid jumping to conclusions though
- Among machines, the digital computer holds most interest
- Components of a typical digital computer:
- Memory (RAM)
- Executive unit (CPU)
- Control (program)
- A program is a series of numbers interpreted as instructions by the executive
- "If position 4505 contains 0 obey next the instruction stored in 6707, otherwise continue straight on"
- Carrying out instructions determines the computer's behaviour
- Two kinds of digital computers:
- Special purpose (e.g., a chess computer)
- General purpose (e.g., a PC)
- A general purpose computer can imitate the activity of any other computer
- If a general purpose computer succeeds at the imitation game, it would be due to its program, not its physical hardware
- Intelligence is highly abstract in nature
- Original prediction: success in 50 years
- Later, 100 years
- Current activity: Loebner prize
- Success is not yet in sight
- Is success out of the question?
- Objections include: Theological, mathematical, consciousness, originality, etc.
- Theological objection:
- Argument: thinking requires a soul, computers have no souls, so computers cannot think
- Reply: it only follows that computers do not think
- Being omnipotent, God could give souls to computers, enabling them to think
- Ultimately, whether computers can think is an empirical matter, determined by empirical tests (e.g., the imitation game)
- Biblical arguments about empirical matters is unreliable
- Mathematical objection:
- Some questions are answerable by humans and not by digital computers:
- Gödel's theorem shows that there are questions of logic not answerable, in principle, by a given computer
- The person framing the question can determine the answer
- Reply: there may be such questions for any given human
- Perhaps a computer could scan your brain and frame a question unanswerable by you, in principle
- The computer could computer the answer though
- The Gödel argument begs the question
- Some questions are answerable by humans and not by digital computers:
- Consciousness objection:
- We can know that something thinks only if we know that it has conscious experiences
- It is like something to be intelligent
- Reply: We know of our own conscious experiences only
- Solipsism: only I am known to be conscious
- To avoid absurdity, we must admit behavioural evidence for intelligence
- E.g., the imitation game
- We can know that something thinks only if we know that it has conscious experiences
- Originality objection:
- Lady Lovelace noted that Babbage's Analytical engine had no pretense to originality
- Perhaps it lacked enough capacity
- A digital computer simply obeys its instructions and so does nothing original, unlike intelligent humans
- Reply:
- The objection begs the question: Perhaps the same is true of humans
- "Machines take me by surprise with great frequency". But they are predictable in principle? See point 1.
- Lady Lovelace noted that Babbage's Analytical engine had no pretense to originality
- Theological objection:
- Turing deemphasizes physical constitution and emphasizes possession of knowledge
- Is hardware truly beside the point?
- Perhaps intelligence requires a brain
- The imitation game is indifferent to experience and learning
- Could a "brain in a vat" be intelligent?
- Perhaps intelligence requires a body
- Logic: the study of arguments
- Assess their strength - when should we be convinced by an argument and when should we not be convinced?
- Represent their content and form for assessment of the strength
- Example: Does the Bear Patrol work?
- Homer: Not a bear in sight. The Bear Patrol must be working like a charm.
- Lisa: That's specious reasoning, Dad.
- Homer: Thank you, dear.
- Lisa: By your logic I could claim that this rock keeps tigers away.
- What problems are there with Homer's argument?
- It is Homer's knowledge about bears, the content of the argument, that is the problem with his logic.
- Homer is relying on an undependable form of argument.
- The strength of the argument rests on its form and the reasons given
- Introduction to modern symbolic logic
- Propositional logic
- Predicate logic
- Probability
- Is formal logic a model for mental representations and procedures?
- Argument Form
- Aristotle (385-322 BC) found that form affects argument assessment
- formal logic
- Homer's argument:
- [There is a gear Patron] (Premise)
- Not a bear in sight (Premise)
- --> (underline)
- Therefore, the Boar Patrol keeps bears away (conclusion)
- Premises, underline, conclusion
- Aristotle (385-322 BC) found that form affects argument assessment
- Argument validity
- Syllogism: 2 premises (and 1 conclusion)
- All geese are birds.
- All birds have Wings.
- -->
- All geese have Wings.
- This argument is valid. If the premises are true, then the conclusion is true, also the next example:
- All lions are cats.
- All cats have fur.
- -->
- All lions have fur.
- This is still valid because if the premises were true, then the conclusion would be true:
- All lions are waffles
- All waffles are birds.
- -->
- All lions are birds.
- Not all argument forms are valid, only certain argument forms are inherently valid; even if the premises are true, the conclusion could be false, i.e. this argument is not valid:
- There is a Bear Patrol
- Not a bear in sight
- -->
- The Bear patrol keeps bears away
- Likewise:
- There is an anti-tiger rock.
- Not a tiger in sight,
- -->
- The anti-tiger rock keeps tigers away.
- Any argument of this form is non-valid
- Syllogism: 2 premises (and 1 conclusion)
- Problem (or limitation): not all valid arguments are syllogisms
- Boole (1815-1864): Treat logic like algebra
- If Socrates is a man, then he is mortal.
- Socrates is a man.
- -->
- Socrates is mortal.
- Becomes:
- S ⊃ M [S = "Socrates is a man"]
- S [M = "Socrates is mortal"]
- -->
- M
- The first premise is a pair of sentences (S, M) connected by "⊃"
- Simple sentences are single letters, e.g.,
- "Jill likes movies" [M], and
- "Jill likes starry skies" [S]
- Complex sentences are single letters combined by connectives, e.g.,
- Conjunction: "Jill likes movies and starry skies" [M & S]
- Disjunction: -Jill ikes movies or starry skies" [M ∨ S].
- Implication: "If Jill likes movies, then Jill likes starry skies" [M ⊃ S]
- Negation: "Jill does not like movies" [~M]
- More examples:
- The rent is due and I have no money. [R & ~M]
- London and Paris are national capitals. [L & P]
- Tme is not on my side. [~T]
- The campers were tired, but they were happy. [T & H]
- I will go hiking if I finish my work first. [F ⊃ H]
- If nominated I will not run, and if elected I will not serve. [(N ⊃ ~R) & (E ⊃ ~S)]
- Exercise: symbolize the following using the symbols given
- A conjunction has two components while a negation has only one. [C, N]
- Answer: [C & N]
- If we attempt this pay then we'll either win big or lose big. [A, W, L]
- Answer: [A ⊃ ( W ∨ L )]
- I will leave town unless you call me. [L, C]
- Answer: [~C ⊃ L]
- Skip class again and you won't pass the course. [S, P]
- Answer: [S ⊃ ~P]
- A conjunction has two components while a negation has only one. [C, N]
- Symbolization exposes form and validity:
- If the gear Patrol works, then no bears are in Sight.
- The Bear Patrol works.
- -->
- No bears in Sight.
- To the symbol, it will be:
- W ⊃ ~B
- W
- -->
- ~B
- An argument Of this form is called modus ponens and is always valid:
- p ⊃ q
- p
- -->
- q
- Not every form of argument is valid, e.g.,
- If the gear Patrol works, then no bears are in Sight.
- No bears in Sight.
- -->
- The Bear Patrol works.
- To the symbol, it will be:
- W ⊃ ~B
- ~B
- -->
- W
- This form is a fallacy: affirming the consequent. It is always non-valid:
- p ⊃ q
- q
- -->
- p
- The fact that an argument is a fallacy does not imply that the conclusion is false
- It simply means that the form of the argument is not enough to guarantee the truth of the conclusion
- Problem (limitation): many arguments valid in English are not valid in propositional logic, e.g.,
- A geese are birds.
- All birds have winqs.
- -->
- A geese have Wings.
- or, symbolicly:
- B
- W
- -->
- G
- Propositional logic does not symbolize content shared among statements
- Predicate logic addresses this deficiency
- Sentences are broken down into predicates and
terms, e.g.,
- "Bill has a great smile."
- "Jill is witty and intelligent."
- "Tina is taller than Jill."
- Becomes
- Sb [Sx: x has a great smile; b = Bill]
- Wj & Ij [Wx: x is witty; Ix: x is intelligent; j = Jill]
- Stj [Txy: x is taller than y; t = Tina]
- Quantifiers
- Quantifiers symbolize English quantity adverbs:
- The universal quantifier (∀): The sentence applies to every individual
- The existential quantifier (∃): The sentence applies to at least some individual.
- Quantifiers symbolize English quantity adverbs:
- For example:
- "Some people just do not listen. [(∃x)(Px & ~Lx)]"
- "All is well that ends well. [(∀x)(Ex ⊃ Wx)]"
- "Nobody likes a smartass. [~(∃x)(Px & (∀y)(Sy ⊃ Lxy)]"
- Exercise
- Symbolize the following using the symbols given
- Some people can't be bought (P, B).
- Answer: (∃x)(Px & ~Bx)
- A penny saved is a penny earned (P, S, E)
- Answer: (&any;x)[(Px & Sx) ⊃ Ex]
- Every dog has day (D, D', H).
- Answer: (&any;x)[(Dx ⊃ (∃y)(D'y & Hxy)]
- Some people can't be bought (P, B).
- Symbolize the following using the symbols given
- A valid predicate argument:
- A geese are birds. [(∀x)(Gx ⊃ Bx)]
- A birds have wings. [(∀x)(Bx ⊃ Wx)]
- -->
- A geese have wings. [(∀x)(Gx ⊃ Wx)]
- Another valid argument (different form):
- All men are mortal. [(∀x)(Mx ⊃ M'x)]
- Socrates is a man. [Ms]
- Socrates is mortal. [M's]
- We need to assess non-valid arguments too
- E.g., weather forecasts
- Apply domain-specific knowledge
- Probabilities can represent such knowledge
- E.g., the probability of rain is 40%
- Extend propositional logic for this purpose
- Probability of proposition p: pr(p) = [0...1]
- pr(p) = 0 if p is certainly false and
- pr(p) = 1 if p is certainly true.
- Examples:
- p = "It Will snow in January." pr(p) = .99
- p = "It Will snow in April." pr(p) = .65
- p = "It will snow in August." pr(p) = .02
- Probability rules
- Rules for probabi ities of complex sentences:
- pr(~p) = 1 - pr(p)
- pr(p∨q) = pr(p) + pr(q)
- pr(p&q) = pr(p) • pr(q)
- pr(p|q) = pr(p&q) / pr(q)
- Examples:
- pr("no snow in January") = 1 - pr("snow in January") = 1 - .99 = .01
- pr("1 or 2 on a die roll") = pr("1 on a die roll") + pr("2 on a die roll") = 1/6 + 1/6 = 1/3
- pr("snow in January and April") = pr("snow in January") • pr("snow in April") = .99 • .65 = .6435
- pr("someone is an artsie given that she's female") = pr("artists is female") / pr("female")
- Rules for probabi ities of complex sentences:
- An argument is convincing if
- pr(c|r) >> pr(~c|r)
- r = reason, c = conclusion
- Example:
- Verv often. it has snowed in January.
- -->
- Probably, it WI I snow next January.
- Symbolicly:
- J
- -->
- N
- Calculations:
- pr(N|J) = pr(N & J)/pr(J) = (.9 • .99)/.99 = .9
- pr(~N|J) = pr(~N & J)/pr(J) = (.1 • .99)/.99 = .1
- The argument is a strong one, probabilistically
- A weak argument?
- Not a bear in sight.
- -->
- The Bear Patrol works.
- Symbolicly:
- ~B
- -->
- W
- Calculations:
- pr(W|~B) = pr(W & ~B)/pr(~B) = (.5 • .99)/.99 = .5
- pr(~W|~B) = pr(~W & ~B)/pr(~B) = (.5 • .99)/.99 = .5
- The argument is weak
- Assuming pr(W) = pr(~W)
- Formal logic provides for the symbolization and evaluation of arguments
- Does logic capture laws of thought?
- Aristotle and Boole agreed, Frege did not
- Why did the pioneers of CogSci look to formal logic?
- Powerful and rigourous
- Amenable to computational modeling
- Logical thinking is a hallmark of intelligence
- Propositional logic captures some valid arguments, e.g., (modus ponens)
- If it rains, then the sidewalk gets Wet.
- It is raining.
- -->
- The sidewalk is wet.
- or symbolicly:
- R ⊃ W
- R
- -->
- W
- Predicate logic captures more valid arguments, e.g.,
- All men are mortal. [(∀x)(Mx ⊃ M'x)]
- Socrates is a man. [Ms]
- Socrates is mortal. [M's]
- Sentences
- To symbolize arguments, formal logic focuses on statements
- There are other kinds of sentences, e.g.,
- Questions: "How do I get to the Bookstore from here?"
- Orders: "Set your phasers to kill!"
- Requests: "Would you pass the salt, please?"
- Texts
- Not all texts are arguments, e.g.,
- I'm sorry but this reading initiative. I'm sorry, I've never been a fan of books. I don't trust them. They're all fact, no heart. I mean, they're elitist, telling us what is or isn't true, or what did or didn't happen. Who's Britannica to tell me the Panama Cana was built in 1914? If I want to say it was built in 1941, that's my right as an American! I'm with the prescient, let history decide what did or did not happen."
- Not all texts are arguments, e.g.,
- Representational limitations
- Predicate logic is specialized where natural languages are generalized
- Doesn't the generalized nature of language reflect the generalized nature of thinking, and so mental representations?
- Formal logic has been extended to address other kinds of sentences
- The extensions are complex and unwieldy in combination
- Argument construction is a model of intelligent thinking
- E.g., when Homer said
- "Not a bear in sight. The Bear patrol must like a charm,"
- was he thinking...?
- There is a Bear patrol
- Not a bear in Sight
- -->
- The Bear Patrol keeps bears away
- Perhaps thinking is applying rules to symbols, e.g.
- p ⊃ q
- p
- -->
- q
- Planning: represent goals and steps to achieve them
- E.g., Go from Guelph to UW:
- travel(I, Hwy-7) --> reach(I, Hwy-85)
- reach(I, Hwy-85) --> travel(I, Hwy-85)
- travel(I, Hwy-85) --> reach(I, University-Ave)
- reach(I, University-Ave) --> travel(I, University-Ave)
- travel(I, University-Ave) --> reach(I, UW)
- Note the change in notation favoured by Cognitive Scientists
- A route could be deduced from these rules
- Pro: if a route exists, deduction will determine it
- Cons: Many valid inferences are not helpful:
- p ["Conjunction"]
- q
- -->
- P & q
- If I travel Hwy-7 and Hwy-85, I could deduce:
- travel(I, Hwy-7) & travel(I, Hwy-85)
- travel(I, Hwy-7) & travel(I, Hwy-85) & travel(I, Hwy-7)
- The relevance of an inference s unconnected with its validity
- Deduction is monotonic
- Planning must often be non-monotonic
- E.g., a route s blocked
- monotonic
- non-monotonic
- monotonic
- E.g., a route s blocked
- Decision: choosing among plans
- Deduction only determines if plans exist
- Preferences need to be added, e.g., travel(I, Hwy-7) --> reach(I, Hwy-85) & prefer-to(I, Hwy-7, Hwy-401)
- Assumptions of this approach:
- I can completely order my preferences, and
- I can know all my preferences before I make my plans.
- Perhaps probabilities could address these assumptions
- E.g., decide among English, German, Philosophy courses
- pr("C is interesting" | "C is english")
- This solution is computationally explosive
- pr(A|B) must be known for every A and B
- For n predicates, there 2n-1 conditional probabilities
- Why doesn't my favorite website load?
- Your browser has a bug;
- Your connection s not working properly;
- Your server at your service provider is not working;
- The Website server is not working;
- The URL is incorrect.
- Some deductions are explanations (Hempel)
- ~respond(Website) --> incorrect(URL)
- ~respond(Website)
- -->
- incorrect(URL)
- Problems:
- Multiple explanations?
- Some deductions are not explanations, e.g., the height of a flagpole
- How do you explain the height of a flagpole?
- Some explanations are abductions (Peirce)
- down(Website-server) --> ~readable(Website)
- -readable(Website)
- -->
- down(Website-server)
- Some explanations are abductions (Peirce)
- Problems:
- Such inductions are risky
- Use conditional probability to determine the best explanation, e.g., pr("Website server is down"|"Website is not readable") = 0.4
- Abduction is a form Of learning
- Inductive generalization also, e.g.,
- Philosophy(Phil-128) & interesting(Phil-128)
- Philosophy(Phil-256) & interesting(Phil-256)
- -->
- (∀x)(Philosophy(x) --> interesting(x))
- Problem: risk jumping to conclusions
- Do you reason in this way? When?
- Subjects agree that modus ponens is deductive, but not affirming the consequent:
- If the Bear Patrol works, then no bears are in sight
- No bears are in sight
- -->
- Therefore, the Bear Patrol works
- Do people think deductively?
- Wason card task:
- given four cards from a deck with numbers on one side & letters on the other: [A] [B] [2] [3].
- Flip which cards to test the rule: If a card has an A on one side, then it has an even number on the Other side
- Most subjects select [A] and [2]; many omit [3]
- Explanations:
- People are not logical, do not apply modus tollens:
- p ⊃ q
- ~q
- -->
- ~p
- People employ schemata
- People employ mental models: represent [A] and even-number; assume only represented items are relevant
- People are not logical, do not apply modus tollens:
- People do not seem to think in accord with the axioms of probability, e.g.,
- pr(a) • pr(b) < pr(b)
- Suppose Frank likes to read French literature, attend foreign films, and discuss world politics
- People often judge that
- pr("college-educated") • pr("carpenter") > pr("carpenter")
- Instead of probability, people employ stereotypes
- Rule: an IF... THEN... structure modeled on implication (⊃), e.g.
- IF a king is in check AND no move can remove it from check THEN the checking player wins.
- Basic idea:
- Preserve the representational power of logic
- Extend and generalize it as needed
- Adaptations include:
- Different meaning for IF... THEN... structures
- Define and apply search strategies for rule use
- Logic Theorist (LT) developed by Newell & Simon (1950s)
- Imitate theorem-proving methods of students
- Used backward chaining and subgoaling
- Generalized Problem Solver (GPS) developed in the 1960s
- Solve any sort of problem
- Used means-ends analysis and difference lists
- Limitations of GPS include:
- Certain problems, e.g., chess, were beyond its capabilities
- Could not learn from experience
- Modern systems address limitations of GPS, e.g.,
- ACT-R (Anderson)
- SOAR (Newell, Laird, Rosenbloom)
- Expert systems, e.g., MYCIN, also
- Three components:
- representation of goal and initial condition,
- a database ot rules, and
- a strategy or algorithm for applying the rules
- Initial condition of Towers ot Hanoi problem:
- Peg 1 contains a, b, and c trom top to bottom
- Peg 2 is empty
- peg 3 is empty
- Disc a < disc b, disc b < disc c
- Goal: peg 3 contains discs a, b, and c from top to bottom
- Knowledge
- The rule database represents the knowledge of the system, e.g., the Towers of Hanoi problem.
- IF disc x is on top of peg i and peg j is empty THEN move disc x on top of peg j
- IF disc x is on top of peg i and disc y is on of peg j and x < y THEN move disc x on top of peg j
- Perhaps one rue would do, e.g., 3. IF peg 1 contains discs a, b and c from top to bottom THEN move a to 3, b to 2, a to 2, c to 3, a to 1, b to 2, a to 3
- Often, we are not so fortunate
- Rules may be combined to arrive at a plan
- IF disc a is on top of peg 1 and 3 is empty THEN move a to 3
- IF disc a is on top of peg 1 and 2 is empty THEN move b to 2
- IF disc a is on top of peg 3 and disc b is on top of peg 2 and a < b THEN move a to 2
- IF disc c is on top of peg 1 and 3 is empty THEN move c to 3
- IF disc b is on top of peg 2 and 1 is empty THEN move a to 1
- IF disc b is on top of peg 2 and disc c is on top of peg 3 and b < c THEN move a to 3
- IF disc a is on top of peg 2 and disc b is on top of peg 3 and a < b THEN move a to 3
- Rules are a so known as productions and rule-based systems as production systems
- Discussion questions
- In what situations to you apply rules? DO you apply Other kind Of knowledge then as well?
- What sorts of knowledge are difficult to capture in terms of rules?
- The rule database represents the knowledge of the system, e.g., the Towers of Hanoi problem.
- Determining which rules to combine and how is accomplished by search
- Search is guided by search strategies
- May be systematic, random, etc.
- Search through a database of rules is a knowledge search
- In a physical search,
- Search occurs in a specified area
- Begins in an initial location
- Visits neighbouring locations until the goal is found
- In a knowledge search
- Search occurs in a knowledge space
- Begins with an initial condition
- Visits neighbour'ng condit'ons until a goal condition is found
- Case: tic-tac-toe
- Initial condition: blank board
- Neighbouring conditions: next move
- Goal state: a win
- Forward strategy: compare current situation with IF conditiom on a match, apply THEN action
- Analogous to modus ponens
- Backward strategy: compare goal state to actions; on a match, adopt the IF condition as a subgoal
- Analogous to affirming the consequent
- UsedbyLT and GPS
- Bidirectional strategy: forward or backward
- Any strategy can be conducted in several ways (heuristics)
- Depth-first search: apply the first rule that matches
- Like traveling 'deep' into the knowlec$e space
- Breadth-first search: apply every rule that matches
- Like traveling across the space
- Best-first search: rank every rule that applies, attempt the best one first
- The ranking function is called the heuristic function
- Used by GPS
- Depends on the distribution of goals
- Tic-tac-toe: fordard, best-first
- Space small, goals plentiful
- Rules can be searched in predetermined order
- Win: IF a blank is flanked by two of my pieces THEN play it
- Block: IF a blank is flanked by two my opponent's pieces THEN play it
- Center: IF the center blank THEN play it
- Cornet: IF a corner is empty THEN play it
- Other: IF a Square is empty THEN it
- Chess: space large, goals sparse
- Opening: use "gambits", forward, breadth-first
- Midgame: adopt strong positions as subgoals, backward
- Endgame: wins available, backward, depth-first
- Strategies vary with expertise
- Experts can employ best-first, novices depth-first
- Main elements of rule-based systems:
- Database of rules
- Strategy for searching knowledge space
- Discussed problem solving
- Evaluation issues include:
- Are rules mental representations?
- Is search a mental procedure?
- Logic is highly specialized
- Rules are more flexible, e.g.,
- (∀)(Bx ⊃ Fx) "All birds fly."
- IF x is a bird THEN x flies. "Usually..."
- In logic, exceptions are disastrous, e.g.,
- (∀)(Bx ⊃ Fx) "All birds fly."
- (∀)(Px ⊃ Bx) "All penguins are birds."
- (∀)(Px ⊃ ~Fx) "No penguins fly."
- Pp "Pete is a penguin"
- Valid arguments would admit contradictions, e.g.,
- Pp
- (∀)(Px ⊃ Bx)
- (∀)(Bx ⊃ Fx)
- -->
- Fp
- or:
- (∀)(Px ⊃ ~Fx)
- Pp
- -->
- ~Fp
- Consider similar rules:
- penguin(Pete)
- IF penguin(x) THEN bird(x)
- IF bird(x) THEN fly(x)
- IF penguin(x) THEN not-fly(x)
- The system could retract 1,2,3 upon 1,4
- Exceptions are acceptable because the system is not absolutely committed to its conclusions
- Rule are interpreted as defaults
- Rule systems are non-monotonic
- They are not engaged in deduction
- On this view, deduction has little/no role in cognition
- Forward chairing (like deduction)
- IF Psych-101 fills up quickly THEN it has a popular professor
- Backward chaining (like abduction)
- IF Psych-101 has a popular professor THEN it up quickly.
- For the best explanation, attach a likelihood to each rule
- Difficult to anticipate
- Explanations could be generated by a rule trace
- IF a patient has some set Of symptoms THEN he has appendicitis
- The system can only say that it applied a given rule, not why the rule is appropriate
- Inductive generalization: use constant conjunctions to acquire rules
- After several instances, conclude IF a class has a popular professor THEN it fills up
- Can lead to inconsistent rules
- Chunking (composition):
- The first string is harder to recall than the second one:
- "r p l b v q m s d g"
- "I am the very model of a modern major-general"
- condense rules together. e.g., IF you travel Hwy-7, Hwy-86, University-Ave THEN you reach UW
- The first string is harder to recall than the second one:
- Specialization: acquire rules for exceptions, e.g.,
- IF you travel Hwy-7, HWY-85, University-Ave AND it'S rush hour THEN you reach UW
- Associative theories (pre-Chomsky)
- Generative theories (Chomsky)
- Apply rules to assign syntactic structure:
- S --> NP VP
- NP --> dogs, cats, cows, grass
- VP --> NP
- V --> chase, eat, admire
- Pinker: some conjugations learned "associatively", e.g., past tense of "sing" and "ring"
- Do expert systems scale up?
- Can specialized knowledge simply be combined to simulate general inte ligence?
- The scaling problem: adding more rules becomes ineffective
- Conflicts among rules increase
- Thrashing: managing conflicts dominates search effort
- Rules do well simulating expert performance
- Domain-specific
- Automatic and quick
- Acquired through training
- Novice performance
- Domain-general
- Tentative and slower
- Requires more effort
- Power law of practice
- Have you observed the power law in effect?
- Newell: combining chunks of knowledge (rules) is central to intelligence
- An expression the classical CogSci view
- Frame problem (Minsky):
- Intelligent beings need to distinguish relevant from irrelevant knowledge
- However, adding more rules to capture this need may become counterproductive
- What solutions might there be?
- Concept: a chunk of more-or-less general knowledge
- An idea or general description
- Functions of concepts include:
- Categorization, e.g., the good (Socrates)
- Configuration of experience, e.g., words (Kant)
- Foundation of inductive inferences, e.g., a growling dog (Smith)
- Epistemology:
- How well do concepts allow us to categorize things e.g., justice?
- Do concepts present a sensible picture of reality? (Is the concept human that of a featherless bped?)
- Psychology: What are concepts?
- Propositions (Hobbes), images (Aristotle), abstractions (Locke), words (Wittgenstein), frames (Minsky), or distributed representations (Hebb)?
- Learning: how are concepts acquired?
- Through experience, e.g., red (Locke)
- Innate (built-in), e.g., physics (Piaget)
- The classical view: concept X is the definition of X, the set of jointly necessary and sufficient conditions that must be had to be an X
- Necessary: properties all X must have
- Sufficient: anything With all the necessary properties is an X
- E.g., bachelor: an unmarried man
- Strengths of the classica theory:
- Applies well to technical (nominal) like bachelor, triangle, contract
- Limitations: natural kinds
- Centra properties seem dispensable, e.g., tiger
- Typicality effects, e.g., bird
- Exercise (pairs): One person define fruit, the Other vegetable (30 seconds)
- Prototypes: list of typical or standard features (Rosch 1970)
- Not all features are necessary
- "family resemblance" (Wittgenstein)
- Exercise: write down the typical features of a game (30 seconds)
- The typicality of instance I to prototype P is computed by a score of similarity (Tversky's contrast rule):
- Sim(I, P) = a•f(I & P) - b•f(P - I) - c•f(I - P)
- Strengths of the theory include:
- Explains Wpicallty effects, e.g., robin vs. penguin
- Applies to other concepts types, e.g., artifacts, psychological and psychiatric terms
- Limitations of the theory inc ude:
- Technical coru:epts exhbit typicality but not fuzziness
- Features do not weigh independently
- People do not discard information about class size or variability (e.g., exemplars)
| Small | Big | ||
|---|---|---|---|
| Wood | Small-wood-spoon | Big-wood-spoon | |
| ↑ ↓ | |||
| Metal | Small-metal-spoon | ↔ | Big-metal-spoon |
- Exemplar: a good instance Of a concept
- A robin is a good bird
- Classification is a score computed by
- Comparison with exemplars, or
- Construction of a prototype from exemplars
- Classification is done "on the fly
- E.g., Arts professor
- Strengths of the exemplar theory:
- Preserves typicality judgements
- Explains how people have access to class size and variability
- Which class is larger or more diverse, vegetable or spoon?
- Accounts for dependence Of features
- Limitations Of the exemplar theory:
- How are exemplars affected by learning general facts?
- Exemplars do not explain the existence of general categories
- Causal theory: X is a C if x obeys theories that apply to C
- E.g., tomatoes as entrees
- Explains natural/artifactual difference, e.g.
- A broom can become a hockey stick
- A goose cannot become a swan
- Explains centrality of some features
- Straight banana more typical than straight boomerang
- Limitations:
- Some causal theories have no effect, e.g., unicorn
- Method may depend upon context, e.g., quick and dirty method
- Exemplar and causal theories most promising
- Examine frames (Minsky, 1974)
- Slots and fillers
- Review procedures on frames:
- Inheritance
- Matching
- Psychological plausibility
- Frames
- Minsky was concerned about the relevance (frame) problem in logic and rules
- Rules do not tell us what not to do
- Proposal: collect relevant information together into frames, a list of slots and various fillers
- Also called schemas
- Frames can be a computational representation of concepts
- See the following example frame and script
- Suitable for stereotyped situations
- Automatism: Hilbert and the go-to-bed script
- Rules would lose relevance structure:
- IF you dine out THEN you get to the location
- IF you get to the location THEN you enter and be seated
- a-kind-of and subtype slots
- capture hierarchical organization of concepts
- Exercise: represent some
- concept as a frame
- The course frame
- Frame: course
- A kind of: educational process (sequence of events)
- Subtypes: lecture*, seminar, lab, correspsndence
- Instructor: ___
- Room: ___
- Meeting: ___
- Evaluation: exams*, quizzes, essays, ...
- Examples: Phil 256, Psych 101, ...
- Relations: 2 affects 4,5,6
- Frame: course
- The dine out script
- Frame: dine out
- A kind Of: dining event
- Subtypes: sit-down*, take-out, fancy-sit-down
- Location:
- Time:
- Get to location
- Enter and be seated
- peruse menu
- Order food
- Eat tcn»d
- Obtain and pay chequæ
- Examples: Burger King, Kooh-I-Noor, Mongolian Gri I
- Relations: 2 affects 5
- Frame: dine out
- Newell: knowledge search is the most basic ability Of an intelligent mind
- Matching is fundamental to concepts:
- Any process that relies on similarity to associate chunks of knowledge
- Concepts also involve inheritance:
- Information inherent in the configuration of knowledge
- Concepts inherit information through hierarchical organization
- Dogs have fur because a dog is a mammal
- Similar to rules (forward usage)
- IF x is a mammal THEN x has fur
- IF x is a dog THEN x is a mammal
- Slots may encode defaults, e.g.,
- Dog: Boppy ears
- Penguin: webbed feet
- Exceptions can be explicitly noted:
- (Mexican hairless) Pans: not-has-tur
- (Peruin) Abilities: not-fly
- Imagine following a link vs. searching Google
- Which concept best fits the current situation?
- Realized by competition for activation
- Contrained by excitatory and inhibitory links
- Concludes when activation pattern ceases to change
- E.g., deciding ona child's name
- Exploits content organization of mernory
- a-kind-of
- a-part-of
- examples
- Scripts can support planning
- E.g., howiwhen to do assignments
- Scripts can be inflexible (e.g., a hockey game?)
- Frames a so support explanations
- Why is there money on a restaurant table?
- Why is a Camero in the ditch? (overgeneralize)
- Why is there a goat in the restaurant? (inflexible)
- Definition
- zythum
- Specialization (zythum again)
- Copying with substitution
- Road rage, air rage, boat rage, computer rage, parking rage, shoppyng rage
- Generalization
- pull a Homer
- Combination
- Mouse potato, wavicle
- Frames replace search with association
- This move mitigates the frame problem
- But, is a concept-based account flexible enough to model intelligent behaviour?
- Analogy applies old concepts to novel situations
- E.g., advising the 80 year-old groom
- Examine and evaluate theories of analogy
- Esp. the multiconstraint theory (Holyoak & Thagard)
- Aristotle: analogies are 4-part proportions
- A:B :: C:D
- E.g., 2:4 :: 6:12
- E.g., wine-cup:Dionysus :: spear:Ares
- Consider the warfare analogy:
- Thebans:Phocians :: Athenians:Thebans
- Works via inductive generalization:
- lhs --> general rule --> rhs
- The generalization occurs in the first step
- Generalization
- The first step infers a rule from a single instance:
- It was wrong for the Thebans to attack the Phocians
- It is wrong for a state to attack its neighbour
- This process is an example of jumping to conclusions
- How to complete the following proportions?
- abc:abd :: xyz:??
- abc:abd :: kji:??
- The first step infers a rule from a single instance:
- The given information does little to constrain the rule in each case
- Extrapolation: continue a trend into an area of sparse data
- Analogy as extrapolation (Mill 1872):
- Items X and Y have features p, q, and r
- Item X has feature s
- -->
- Probably, item Y has feature s
- Robert Plot (ca. 1600) on arrowheads
- British artifacts and Indian artifacts are triangular, sharpened, and worn.
- Indian artifacts are used for war and hunting.
- -->
- Probably, British artifacts were used for war and hunting.
- Extrapolation
- The arrowheads analogy seems strong
- Consider the Earth and Moon
- There are many similarities
- The Earth is inhabited
- -->
- Probably, the moon is inhabited
- Is this argument strong?
- The extrapolation seems indifferent to relevance
- The Multiconstraint theory
- Rule-based accounts emphasize the weaknesses Of analogy
- Associative accounts identify its strengths
- E.g the Multiconstraint theory (MT) of Holyoak and Thagard (legs)
- Analogy: an alignment of structured sets of concepts
- Analogical mappings
- The Ford Excursion, per Dan Becker:
- "It's basically a garbage truck that dumps into the sky."
- The Ford Excursion, per Dan Becker:
- Analogical coherence
- The goodness of an analogy is its coherence:
- Structural consistency: the analogy should exhibit a one-to-one correspondence (systematicity in Gentner).
- Similarity: corresponding items should be similar.
- Pragmatic utility: the analogy addresses the problem at hand.
- Coherence is a matter of degree
- There are severa possible system relations, e.g., etc.
- The goodness of an analogy is its coherence:
- Analogy is important to a concept-based account of cognition
- Examine the power of analogical thinking
- As per the MT of analogy
- Explore its possibilities and limitations
- Some analogies are essentially verbal, e.g.,
The English Department will not accept that a fine novelist like Stone has anything to contribute to my literary education. Having Stone teach literature is, in their eyes, like having a gorilla teach zoology.
-
Analogical locutions include:
- "be like", properly conjugated (as above),
- "likewise", "similarly", or
- "x is the equivalent of y", "x is the y of z"
-
For example:
Writing about mUSic is like dancing about architecture—it's a really stupid thing to want to do." (Elvis Costello)
- Some visual analogies concern static spatial relationships
- E.g., inside-of(A,B), left(C,D)
- Visual analogies can a so involve dynamic (changing) relationships
- E.g., Duncker's (1926) tumor/fortress problem
- Analogies can capture an emotional experience
- E.g., someone "letting off steam"
- Analogies can induce an emotional experience
- E.g. David Wolf being left at Mir
- Analogies are typically specific
- They link two specific situations
- If analogies are information-poor, then can they be reliable?
- Mil: "no"
- MT: "somewhat"
- Analogy evaluation involves more than just similarity
- Often begins With an impasse (missing information)
- Possibe so utions include
- Being given a source analog, or
- Locating and retrieving one trom memory
- Retrieval is affected by similarity
- E.g., Copernicus's Earth/ship analogy On the MT, resemblance does not much affect analogy evaluation When aligning ana ogs, structural consistency (systematicity) is paramount
- Often involves copy with substitution
- Results in a candidate inference The candidate inference may require adaptation
- E.g., CHEF'S Stir-fry recipes
- Analogies may be used to explain, e.g.,
- Copernicus's Earth/ship analogy
- Darwin's analogy between human and animal population growth
- Consider the "other-minds problem
- How do you know that other people have minds like yours?
- Abduction:
- X has a mind --> X behaves intelligently
- You intelligently
- -->
- you probably have a mind
- An analogical abduction: My mind causes my behaviour. You behave similarly, so you have a simllar mind that causes your behaviour
- Do you explain other people's behaviour analogically? When is this practice convincing?
- People may learn analogically
- E.g., writing an essay Napoleon might like writing an essay on Julius Caesar
- The re-application of analogies may lead to schema induction
- E.g., the problem with a fire-fighter analog added
- Some metaphors are based on analogies (Aristotle):
- E.g., "my job is a jail"
- In what way is a job a jail?
- Not all metaphors are analogical
- E.g., "Ottawa says ..."
- Analogy processing is voluntary; metaphor processing seems obligatory
- E.g., "some desks are junkyards" vs. "some desks are roads"
- False analogy: A comparison that misrepresents a situation
- E.g. Quebec has the right to secede from Canada, just as Palestine has the right to break away from Israel.
-
Limits: counteranalogies
-
Limits: counteranalogies Counteranalogy: a comparison that contradicts another comparison
- E.g., miracles and theology (ca. 1700):
- God is like a perfect watchmaker (Leibniz)
- God is like a model king (Clarke)
- Much classic CogSci concerns symbols
- Another possible mental representation would be images
- A representation that preserves qualities of perception, e.g., seeing an apple
- Images correspond to perceptual modalities
- Difficult questions:
- What kind of thing is a mental image?
- Do mental images really participate in cognition?
- Examine the imagery debate
- The experience of the "mind's eye" is commonplace
- People can answer questions using visual mental imagery
- E.g., what did you have for breakfast? (Galton, 1822-1911)
- Aristotle introduced the concept of the faculty of imagination
- claimed that all thought involves images
- Criticisms of the importance of imagery:
- Descartes (1569—1650): imagine a chiliagon!
- Berkeley (1683—1753): an image Of a triangle is always of a part'cular kind, e.g., 'sosceles
- These points ead to skeptic'sm about images as concepts
- Supporters and skeptics
- A pictorialist account of visual mental imagery emerged in the 1960s (Kosslyn)
- Visual mental images the things they represent, and
- Visual mental images can play a substantial role in intelligent thinking.
- A descriptionist account is supported by critics (Pylyshyn)
- Visual mental images are really descriptions, propositions, that contain symbolic information about the things they represent, and
- The phenomenon that we experience as visual mental imagery, i.e., the mind's eye, plays no role substantial role in cognition.
- A pictorialist account of visual mental imagery emerged in the 1960s (Kosslyn)
- How does an image resemble what it represents?
- "tiger" does not resemble a tiger
- Every part of the image corresponds to a part of what the image represents, and
- Proximity and adjacency relations among parts of an image correspond to the relations among the parts of what the image represents.
- "tiger" does not resemble a tiger
- Mental images are ana ogous to graphical files on a computer
- JPEG --> bitmap in a display buffer
- LTM --> STM in a visual buffer (but no dsplay!)
- Shepard & Metzler (1971) asked subjects if pairs of figures were both the same object
- Result: a linear relationship between rotation angle and decision time
Kosslyn, Ball & Reiser (1978) asked subjects to memorize a map and answer questions about locations on it - Control: consider whole map first - Experimental: focus on one location first
- Result: response time in experimental group was a linear function Of the distance between locations
- Kosslyn (1975) asked subjects to imagine either
- a rabbit next to an elephant, or
- a rabbit next to a mouse.
- Result: When asked if the rabbit had red eyes, subjects were quicker with the larger rabbit (2)
- In these experiments, subjects are instructed to think visually
- Perhaps the regults are due to the subjects trying to please the experimenters (demand characteristic)
- Finke and Pinker (1982) asked subjects to say whether an arrow points to a dot
- Result: response time was a linear function Of the distance between arrow and dot (when they aligned)
- Also, more errors occurred when arrow and dot were close together
- Some fMRI studies suggest V1 is connected With visual mental imagery (e.g., Kosslyn et al. 995)
- Toote et al. (lg82) suggest that monkey V1 is retinotopically mapped
- perhaps V1 is (part of) the visual buffer
- However, many fMRI studies do not corroborate this account
- E.g. , V1 was not active while subjects located a dot within an imagined figure (Knauff et al. 2000)
- Perhaps vision and visual mental imagery can interfere (compete for the same resource)
- E.g., daydreaming might prevent visual memory
- If so, then imagery and perception share the visual buffer
- Segal and Fusella (1970) showed that imagery could interfere with same-modality perception
- Pylyshyn argues that both tasks demand application of similar concepts, eading to confusion
- Main issues in the imagery debate:
- What kind of thing is a mental image?
- Do mental images really participate in cognition?
- Pictorialist versus descriptionist answers
- Philosophical question: Is imagery too tied to perception to represent general knowledge?
- E.g., "triangularity"
- Review the case for descriptionism
- Examine array theory of imagery
- Defenses of descriptionism are often attacks on pictorialism (e.g., Dennett)
- Infinite regress
- For an image to be a representation, it must be perceived
- This requirement leads to an infinite regress of perceivers
- Pictorialism is a conceptual muddle
- This argument conflates two issues:
- A representation is something manipulated by procedures
- Images and symbols are in the same position here
- Intentionality: the aboutness of representations
- This problem applies to any representation
- There is nothing muddled about preserving perceptual information instead of eliminating it
- A representation is something manipulated by procedures
- Pictorialists hedge the concept of image until it means nothing
- What is a quasi-image but a weasel word?
- Perhaps pictorialism is unscientific
- Why would image be a complex concept?
- The phenomenon is complex
- Available methods Of inquiry are not adequate
- Imagery often omits details
- An a tiger versus a picture
- A description might just gay "numerous"
- An image is just a description
- Reply:
- A sketch is pictoria whilo omitting details
- may be represented separately, e.g., shape and texture
- Counterargument: Unlike descriptions, images have obligatory features, e.g., posture
- What are some obligatory features of images?
- people have trouble reinterpreting ambiguous figures, e.g. Wittgenstein's duck/rabbit
- Memorization and mental rotation of figures (Slezak 1995).
- Descriptionist view: images are just symbolic interpretations of perceptions
- Pictorialist reply:
- people can sometimes perform this such tasks
- Kossvn (1994): image parts fade quickly from the visual buffer unless we attend to them
- Tsal and Kolbet (1985): interpretation tends to attention to central features
- Chambers: When we generate a mental image, we attend to central features and others fade, preventing reinterpretation
- Which features are central to the duck or rabbit interpretation of Wittgenstein's figure?
- Anderson (1978) shows that pictorialism is behaviourally equivalent to descriptionism
- Can evidence ever be conclusive in the debate?
- Perhaps pictorialism is simpler
- Pictorialism also involves claims about the brain, e.g., the visual buffer
- Array theory (Glasgow and Papadias 1992):
- Deep, spatial, and visual representations
- Deep representation: a frame, e.g.,
- Frame: Map-of-Europe
- a-kind-of: map-of-continent
- parts: Sweden (0,4), Britain (1,0), ...
- procedures: find-population, ...
- Frame: Map-of-Europe
- An array capturing adjacency relationships
- Permits visual solutions to problems
- E.g., is Sweden north Of Germany?
- Contrast with a rule-based model:
- IF north-of(x, y) and north-of(y, z) THEN north-of(x, y)
- north-of(Britain, Portugal)
- north-of(Denmark, Germany)
- north-of(Sweden, Denmark)
- ...
- Arrays can also be 3D
- E.g., represent the physical structure of chemicals
- Possible alternative: graphs
- An occupancy array approximates the shape of an object
- Perspective dependent
- Supports procedures likes zoom, rotate, translate
- Neurological evidence?
- Spatial arrays imitate the "what" system
- Visual arrays imitate the "where" system
- Visual mental imagery lacks generality
- Portugal is other north or south of France.
- There is no duck on the table.
- Imagery cannot unambiguously represent these situations
- Scientific discovery, e.g.
- Continenta drift (Wegener 1920)
- Special relativity (Einstein 1905)
- Technological innovation, e.g.
- Nikola Tesla (1856-1943)
- Temple Grandin