topic
my notes
Combinatorial Explosion
- Diagram is misleading because by God’s eye view, but in life we’re not out there, we’re at the initial state - ignorant
- Problem solving method is any method finding the sequence while obeying the path constraints
- Diagram not complete
- Can calculate number of pathways by calculating F^D
- F = number of operators I’m applying at any stage
- D = number of stages
- Ex: chess:
- on any turn number of operations is 30 on any turn
- On average 60 turns
- Combinatorial explosion
- Vast number 4.29 X 10^88
- Incomprehensibly large
- Have about 10^10 neurons
- Have 5 x 10^15 neuronal paths
- Greater than the number of particles estimated to exist in the universe
- So can’t search the whole space
- Since can’t search the whole space - what we do is zero in on a small subspace, and often find a solution. Zero in on the relevant information.
- How do we do that?
- Issue of avoiding combinatorial explosion is a central way of understanding intelligence
- Experience in two ways:
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- generation of obviousness: what we have to explain
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- how things are salient to us, how they stand out to us, grab our attention
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- That process isn’t static, sometimes one is more than the other
- Dynamically self-organize what we find relevant and salient
- Newell and Simon realized have to deal with combinatorial explosion
- Proposed: heuristic and algorithm
Algorithm
- Problem solving technique guaranteed to find a solution or prove one can’t be found
- Standard of certainty
- So have to search the whole problem space to guarantee certainty
- Cognitive suicide if want certainty because space is combinatorially explosive
- That’s why Descartes doomed
- Deductive logic is certainty
- Logic and math operate algorithmically - we can’t be comprehensively logical
- Therefore rationalinity =/ being logical
Rationality
- Being rational means knowing when, where, how much, and what degree to be logical in order to overcome self-deception and optimally achieve the goals that we want to achieve
- Not just logic or consistency
Heuristic
- Problem solving method not guaranteed to find a solution, reliable for increasing chances of achieving your goal
- Can’t play chess algorithmically, can play chess heuristically
- Get queen out early
- Control board
- castle
- Try to pre-spcifiy where you should search for the relevant info: limits the space searching
- Prejudge what is going to be relevant
- Heuristic = bias
- Bias where we’re paying attention
- No free lunch: have to use heuristics to avoid combinatorial explosion, price you pay is you fall prey to bias.
(This makes a point that I’ve made a lot: bias is not inherently bad. It is part of the human condition. It is often useful.there’s just too much information to take into account. But it has limits and can lead us astray.)
- Bias is adaptive: very thing that makes us adaptive makes us prone to self-deception
- Ex: representative and availability heuristics: can’t calculate exactly so how many plane crashes can I remember
Newell and Simon
- V respects NandS:
- What makes us intelligent is our ability to use heuristics
- Necessary part: powerful work. Add one more dimension to what it is to do good cognitive science.
- All the great changes that make the scientific way possible is exemplified by them:
- Analyzing, taking complex phenomena breaking it down
- Like Descartes trying to formalize it: graphical mathematical
- Trying to mechanize: I’ve got this right if I can make a machine to carry out my formal analysis
- Trying to explain the mind often fall into a particular fallacy:
- See a triangle, light comes off of it, goes into eye. Into the space (working memory), projected inner screen, homunculus (little man) says triangle
- Sounds like giving mechanical explanation but how does the little man see - well inside his head, etc.
- Gets an infinite regress. Doesn’t explain anything
- Using vision to explain vision
- Circular
- N+S taking a mental term trying to formalize using non-mental terms.
- Naturalistic imperative: try to explain things naturalistically
- Doing this to try and avoid circular explanations of intelligence
- Scientific revolution seems to explain everything except for how I generate scientific explanations - consciousness
- Hole in the naturalistic worldview that’s why many zero in on our capacity to make meaning have consciousness as the thing that’s not being explained
- Right to do that, but wrong to conclude it legitimates other worldviews
- Need to show this project is failing, that not making progress on it.
- Can’t defeat a scientific program by showing problems - what have to do is point to fact that not making any progress in coming up with explanation
- Hard to say we’re not making progress in explaining intelligence by trying to analyze, formalize, mechanize it
- Critique:
- N+S notion of heuristic while necessary is insufficient.
- Didn’t pay attention to other ways we constrain the problem space and zero in on relevant information
- Didn’t notice that they had an assumption in their attempt to come up with a theoretical construct for problem solving. Assumed all problem solving the sam
- Heuristic of essentialism:
- Essentialism: when group a bunch of things with a term, they must all share some core property (essence)
- Some things fall into that (ex: triangles), but not all
- Not everything we group together has an essence
- Ex: call many things games: what set of necessary conditions meet all games - won’t find a definition that meets all and only games
- Science discovers things that have an essence
- Treat any category as if has an essence (heuristic), but many categories don’t have essences
- We look for essences because it allows us to generalize. We can overgeneralize, but also undergeneralize - also a mistake
- N+S thought that problems had an essence, that all problems essentially the same, so to make a general problem solver need one essential problem solving strategy
- essentialism is not a bad thing, we need it
- Fundamentally different kinds of problems:
Well defined vs. ill defined problems
- Well-defined problem: good meaning and effective guiding representation of the initial state and goal state
- Psycho-technologies make well-defined problems for us
(how has this been scientifically established?)
- Can get blinded that that’s how most problems are like
- Most problems ill-defined: don’t know what the relevant information about the initial stat or goal state are, or relevant operators, or even path constraints
- Problem: take good notes.
- Initial state;Don’t have good notes
- Pay attention to relevant information
- But how? How make a machine to do that?
- Operations: write stuff down: do you? What stuff write down? Everyone’s notes look different
- What does the goal state look like? What do good notes look like?
- What’s missing in an ill-defined problem is how to formulate the problem: Zero in on the relevant information constrain the problem to solve it
- Good problem formulation
- If they had noted this, they would have realized that the important work being done by problem formulation
- Following a conversation is an ill-defined problem, go on a successful first date
- Need to be able to deal with ill-defined problems to be considered intelligent
Mutilated Chessboard Problems
- If have dominos, covers 2 squares, need 32
- Now mutilate chessboard, remove a couple squares
- Now, can cover without overlap
- Many find it a hard problem because formulate it as a covering problem. Trying to imagine the chessboard and possible configurations of dominos
- Covering strategy, try and imagine it.
- Combinatorially explosive
- Two squares taken off are the same colour, but not standing out in a way that makes the solution obvious.
- If put piece down always covering one black and one white, no way to put it on the board without covering a black and white square. To cover the whole board need an equal number of - there isn’t an equal number of black and white squares
- Can prove that it is impossible.
- Parity strategy - now solution is obvious, covering strategy combinatorially impossible
- This is why flow and higher states of consciousness so relevant. Capacity to come up with good problem solving formulation - that’s insight
- Insight in addition to logic is central to rationality
- In addition to logical techniques to improve inference, need to have other kinds of psycho-technologies to improve capacity for insight.
(Bam! Brought it home!)
- Involves mindfulness - gives ability to restructure your salience landscape
- Starting to see how problem formulation and relevance realization central to being a real world problem solver, avoiding combinatorial explosions, avoiding ill definedness