Instead of representing our beliefs and uncertainty via probability distributions, we represent them via something called an “ultracontribution”. An ultracontribution is a set of “contributions”, where a contribution is like a probability distribution but the sum of probabilities doesn’t add up to 1. The point of the ultracontribution is to represent being completely uncertain about which contribution is true. For now, you can pretend ultracontributions are just probability distributions, but we’ll need to come back to what they actually are later. The space of ultracontributions over a set X is denoted , so this is the set of all beliefs you could have about X.
The most obvious thing I can be uncertain about is what the physical world is like. For example, does the store have strawberries? Do opposite charges attract or repel? We’ll take the big picture and say I have beliefs about the entire state of the physical universe. (“Phi”, presumably for “physical”) is the set of possible ways the physical world could be, and an element of it is a possible physical world. This might look like a specification of the fundamental physical laws along with an initial state (e.g. if the world was Conway’s Game of Life, a ‘possible world’ could mean specifying the birth and death rules of the cellular automata along with what cells were alive at the initial time). Of course, most possible physical worlds aren’t true.
There’s something else I could be uncertain of. I am incapable of performing infinite computation; so I don’t know all the consequences of my beliefs, don’t know what the 100th digit of pi is, can’t just tell you what every program outputs. Humans in practice reason under uncertainty about logical propositions all the time [citation needed], but it’s not so easy to figure out a mathematical theory of this that works.
The proposal here is to look at ways that “math” could be. Here, we will just look at outputs of possible programs, for example I might think that the python program print(1 + 1) will output 2, or maybe I made a mistake and think it’ll output 3. Taking the big picture again, I have beliefs about what every possible computation would output. So we have these “computational worlds” that specify an output for every possible program. The set of possible computational worlds is denoted . Of course, most possible computational worlds aren’t true.
My beliefs about the physical and computational worlds might be related. For example, maybe I believe that if bubbles form spheres in the physical world, that it must then be the case that spheres have the smallest surface area to volume ratio (so a program that tries to prove that fact should succeed). Or maybe I only think the inverse square law for gravity is likely if when I plug in the properties of the Earth’s orbit I get enough force to keep it there.
We therefore want to look at “joint” beliefs about how likely a pair is, where is a possible computational world and is a possible physical world. Let be such a belief—it lives in .
We will now want to talk about computational worlds that are “consistent” with a certain physical world (like perhaps you live in a weird world where only the 1st digit of pi matters for physics, so that a consistent set of computational worlds is all worlds that have the right 1st digit). Let be the set of subsets of computational worlds, so an element in it is a set of possible computational worlds.
We want to have some fancy function, called the bridge transform, that: - Takes in our joint belief about what computational and physical world we’re in, that is, a belief over . These beliefs are talking about cases that look like where are a computational and physical world respectively. A case like this is saying “I’m in computational world and physical world “ —Spits out a belief about what computational and physical world we’re in, and also what computational worlds are consistent with the physical world, that is, a belief over . These beliefs are talking about cases that look like where is a set of computational worlds. A case like this is saying “I’m in this computational and physical world, and also are the computational worlds that are consistent with this physical world”.
We’ll make our function by specifying what the output belief should look like. Remember that “belief” here means a set of contributions (which are each like a probability distributions), so to specify the output belief we need to specify what contributions are in our output set.
We’re going to require two things of the contributions in our output set (the post uses but I want to use a different symbol so it’s clear that lives in the output belief instead of the input belief ): 1. If we say the computational world is and the physical world is and the consistent computational worlds are … then obviously should be in . Because otherwise you’re saying “The correct outputs of computations aren’t consistent with the physical world I’m in”. Therefore we want our contribution to only assign nonzero “probability” when we have this consistency. This is what - means “place where it’s nonzero”, and is the subset of pairs where actually holds. 2. Our contribution in our belief about consistency should “give rise” to (a contribution in) our input belief on possible computational and physical worlds. By “give rise”, we mean that if we:
I. “shuffle” the computational worlds by some function (while we leave the physical world and the consistency set the same) II. remove the bits of our shuffle that assign probability mass to computational worlds that aren’t in the consistent set
then we should get something that is a contribution in our original beliefs (interpreted by throwing away the consistency set at this point). The idea is (I think) that if we say the set of consistent computational worlds is for some physical world , then what this means is that we believe it possible to have any computational world from be the case along with .
To see this interpretation: note that this process is linear in the input contributions, and so it’s determined by what happens to “single possibility” contributions (this assigns all probability to the case ). For these contributions, the effect of any “shuffle function” is just to change what we put all our mass on (and we only get nonzero contributions from the formula when we change it to something that’s still consistent, that is, is still in ).
So the condition for these single case contributions is that a (computational world, consistent set, and physical world) is allowed if no matter what other consistent computational world we could be in, we think there’s a possibility we are in that (computational, physical) world.
As an example, if our input belief ultracontribution has the single possibility (delta) contributions: 1. We could have computational world X and physical world A, or 2. We could have Y and A 3. We could have Z and A but for B, we only have X and B or Y and B. (and then we find the smallest closed convex downward closed set with these 5 delta contributions so its actually an ultracontribution)
Then our bridge transform should output (the closed convex downward closed set generated by, though I’m only guessing whether the closed condition actually transfers over nicely) the delta contributions etc. for B. The point here is that the possible consistency sets for a and a are those where any in that consistency set is believed by us to be possible to happen along with , and so for example we don’t have because (Z,B) is not something we believe is possible, so we shouldn’t say that Z’s compatible with B.
To get arbitrary contributions you need to assign probability weights to the deltas and then add them up, but I think the intuition is the same.
Can someone break Definition 1.1 down for me? I got lost in all the notation and what acts on what, what is projected to where..
Here’s my attempt:
Instead of representing our beliefs and uncertainty via probability distributions, we represent them via something called an “ultracontribution”. An ultracontribution is a set of “contributions”, where a contribution is like a probability distribution but the sum of probabilities doesn’t add up to 1. The point of the ultracontribution is to represent being completely uncertain about which contribution is true. For now, you can pretend ultracontributions are just probability distributions, but we’ll need to come back to what they actually are later. The space of ultracontributions over a set X is denoted
The most obvious thing I can be uncertain about is what the physical world is like. For example, does the store have strawberries? Do opposite charges attract or repel? We’ll take the big picture and say I have beliefs about the entire state of the physical universe.
There’s something else I could be uncertain of. I am incapable of performing infinite computation; so I don’t know all the consequences of my beliefs, don’t know what the 100th digit of pi is, can’t just tell you what every program outputs. Humans in practice reason under uncertainty about logical propositions all the time [citation needed], but it’s not so easy to figure out a mathematical theory of this that works.
The proposal here is to look at ways that “math” could be. Here, we will just look at outputs of possible programs, for example I might think that the python program
print(1 + 1)will output 2, or maybe I made a mistake and think it’ll output 3. Taking the big picture again, I have beliefs about what every possible computation would output. So we have these “computational worlds” that specify an output for every possible program. The set of possible computational worlds is denotedMy beliefs about the physical and computational worlds might be related. For example, maybe I believe that if bubbles form spheres in the physical world, that it must then be the case that spheres have the smallest surface area to volume ratio (so a program that tries to prove that fact should succeed). Or maybe I only think the inverse square law for gravity is likely if when I plug in the properties of the Earth’s orbit I get enough force to keep it there.
We therefore want to look at “joint” beliefs about how likely a pair
We will now want to talk about computational worlds that are “consistent” with a certain physical world (like perhaps you live in a weird world where only the 1st digit of pi matters for physics, so that a consistent set of computational worlds is all worlds that have the right 1st digit). Let
We want to have some fancy function, called the bridge transform, that:
- Takes in our joint belief about what computational and physical world we’re in, that is, a belief
—Spits out a belief about what computational and physical world we’re in, and also what computational worlds are consistent with the physical world, that is, a belief over
We’ll make our function by specifying what the output belief should look like. Remember that “belief” here means a set of contributions (which are each like a probability distributions), so to specify the output belief we need to specify what contributions are in our output set.
We’re going to require two things of the contributions
1. If we say the computational world is
2. Our contribution
I. “shuffle” the computational worlds by some function
II. remove the bits of our shuffle that assign probability mass to computational worlds that aren’t in the consistent set
then we should get something that is a contribution in our original beliefs (interpreted by throwing away the consistency set at this point). The idea is (I think) that if we say the set of consistent computational worlds is
To see this interpretation: note that this process is linear in the input contributions, and so it’s determined by what happens to “single possibility” contributions
So the condition for these single case contributions is that a (computational world, consistent set, and physical world) is allowed if no matter what other consistent computational world we could be in, we think there’s a possibility we are in that (computational, physical) world.
As an example, if our input belief ultracontribution
1. We could have computational world X and physical world A, or
2. We could have Y and A
3. We could have Z and A
but for B, we only have X and B or Y and B.
(and then we find the smallest closed convex downward closed set with these 5 delta contributions so its actually an ultracontribution)
Then our bridge transform should output (the closed convex downward closed set generated by, though I’m only guessing whether the closed condition actually transfers over nicely) the delta contributions
etc. for B. The point here is that the possible consistency sets for a
To get arbitrary contributions you need to assign probability weights to the deltas and then add them up, but I think the intuition is the same.