[Link] Quantum theory as the most robust description of reproducible experiments
The paper: http://www.sciencedirect.com/science/article/pii/S000349161400102X
Authors: Hans De Raedt, Mikhail I. Katsnelson, Kristel Michielsen
Abstract
It is shown that the basic equations of quantum theory can be obtained from a straightforward application of logical inference to experiments for which there is uncertainty about individual events and for which the frequencies of the observed events are robust with respect to small changes in the conditions under which the experiments are carried out.
Logical Uncertainty as Probability
This post is a long answer to this comment by cousin_it:
Logical uncertainty is weird because it doesn't exactly obey the rules of probability. You can't have a consistent probability assignment that says axioms are 100% true but the millionth digit of pi has a 50% chance of being odd.
I'd like to attempt to formally define logical uncertainty in terms of probability. Don't know if what results is in any way novel or useful, but.
Let X be a finite set of true statements of some formal system F extending propositional calculus, like Peano Arithmetic. X is supposed to represent a set of logical/mathematical beliefs of some finite reasoning agent.
Given any X, we can define its "Obvious Logical Closure" OLC(X), an infinite set of statements producible from X by applying the rules and axioms of propositional calculus. An important property of OLC(X) is that it is decidable: for any statement S it is possible to find out whether S is true (S∈OLC(X)), false ("~S"∈OLC(X)), or uncertain (neither).
We can now define the "conditional" probability P(*|X) as a function from {the statements of F} to [0,1] satisfying the axioms:
Axiom 1: Known true statements have probability 1:
P(S|X)=1 iff S∈OLC(X)
Axiom 2: The probability of a disjunction of mutually exclusive statements is equal to the sum of their probabilities:
"~(A∧B)"∈OLC(X) implies P("A∨B"|X) = P(A|X) + P(B|X)
From these axioms we can get all the expected behavior of the probabilities:
P("~S"|X) = 1 - P(S|X)
P(S|X)=0 iff "~S"∈OLC(X)
0 < P(S|X) < 1 iff S∉OLC(X) and "~S"∉OLC(X)
"A=>B"∈OLC(X) implies P(A|X)≤P(B|X)
"A<=>B"∈OLC(X) implies P(A|X)=P(B|X)
etc.
This is still insufficient to calculate an actual probability value for any uncertain statement. Additional principles are required. For example, the Consistency Desideratum of Jaynes: "equivalent states of knowledge must be represented by the same probability values".
Definition: two statements A and B are indistinguishable relative to X iff there exists an isomorphism between OLC(X∪{A}) and OLC(X∪{B}), which is identity on X, and which maps A to B.
[Isomorphism here is a 1-1 function f preserving all logical operations: f(A∨B)=f(A)∨f(B), f(~~A)=~~f(A), etc.]
Axiom 3: If A and B are indistinguishable relative to X, then P(A|X) = P(B|X).
Proposition: Let X be the set of statements representing my current mathematical knowledge, translated into F. Then the statements "millionth digit of PI is odd" and "millionth digit of PI is even" are indistinguishable relative to X.
Corollary: P(millionth digit of PI is odd | my current mathematical knowledge) = 1/2.
Against the Bottom Line
In the spirit of contrarianism, I'd like to argue against The Bottom Line.
As I understand the post, its idea is that a rationalist should never "start with a bottom line and then fill out the arguments".
It sounds neat, but I think it is not psychologically feasible. I find that whenever I actually argue, I always have the conclusion already written. Without it, it is impossible to have any direction, and an argument without any direction does not go anywhere.
What actually happens is:
- I arrive at a conclusion, intuitively, as a result of a process which is usually closed to introspection.
- I write the bottom line, and look for a chain of reasoning that supports it.
- I check the argument and modify/discard it or parts of it if any are found defective.
It is at the point 3 that the biases really struck. Motivated Stopping makes me stop checking too early, and Motivated Continuation makes me look for better arguments when defective ones are found for the conclusion I seek, but not for alternatives, resulting in Straw Men.
Difference between CDT and ADT/UDT as constant programs
After some thinking, I came upon an idea how to define the difference between CDT and UDT within the constant programs framework. I would post it as a comment, but it is rather long...
The idea is to separate the cognitive part of an agent into three separate modules:
1. Simulator: given the code for a parameterless function X(), the Simulator tries to evaluate it, spending L computation steps. The result is either proving that X()=x for some value x, or leaving X() unknown.
2. Correlator: given the code for two functions X(...) and Y(...), the Correlator checks for proofs (of length up to P) of structural similarity between the source codes of the functions, trying to prove correlations X(...)=Y(...).
[Note: the Simulator and the Correlator can use the results of each other, so that:
If simulator proves that A()=x, then correlator can prove that A()+B() = x+B()
If correlator proves that A()=B(), then simulator can skip simulation when proving that (A()==B() ? 1 : 2) = 1]
3. Executive: allocates tasks and resources to Simulator and Correlator in some systematic manner, trying to get them to prove the "moral arguments"
Self()=x => U()=u
or
( Self()=x => U()=ux AND Self()=y => U()=uy ) => ux > uy,
and returns the best found action.
Now, CDT can be defined as an agent with the Simulator, but without the Correlator. Then, no matter what L it is given, the Simulator won't be able to prove that Self()=Self(), because of the infinite regress. So the agent will be opaque to itself, and will two-box on Newcomb's problem and defect against itself in Prisoner's Dilemma.
The UDT/ADT, on the other hand, have functioning Correlators.
If it is possible to explicitly (rather than conceptually) separate an agent into the three parts, then it appears to be possible to demonstrate the good behavior of an agent in the ASP problem. The world can be written as a NewComb's-like function:
def U():
box2 = 1000
box1 = (P()==1 ? 1000000 : 0)
return (A()==1 ? box1 : box1 + box2)
where P is a predictor that has much less computational resources than the agent A. We can assume that the predictor has basically the same source code as the agent, except for the bound on L and a stipulation that P two-boxes if it cannot prove one-boxing using available resources.
Then, if the Executive uses a reasonable strategy ("start with low values of L and P, then increase them until all necessary moral arguments are found or all available resources are spent"), then the Correlator should be able to prove A()=P() quite early in the process.
Anthropic Reasoning by CDT in Newcomb's Problem
By orthonormal's suggestion, I take this out of comments.
Consider a CDT agent making a decision in a Newcomb's problem, in which Omega is known to make predictions by perfectly simulating the players. Assume further that the agent is capable of anthropic reasoning about simulations. Then, while making its decision, the agent will be uncertain about whether it is in the real world or in Omega's simulation, since the world would look the same to it either way.
The resulting problem has a structural similarity to the Absentminded driver problem1. Like in that problem, directly assigning probabilities to each of the two possibilities is incorrect. The planning-optimal decision, however, is readily available to CDT, and it is, naturally, to one-box.
Objection 1. This argument requires that Omega is known to make predictions by simulation, which is not necessarily the case.
Answer: It appears to be sufficient that the agent only knows that Omega is always correct. If this is the case, then a simulating-Omega and some-other-method-Omega are indistinguishable, so the agent can freely assume simulation.
[This is a rather shaky reasoning, I'm not sure it is correct in general. However, I hypothesise that whatever method Omega uses, if the CDT agent knows the method, it will one-box. It is only a "magical Omega" that throws CDT off.]
Objection 2. The argument does not work for the problems where Omega is not always correct, but correct with, say, 90% probability.
Answer: Such problems are underspecified, because it is unclear how the probability is calculated. [For example, Omega that always predicts "two-box" will be correct in 90% cases if 90% of agents in the population are two-boxers.] A "natural" way to complete the problem definition is to stipulate that there is no correlation between correctness of Omega's predictions and any property of the players. But this is equivalent to Omega first making a perfectly correct prediction, and then adding a 10% random noise. In this case, the CDT agent is again free to consider Omega a perfect simulator (with added noise), which again leads to one-boxing.
Objection 3. In order for the CDT agent to one-box, it needs a special "non-self-centered" utility function, which when inside the simulation would value things outside.
Answer: The agent in the simulation has exactly the same experiences as the agent outside, so it is the same self, so it values the Omega-offered utilons the same. This seems to be a general consequence of reasoning about simulations. Of course, it is possible to give the agent a special irrational simulation-fearing utility, but what would be the purpose?
Objection 4. CDT still won't cooperate in the Prisoner's Dilemma against a CDT agent with an orthogonal utility function.
Answer: damn.
1 Thanks to Will_Newsome for pointing me to this.
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