Ramana Kumar

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Eight Definitions of Observability

Next, assume that 's agent can observe  according to the additive definition. We will show that 's agent can observe .

I might be misunderstanding this, but the proof suggests you're actually assuming the assuming definition here, not the additive definition. In which case we may be missing the proof of implication of any of the other definitions from the additive definition.

Eight Definitions of Observability

Definition: We say that 's agent can observe a finite partition  of  if for all functions , there exists an element  such that for all .

Claim: This definition is equivalent to the definition from subsets.

This doesn't hold in the degenerate case , since then we have an empty function  but no elements of . (But the definition from subsets holds trivially.)

Committing, Assuming, Externalizing, and Internalizing

Claim: For any Cartesian frame  over  and partition  of , let  send each element of  to its part in . If for all  and  we have , then .

I think this may also need to assume that  is non-empty.

Committing, Assuming, Externalizing, and Internalizing

If

The  argument is missing in several places like this from 4.2 onwards.

Committing, Assuming, Externalizing, and Internalizing

is given by

There's a prime missing on . I'd also have expected  instead of  as the variable (doesn't affect correctness).

Committing, Assuming, Externalizing, and Internalizing

while  is given by

should be applied to  above, I think

Committing, Assuming, Externalizing, and Internalizing

Let , send each element of  to its part in , so .

Presuming the  here should be a

Committing, Assuming, Externalizing, and Internalizing

Oh I see this has also been fixed. Thanks!

Committing, Assuming, Externalizing, and Internalizing

This is also suspicious in section 2.2 about Assuming. I think it should be the other way around and about Assume rather than Commit, and I don't think that's equivalent to what's written here. (But I'm not confident about this.)

Claim: For all  and .

Committing, Assuming, Externalizing, and Internalizing

Claim: For all  and

Are these the wrong way around?

I believe  is indeed trivial, but the opposite is less obvious.