I've recently come across the Pinocchio Paradox:
If Pinocchio says "my nose will grow."
Does his nose grow or not grow?

Tracing through this scenario, we can see this is related to the Epimenides Paradox.

The scenario in the Pinnochio Paradox assumes Pinnochio can predict when his nose will grow or not grow with perfect accuracy and achieves 100% certainty (an assumption that is a poor representation of real world agentic systems). This is essentially a scenario where Pinnochio has become self-aware that deception causes his nose to grow and has honed his predictive model on what threshold of deception triggers nose growth.

The Pinocchio Paradox is resolved by acknowledging that the word "will" in "will grow " indicates an anticipated future nose growth event; intuitively in the imminent future. The "my nose will grow" message must be in reference to some other action that Pinocchio has taken or a message he transmitted prior to the "my nose will grow" message. When Pinocchio transmits a message, he knows with complete certainty if that message is an act of deception or not. Thus, each message transmission event provides the opportunity for nose growth/non-growth to occur.

Deception exists where there is a difference between Pinnochio's internal model and the model that will be compiled out of the transmitted message upon being received by another agent on the social graph. The difference between 2 models is commonly known as relative entropy or KL Divergence.

For this scenario to work, it also requires an assumption that nose growth lags by at least 2 message passing events, giving room for both the 1st message and 2nd message to be transmitted before nose growth happens. Without this assumption, both Pinocchio and the other agent will have the empirical information of whether Pinocchio's nose grew upon the first event, which would invalidate the purpose of sending the 2nd message that says "my nose will grow."

This also means that we should assume that nose growth happens proportional to the number of deceptive messages transmitted within this trailing 2 message window (else, we wouldn't be able to distinguish the number of messages within this window that contain the deception) Here, nose growth is proportional to the total relative entropy deliberately transmitted within the previous 2 message events.

There are 3 sequential events that occur in this scenario.
Breaking down the event sequence:
1) The first message is transmitted that reflects some information Pinnochio has stored in his memory that he has chosen to speak about.
2) When Pinocchio transmits the 2nd message "my nose will grow", having knowledge of the previous message in his memory, he knows if both the previous message and this present message is an act of deception or not.
3) Nose growth happens in proportion to deceptive messages transmitted; no nose growth if no deception.

To describe the model of Pinnochio's nose growth system logic in simplified terms we can assume the messages transmitted in this scenario contain answers to yes/no questions, which yields the following logical tree:

Deliberate True Positive (honesty)
    no nose growth
Deliberate True Negative (honesty)
    no nose growth
Deliberate False Positive (deception)
    nose growth
Deliberate False Negative (deception)
    nose growth
    
Unintentional True Positive (lucky guess)
    no nose growth
Unintentional True Negative (lucky guess)
    no nose growth
Unintentional False Positive (unlucky guess)
    no nose growth
Unintentional False Negative (unlucky guess)
    no nose growth

This shows that nose growth happens where messages that contain deliberate false negatives or deliberate false positives are transmitted by Pinnochio. This logic tree would be run twice for this scenario in response to each message passing event, with the nose growth output occurring subsequent to both message transmissions.

If we play into the model of this scenario that leads to the paradox, we would assume there is no information available about the events or messages that precede the "my nose will grow" message, which would require the assumption that Pinnochio has instantaneous memory loss, and is a essentially memoryless system. Here, Pinocchio doesn't have a predictive model or any kind of world model in this case since he can't store any information in memory, meaning that any outputs he generates and transmits are done with a genuine lack of knowledge. This results in the latter half of the logic tree with the cases of unintentional outcomes.
If he would portray a lucky guess as having genuine knowledge, then this would be an act of deception, causing his nose to grow. However, with the assumption of a memoryless system in place, Pinocchio wouldn't have the memory to be able to have the opportunity to commit an act of deception. Since he doesn't store any information, there's no information to misrepresent.
The memoryless system assumption is generally not a very practical assumption because most agentic systems require memory to store a world model to make decisions from. A memoryless agent, would be an aimless agent. Thus, the assumption that Pinocchio is agentic is mutually exclusive with the assumption that Pinocchio is memoryless system.
The way this scenario is presented might seem to encourage an attempt to cause a person to make both of these assumptions simultaneously, which leads to the logical conflict that causes the purported paradox.

Some takeaways from parsing this scenario are:
- All information processing contains a relativistic component.
- The memoryless system assumption is generally not a very practical assumption for models of agentic systems since most agentic systems require memory to store a world model to make decisions from. There could exist submodules within an agent that function memorylessly for decision making, but if the entirety of the agent's internal model and information processing is memoryless, then the agent would be aimless.