Answers.Body: <p>State space in contextual bandits or reinforcement learning covers the input space to a policy or predictive model. The question also specifies a <em>representation</em> for this input, which is different in some ways from the meaningful space that the problem presents to you (most practical representations are larger spaces that include the whole problem plus a lot of possible values that are not relevant), but from the wording of the question, I don't think you are expected to consider that.</p>
  <p>With that in mind, I think the question is expecting your first answer, <span class="math-container">$2^{784}$</span>.</p>
  <p>The larger space you consider in your second solution, I would not call the &quot;state space&quot;. It is instead <em>either</em>:</p>
  <ul>
  <li>The space of all possible input/output pairs, irrespective of correctness or usefulness to the problem. You could consider that as your search space for finding the agent's prediction function, and I think <a href="https://ai.stackexchange.com/a/44904/1847">another answer explains the correct size for that</a>, <span class="math-container">$10 \times 2^{784}$</span>.</li>
  </ul>
  <p><em>or</em>:</p>
  <ul>
  <li>The value <span class="math-container">$10^{2^{784}}$</span> is also meaningful, and is the space of all
  possible agents, that can arbitrarily choose an output in <span class="math-container">$[0,9]$</span> for
  each of the <span class="math-container">$2^{784}$</span> representable inputs.</li>
  </ul>
  <p>I cannot rule out that the question is considering <em>state</em> from the perspective of something searching for an agent. In which case the other answer is correct. It is not the definition I would use, coming from a reinforcement learning or machine learning background. But it may depend on the learning materials which would be the preferred answer.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-02-28T14:04:04.047Z
Answers.Text: Can you please elaborate how 10^(2^784)) is the state space of all possible agents?
Answers.UserDisplayName: Dawood Ahmad
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Answers.Text: @DawoodAhmad: Each agent has a free choice out of 10 answers for each possible input combination. So you can tabulate an exact description of an agent as a string of digits 0-9 for each input state. This input-to-output table is $2^{784}$ entries long, essentially a base-10 number with that number of digits. Two agents can be called different even if they disagree only on a single specific input.
Answers.UserDisplayName: Neil Slater
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Answers.LastEditDate: 2024-02-24T19:49:08.153Z
Answers.UserDisplayName: Neil Slater
Answers.Body: <p>It's an interesting question!</p>
  <p>I would say, that the element of the state space is an example of observation. In your problem an observation is a pair of image and label, say ( X, y ). Image lies in space <span class="math-container">$\{0, 1\}^{28 x 28}$</span> and the label lies in space <span class="math-container">$\{0, 1, 2, 3, ..., 9 \}$</span>. Therefore, you have
  <span class="math-container">$$
  (X, y) \in \{0, 1\}^{784} \times\{0, 1, 2, 3, ..., 9\}
  $$</span>
  <span class="math-container">$$
  |(X, y)| = 10*2^{784}
  $$</span></p>
  
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Answers.UserDisplayName: vl_knd
Body: <p>I encountered the following problem as part of a quiz:</p>
  <p>Imagine you are designing a digit recognition agent and you want to analyze the state space of this problem. Given a binary image of 28 x 28 grid of pixels where a 1 represents black color and a 0 represents white color. The agent has to determine the digit from 10 possible values, 0, 1, ..., 9. Compute the state space.</p>
  <p>I am confused between two approaches to solve this problem.</p>
  <ol>
  <li><p>The state space consists of all possible configurations of the grid. The number of these combinations is 2^(28x28)= 2^(784). So the size of the state space is 2^(784).</p>
  </li>
  <li><p>Because there are 10 digits and the agent must find search all combinations for each digit, the state space is 10^(2^(784)).</p>
  </li>
  </ol>
  <p>Is the size of the state space (2^(784) or 10^(2^(784))?</p>
  
Tags: <search><state-spaces><combinatorics><goal-based-agents>
Title: What is the state space of digit recognition agent on a grid of 28 x 28 pixels?
UserDisplayName: Dawood Ahmad
ThreadText: # What is the state space of digit recognition agent on a grid of 28 x 28 pixels?
  
  (Asked by: username_0 on 2024-02-24)
  
  I encountered the following problem as part of a quiz:
  
  Imagine you are designing a digit recognition agent and you want to analyze the state space of this problem. Given a binary image of 28 x 28 grid of pixels where a 1 represents black color and a 0 represents white color. The agent has to determine the digit from 10 possible values, 0, 1, ..., 9. Compute the state space.
  
  I am confused between two approaches to solve this problem.
  
  1. The state space consists of all possible configurations of the grid. The number of these combinations is 2^(28x28)= 2^(784). So the size of the state space is 2^(784).
  2. Because there are 10 digits and the agent must find search all combinations for each digit, the state space is 10^(2^(784)).
  
  Is the size of the state space (2^(784) or 10^(2^(784))?
  
  **Tags:** search, state-spaces, combinatorics, goal-based-agents
  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_1 on 2024-02-24. Score: 2
  
  State space in contextual bandits or reinforcement learning covers the input space to a policy or predictive model. The question also specifies a *representation* for this input, which is different in some ways from the meaningful space that the problem presents to you (most practical representations are larger spaces that include the whole problem plus a lot of possible values that are not relevant), but from the wording of the question, I don't think you are expected to consider that.
  
  With that in mind, I think the question is expecting your first answer, $2^{784}$.
  
  The larger space you consider in your second solution, I would not call the "state space". It is instead *either*:
  
  * The space of all possible input/output pairs, irrespective of correctness or usefulness to the problem. You could consider that as your search space for finding the agent's prediction function, and I think [another answer explains the correct size for that](https://ai.stackexchange.com/a/44904/1847), $10 \times 2^{784}$.
  
  *or*:
  
  * The value $10^{2^{784}}$ is also meaningful, and is the space of all
    possible agents, that can arbitrarily choose an output in $[0,9]$ for
    each of the $2^{784}$ representable inputs.
  
  I cannot rule out that the question is considering *state* from the perspective of something searching for an agent. In which case the other answer is correct. It is not the definition I would use, coming from a reinforcement learning or machine learning background. But it may depend on the learning materials which would be the preferred answer.
  
  > **Comments:**
  > **username_0**: Can you please elaborate how 10^(2^784)) is the state space of all possible agents?  
  > **username_1**: @username_0: Each agent has a free choice out of 10 answers for each possible input combination. So you can tabulate an exact description of an agent as a string of digits 0-9 for each input state. This input-to-output table is $2^{784}$ entries long, essentially a base-10 number with that number of digits. Two agents can be called different even if they disagree only on a single specific input.  
  
  ---
  
  ### Answer by username_2 on 2024-02-24. Score: 0
  
  It's an interesting question!
  
  I would say, that the element of the state space is an example of observation. In your problem an observation is a pair of image and label, say ( X, y ). Image lies in space $\{0, 1\}^{28 x 28}$ and the label lies in space $\{0, 1, 2, 3, ..., 9 \}$. Therefore, you have
  $$
  (X, y) \in \{0, 1\}^{784} \times\{0, 1, 2, 3, ..., 9\}
  $$
  $$
  |(X, y)| = 10\*2^{784}
  $$
  
  
  ---
  
  