Based on the provided case study, analyze the engineer's implemented formula. Explain why this specific implementation leads to the observed problem of the system failing to retain information from previous steps.

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When a memory component operates as a neural network, its update mechanism can be formulated as a general mathematical function: $$\mathrm{Mem} = \mathrm{Update}(S_{\mathrm{kv}}, \mathrm{Mem}_{\mathrm{pre}})$$. Here, $$\mathrm{Mem}$$ and $$\mathrm{Mem}_{\mathrm{pre}}$$ denote the memory outputs at the current and prior steps, while $$S_{\mathrm{kv}}$$ is a set of key-value pairs representing the recent states of the model. This flexible formulation enables the creation of diverse memory models by altering the $$\mathrm{Update}(\cdot)$$ function and the contents of $$S_{\mathrm{kv}}$$.

Formula for Neural Network Memory Update

A system processes information in discrete steps and updates its internal memory at each step. The update is performed by a neural network that follows the general rule: `New_Memory = Update_Function(Current_State, Previous_Memory)`. If the system is currently processing Step 10, what specific inputs are required by the `Update_Function` to calculate the memory for this step?

Given the general formula for updating a neural network's memory, `Mem = Update(S_kv, Mem_pre)`, explain the distinct roles of the `S_kv` and `Mem_pre` inputs in the calculation of the new memory state, `Mem`.

Analysis of Memory Update Inputs

Diagnosing a Faulty Memory Update Mechanism

A general neural network memory formulation, $$\mathrm{Mem} = \mathrm{Update}(S_{\mathrm{kv}}, \mathrm{Mem}_{\mathrm{pre}})$$, can be configured to act as an RNN-like model. This occurs when the set of recent model states, $$S_{\mathrm{kv}}$$, is restricted to contain only the most recent key-value pair, $$(\mathbf{k}_i, \mathbf{v}_i)$$, and the $$\mathrm{Update}(\cdot)$$ mechanism is defined as a recurrent cell, denoted as $$f$$. This specific configuration can be expressed with the formula:

$$\mathrm{Mem} = f((\mathbf{k}_i, \mathbf{v}_i), \mathrm{Mem}_{\mathrm{pre}})$$

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