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Counter machines are abstract computational devices utilized in the theory of computation to examine the capabilities and limitations of computing systems.  A counter machine consists of a finite set of counters, which are non-negative integer variables. These counters can be incremented, decremented, and tested for zero, serving as data storage and manipulation during computation. Additionally, a control unit dictates the sequence of instructions based on the current state and counter values . Instructions in a counter machine generally fall into three categories:  increment (increase a counter by one),  decrement (decrease a counter by one), and  conditional branch ( Check to see if the counter is zero) . Conditional branching allows the machine to make decisions and modify its behaviour accordingly. Counter machines operate in discrete steps, where each step corresponds to the execution of a single instruction. The process initiates from an initial state, and ...