When we have two or more acceptable paths in a non-deterministic machine, which one do we choose and what criteria can we use?
In the realm of computational complexity theory, specifically in the study of finite state machines, the concept of non-determinism plays a important role.
Non-deterministic finite state machines (NFSMs) are theoretical models that allow for multiple acceptable paths to be taken at any given state. However, when faced with such a situation, the question arises: which path should be chosen?
This query touches upon the notion of "acceptance" in NFSMs and the criteria that can be employed to make a decision.
To understand the selection process, let us first explore the nature of non-determinism in NFSMs. Unlike deterministic finite state machines (DFSMs), NFSMs do not possess a unique transition for every possible input symbol at each state. Instead, they allow for the existence of multiple transitions for the same input symbol. This characteristic leads to the possibility of having multiple paths to follow from a single state, potentially resulting in different outcomes.
When confronted with such a situation, NFSMs employ a mechanism called "branching" to explore all possible paths simultaneously. This means that the machine creates multiple copies of itself, each following a different path. As a result, the NFSM can be seen as exploring a tree-like structure, where each branch represents a different computation path. This branching technique is fundamental in the analysis of NFSMs and their computational complexity.
Now, let us consider the criteria that can be employed to choose a specific path among the multiple acceptable ones. One common approach is to consider the concept of "acceptance" in NFSMs. Acceptance refers to the condition that determines whether a given input is considered valid or not by the machine. In NFSMs, acceptance can be defined in two main ways: "acceptance by final state" and "acceptance by empty stack."
Acceptance by final state occurs when, upon consuming the entire input string, the NFSM ends up in a state designated as a final state. This criterion implies that the machine accepts the input if there exists at least one computation path that leads to a final state. Conversely, if no path leads to a final state, the input is rejected.
Acceptance by empty stack, on the other hand, is relevant when NFSMs incorporate a stack as an additional component. In this scenario, acceptance occurs when the input string is fully processed, and the stack becomes empty. Similar to acceptance by final state, if there exists at least one computation path that results in an empty stack, the input is accepted; otherwise, it is rejected.
Given these criteria, the selection of a specific path among the multiple acceptable ones in a non-deterministic machine can be determined by prioritizing the acceptance conditions. For example, if acceptance by final state is the primary criterion, the machine would choose the path that leads to a final state, regardless of other potential paths. Conversely, if acceptance by empty stack is the primary criterion, the machine would prioritize the path that results in an empty stack.
It is important to note that the choice of path in NFSMs does not affect the computational power of the machine. Regardless of the chosen path, the NFSM can still recognize the same set of languages as any other NFSM for a given input. The selection process merely determines the acceptance or rejection of the input based on the specified criteria.
When faced with multiple acceptable paths in a non-deterministic machine, the choice of path can be determined by prioritizing acceptance conditions, such as acceptance by final state or acceptance by empty stack. The selection process does not impact the computational power of the machine but influences whether the input is accepted or rejected.
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