Describe the process of constructing an equivalent deterministic FSM given a non-deterministic FSM.
The process of constructing an equivalent deterministic finite state machine (FSM) from a non-deterministic FSM involves several steps that aim to transform the non-deterministic behavior into a deterministic one. This transformation is important in the field of computational complexity theory as it allows for the analysis and comparison of different FSMs based on their computational power and complexity.
To begin with, let us define a non-deterministic FSM. A non-deterministic FSM is a mathematical model that consists of a set of states, a set of input symbols, a transition function, an initial state, and a set of accepting states. The transition function maps a state and an input symbol to a set of possible next states. Moreover, the non-deterministic FSM can have multiple transitions for the same input symbol from the same state, leading to different sets of possible next states.
The construction of an equivalent deterministic FSM from a non-deterministic FSM can be achieved through the following steps:
Step 1: Determine the set of states for the deterministic FSM. Each state in the deterministic FSM corresponds to a set of states in the non-deterministic FSM. The power set construction is used to generate all possible combinations of states in the non-deterministic FSM. Each combination represents a state in the deterministic FSM.
Step 2: Identify the initial state of the deterministic FSM. The initial state of the deterministic FSM is the set of states in the non-deterministic FSM that contains the initial state of the non-deterministic FSM.
Step 3: Define the transition function of the deterministic FSM. For each state in the deterministic FSM and each input symbol, determine the set of states in the non-deterministic FSM that can be reached from any state in the current set of states using the given input symbol. This set of states becomes the next state in the deterministic FSM.
Step 4: Determine the set of accepting states for the deterministic FSM. A state in the deterministic FSM is considered accepting if it contains at least one accepting state from the non-deterministic FSM.
By following these steps, we can construct an equivalent deterministic FSM from a non-deterministic FSM. The resulting deterministic FSM will have a single transition for each input symbol from each state, eliminating the non-deterministic behavior of the original FSM. This allows for a more systematic analysis and comparison of FSMs based on their computational complexity and power.
To illustrate this process, let's consider an example. Suppose we have a non-deterministic FSM with three states (A, B, C), two input symbols (0, 1), and the following transitions:
– A, 0 -> {A, B}
– A, 1 -> {A}
– B, 0 -> {C}
– B, 1 -> {B}
– C, 0 -> {C}
– C, 1 -> {A}
Step 1: The power set construction generates the following states for the deterministic FSM: {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C}.
Step 2: The initial state of the deterministic FSM is {A}.
Step 3: The transition function of the deterministic FSM is as follows:
– {A}, 0 -> {A, B}
– {A}, 1 -> {A}
– {B}, 0 -> {C}
– {B}, 1 -> {B}
– {C}, 0 -> {C}
– {C}, 1 -> {A}
– {A, B}, 0 -> {A, B, C}
– {A, B}, 1 -> {A, B}
– {A, C}, 0 -> {A, C}
– {A, C}, 1 -> {A}
– {B, C}, 0 -> {C}
– {B, C}, 1 -> {B}
– {A, B, C}, 0 -> {A, B, C}
– {A, B, C}, 1 -> {A, B}
Step 4: The accepting states of the deterministic FSM are {A, B, C}.
The resulting deterministic FSM is now fully defined and can be analyzed and compared with other FSMs using computational complexity theory.
Constructing an equivalent deterministic FSM from a non-deterministic FSM involves determining the set of states, identifying the initial state, defining the transition function, and determining the set of accepting states. This transformation allows for the analysis and comparison of FSMs based on their computational power and complexity.
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