Construct a finite-state machine for a restricted telephone...

Construct a finite-state machine for a restricted telephone switching system that implements these rules. Only calls to the telephone numbers 0, 911, and the digit 1 followed by 10-digit telephone numbers that begin with 212, Construct a finite-state machine for a restricted telephone switching system that implements these rules. Only calls to the telephone numbers 0, 911, and the digit 1 followed by 10-digit telephone numbers that begin with 212,

Key Concepts

Finite State Machine

A finite state machine is a mathematical model used to design and analyze systems that transition between a limited number of states based on input symbols. It consists of states, transitions, an initial state, and one or more accepting states, and is used to recognize patterns in strings.

Deterministic Finite Automaton

A deterministic finite automaton (DFA) is a type of finite state machine where every state has exactly one transition for each input symbol, ensuring a unique computation path for any given input. DFA are used for pattern matching and validating sequences such as telephone numbers.

Regular Languages

Regular languages are sets of strings that can be defined by regular expressions or accepted by finite state machines. They are used to formalize patterns, like specific telephone number formats, and ensure that inputs conform to a given structure.

Pattern Matching in Input Validation

This concept involves constructing state transitions that only accept strings matching certain predefined patterns, such as specific telephone numbers. For example, building a finite state machine to detect the exact sequences '0', '911', and numbers starting with '1' followed by a '212'-prefixed 10-digit number.

Transition Diagrams

Transition diagrams visually represent the states of a finite state machine and the transitions between them based on input symbols. They provide a clear overview of the system design, showing how an input is processed sequentially from the start state to an accepting state, or rejected otherwise.

Transcript

00:01 We're asked to construct a finite state machine for a restricted telephone switching system that implements the following rules.

00:09 We're told the only calls to the telephone numbers 0, 911, and the digit 1, followed by 10 -digit telephone numbers that begin with 212.

00:23 We're going to go through.

00:38 Sorry.

00:38 I mean, in the digit one followed by 10 -digit telephone numbers that begin with 212 -80 -866 -867 and 888 are sent to the network.

00:54 All those strings of digits are going to be blocked by the system, and the user is going to hear an error message.

01:03 So, assume the possible inputs are the 10 digits.

01:12 Now let s0 be the start state.

01:56 Let s1 be the state representing a successful call.

02:10 So a call that goes through to the network.

02:15 And therefore it follows that there's not going to be any outputs from s1.

02:25 Now, from s0 we have the inputs of 2, 3, 4, 5, 6, 7, or 8.

02:57 So if we put in those inputs, then the machine goes back to s0.

03:10 And we'll output an error message for the user.

03:36 Now suppose that the input is zero at the state s0, then we're going to move to state s1, and we're going to output that zero is sent to the network.

04:24 Now, suppose that we input 9 at s0, then we're going to move to state s2, with no output.

05:27 Now, suppose that input 1 at s2, then we'll move to state s3 with no output...

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