CSE 311 Lecture 23: Finite State Machines. Emina Torlak and Kevin Zatloukal

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1 CSE 3 Lecture 3: Finite State Machines Emina Torlak and Kevin Zatloukal

2 Topics Finite state machines (FSMs) Definition and examples. Finite state machines with output Definition and examples.

3 Finite state machines (FSMs) Definition and examples.

4 3

5 Finite state machines by example An FSM recognizes (or accepts) a set of strings. start zero one 4

6 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. start zero one 4

7 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( is empty), stop. s start zero one 4

8 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. zero s w The FSM accepts double circle. s iff it stops in a final state, denoted by a one start 4

9 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. s w The FSM accepts double circle. Which strings are recognized by the example FSM? ε s iff it stops in a final state, denoted by a start zero one 4

10 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. s w The FSM accepts double circle. Which strings are recognized by the example FSM? no ε s iff it stops in a final state, denoted by a start zero one 4

11 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. s w The FSM accepts double circle. Which strings are recognized by the example FSM? no yes ε s iff it stops in a final state, denoted by a start zero one 4

12 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. s w The FSM accepts double circle. Which strings are recognized by the example FSM? no yes yes ε s iff it stops in a final state, denoted by a start zero one 4

13 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. s w The FSM accepts double circle. Which strings are recognized by the example FSM? no yes yes no ε s iff it stops in a final state, denoted by a start zero one 4

14 Finite state machines by example An FSM recognizes (or accepts) a set of strings. () Given a string s, begin at the start state, denoted by an incoming arrow with no source. () If s = aw, take the edge labeled a to get to the next state. (3) Otherwise ( s is empty), stop. (4) Let be and repeat ()-(4) until the machine stops. s w The FSM accepts double circle. s iff it stops in a final state, denoted by a Which strings are recognized by the example FSM? ε no yes yes no All binary strings that end in. start zero one 4

15 Defining an FSM Finite state machine (FSM) A finite state machine (FSM) M = (S, Σ in, f, s, F) consists of a finite set of states S, a finite input alphabet Σ in, a transition function that maps each state in and input in to a state in, a start state, and a set of final states. s f S Σ in S S F S 5

16 Defining an FSM Finite state machine (FSM) or deterministic finite automaton (DFA) A finite state machine (FSM) M = (S, Σ in, f, s, F) consists of a finite set of states S, a finite input alphabet Σ in, a transition function that maps each state in and input in to a state in, a start state, and a set of final states. s f S Σ in S S F S 5

17 Defining an FSM Finite state machine (FSM) or deterministic finite automaton (DFA) A finite state machine (FSM) M = (S, Σ in, f, s, F) consists of a finite set of states S, a finite input alphabet Σ in, a transition function that maps each state in and input in to a state in, a start state, and a set of final states. s f S Σ in S S F S state s s s s s s, s 3 What language does this FSM accept? s s s s s s 3 s 3 s 3 s 3 5

18 Defining an FSM Finite state machine (FSM) or deterministic finite automaton (DFA) A finite state machine (FSM) M = (S, Σ in, f, s, F) consists of a finite set of states S, a finite input alphabet Σ in, a transition function that maps each state in and input in to a state in, a start state, and a set of final states. s f S Σ in S S F S state s s s s s s s s s 3 s s s, s 3 What language does this FSM accept? The set of all binary strings that contain or end in. s 3 s 3 s 3 5

19 Applications of FSMs Implementation of string matching. For example, in grep. Algorithms for communication and cache-coherence protocols. Each agent runs its own FSM. Specifications for controllers. For example, device drivers. Specifications for reactive systems. Components are communicating FSMs. Verification. Is an unsafe state reachable? 6

20 Example: an FSM over binary strings What language does this FSM recognize? s s s 3 s 7

21 Example: an FSM over binary strings What language does this FSM recognize? The set of all binary strings where the numbers of s and s are congruent mod. That is, both numbers are even or both are odd. s s s 3 s 7

22 Example: FSMs over {,, } M : strings with an even number of s s s M : strings where the sum of digits mod 3 is t t t 8

23 Example: FSMs over {,, } M M : strings with an even number of s,, s s : strings where the sum of digits mod 3 is t t t 8

24 Example: FSMs over {,, } M M : strings with an even number of s,, s s : strings where the sum of digits mod 3 is t t t 8

25 Example: a combined FSM over {,, } M M and : strings with an even number of s where digit sum mod 3 is s t s t s t s t s t s t,, s s t t t 9

26 Example: a combined FSM over {,, } M M and : strings with an even number of s where digit sum mod 3 is s t s t s t s t s t s t,, s s t t t 9

27 Example: another combined FSM over {,, } M M or : strings with an even number of s or digit sum mod 3 is s t s t s t s t s t s t,, s s t t t

28 Finite state machines with output Definition and examples.

29 Adding output to FSMs FSMs can do more than just accept or reject strings. So DFAs aren t the only kind of FSM! Another useful kind of FSM can also return output. Whenever you enter specific states, the FSM produces an output. These FSMs (Moore machines) are used as controllers.

30 Adding output to FSMs FSMs can do more than just accept or reject strings. So DFAs aren t the only kind of FSM! Another useful kind of FSM can also return output. Whenever you enter specific states, the FSM produces an output. These FSMs (Moore machines) are used as controllers. Finite state machine (FSM) with output A finite state machine (FSM) M = (S, Σ in, Σ out, f, g, s ) consists of a finite set of states S, a finite input alphabet Σ in, a finite output alphabet Σ out, a transition function that maps each state in and input in to a state in, an output function g that maps each state in Sto an output symbol in Σ out, and start state. s f S Σ in S S

31 Example: vending machine Consider a simple vending machine. Enter 5 cents in dimes or nickels. Press S (Snickers) or B (Butterfinger) for a candy bar. 3

32 Example: vending machine Consider a simple vending machine. Enter 5 cents in dimes or nickels. Press S (Snickers) or B (Butterfinger) for a candy bar. D N 5 N N, D 5 D B, S Basic transitions on N (nickel), D (dime), B (Butterfinger), and S (Snickers). 3

33 Example: vending machine with output N D ' [B] B N 5 D N D N D N 5 5' [N] D B S S '' [S] Adding output to states: N (nickel), B (Butterfinger), and S (Snickers). 4

34 Example: complete vending machine with output B, S N ' [B] D B B B, S B, S D B, S 5 N D N N 5' [N] D N D D 5'' [D] S B S N 5 D N D S N '' [S] B, S Add transitions to cover all symbols for each state. 5

35 Summary A finite state machine (FSM) consists of states and transitions. States include the start state and final states. Transitions map states and input symbols to states. Also known as deterministic finite automata (DFAs). An FSM recognizes a set of strings (language). These are all strings that reach a final state from the start. FSMs have many applications. Regular expression matching, cache coherence protocols. Verification (e.g., of OS kernels and devices). Controllers (FSMs with output). 6

CSE 311: Foundations of Computing. Lecture 23: Finite State Machine Minimization & NFAs

CSE 311: Foundations of Computing. Lecture 23: Finite State Machine Minimization & NFAs CSE : Foundations of Computing Lecture : Finite State Machine Minimization & NFAs State Minimization Many different FSMs (DFAs) for the same problem Take a given FSM and try to reduce its state set by