CSE450. Translation of Programming Languages. Lecture 20: Automata and Regular Expressions

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1 CSE45 Translation of Programming Languages Lecture 2: Automata and Regular Expressions

2 Finite Automata Regular Expression = Specification Finite Automata = Implementation A finite automaton consists of: An input alphabet A set of states S A start state n in S A set of accepting states F S A set of transitions: (state, input) state Execution starts in the "start state" and then inputs are read in, changing state based on the transition rules. Accept if the state at the end of input is an accepting state, else reject.

3 Finite Automata State Graphs A state: The start state: An accepting state: A transition: a

4 A Simple Example A finite automaton that accepts only "":

5 Another Simple Example Alphabet: {, } A finite automaton accepting any number of 's followed by a single

6 And Another Example Alphabet: {, } What language does this recognize? Warning: Very Hard. *?* ( )* (* ) ( *)+

7 Another Simple Example Alphabet still {, } The operation of the automaton is not completely defined by the input On input "" the automaton could be in either state

8 Epsilon Moves Another kind of transition: -moves

9 Deterministic and Non-deterministic Automata Deterministic Finite Automata (DFA) One transition per input per state No -moves Non-deterministic Finite Automata (NFA) Can have multiple transitions for one input in a given state Can have -moves Finite automata have finite memory Need only to encode the current state

10 Execution of Finite Automata A DFA can take only one path through the state graph Completely determined by the input NFSs can choose Whether to make -moves Which of multiple transitions for a single input to take

11 Acceptance of NFAs An NFA can get into multiple states Input: Rule: NFA accepts if it can get in a final state.

12 NFA vs. DFA () NFAs and DFAs recognize the same set of languages (regular languages) NFAs are easier to design You can just list all the rules you want to allow DFAs are easier to implement There are no choices to consider

13 NFA vs. DFA (2) For a given language the NFA can be simpler than the DFA NFA DFA DFA can be exponentially larger than the NFA

14 Automata to Regular Expression ( )*. Add new start and single end state 2. Connected new states with epsilon transitions 3. Remove one state at a time A. Replace each incoming transition (to the removed state) with a regex to possible outputs 4. Simplify regex if possible

15 Lexical Analysis Overview Non-deterministic Finite Automata Regular Expressions Deterministic Finite Automata Lexical Specification Table-driven Implementation of DFA

16 Regular Expressions to NFA () For each kind of regular expression, we can define an NFA. We wil do this by using recursive definition. Notation: NFA for regular expression A: A Base cases: For For input 'a' a

17 Regular Expressions to NFA (2) For AB A B For A B A B

18 Regular Expressions to NFA (3) For A* A How do we do "A+"? What about "A?"?

19 Example of RE to NFA Conversion Consider the regular expression ( )* The NFA is: A B C E D F G H I J

20 Lexical Analysis Overview Non-deterministic Finite Automata Regular Expressions Deterministic Finite Automata Lexical Specification Table-driven Implementation of DFA

21 NFA to DFA: The Trick Simulate the NFA Each state of DFA = a non-empty subset of states of the NFA Start state = the set of NFA states reachable through -moves from the NFA start state Add a transition (S, a) S' to DFA if and only if: S' is the set of NFA states reachable from the states in S after seeing the input a (remember to consider -moves as well!)

22 NFA to DFA Example A B C E D F G H I J FGABCDHI ABCDHI EJGABCDHI

23 NFA to DFA Remark An NFA may be in many states at any time How many different possible states are there? If there are n states, the NFA must be in some subset of those n states How many non-empty subsets are there? 2 n - = finitely many

24 Lexical Analysis Overview Non-deterministic Finite Automata Regular Expressions Deterministic Finite Automata Lexical Specification Table-driven Implementation of DFA

25 Table Implementation A DFA can be implemented by a 2-dimensional table T One dimension is "states" Other dimension is "input symbols" For every transition (Si, a) Sk define T[i, a] = k DFA "execution" If in state Si and input a, read T[i, a] = k and go to state Sk Very efficient

26 Table Implementation of DFA T S U S T U T T U U T U

27 Final notes about Implementation NFA to DFA conversion is at the heart of tools such as Flex But, DFAs can be huge In practice, lex-like tools trade off speed for space in the choice of NFA and DFA representations

CSE450. Translation of Programming Languages. Automata, Simple Language Design Principles

CSE450. Translation of Programming Languages. Automata, Simple Language Design Principles CSE45 Translation of Programming Languages Automata, Simple Language Design Principles Finite Automata State Graphs A state: The start state: An accepting state: A transition: a A Simple Example A finite