Decision, Computation and Language

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1 Decision, Computation and Language Context-free Grammar (CFG) Dr. Muhammad S Khan (mskhan@liv.ac.uk) Ashton Building, Room G22

2 The Chomsky Hierarchy Languages exist which are not regular; Noam Chomsky categorised regular and other languages as follows: Language Class Grammar Automaton 3 Regular NFA or DFA 2 Context-Free Push Down Automaton 1 Context-Sensitive Linear-Bounded Automaton 0 Unrestricted Turing machine M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 2

3 Type 3 - Regular Languages A regular language is one which can be: represented by a regular grammar, described using a regular expression, or accepted using an NFA or a DFA. M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 3

4 Type 2 - Context-Free Languages A Context-Free Grammar (CFG) is one whose production rules are of the form: A α where A is any single non-terminal, and α is any combination of terminals and non-terminals. A NFA/DFA cannot recognise strings from this type of language since we must be able to "remember" information somehow. CFG is accepted using Push-Down Automaton which is like a DFA except that we are also allowed to use a stack (memory). M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 4

5 Type 1 - Context-Sensitive Languages Context-Sensitive grammars may have more than one symbol on the left-hand-side of their production rules (provided that at least one of them is a non-terminal). However, the production rules must now obey the following: The number of symbols on the left-hand-side must not exceed the number of symbols on the right-hand-side We do not allow rules of the form A ε unless A is the start symbol and does not occur on the right-hand-side of any rule. M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 5

6 Type 0 - Unrestricted (Free) Languages Free grammars have absolutely no restrictions on their grammar rules, (except, of course, that there must be at least one non-terminal on the left-hand-side). The type of automata which can recognise such a language is basically a NFA/DFA with an infinitely-long list at its disposal to use as a store; this is called a Turing machine. M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 6

7 Context-free Grammars (CFG) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 7

8 Grammar M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 8

9 Context-free Grammars Show how CFGs can be converted into normal forms, i.e. equivalent CFGs that have additional syntactic restrictions Use normal form to show that pushdown automata are the class of machines that accept CFLs. Parsing is the process of checking that a sequence of symbols is generated by a context-free grammar Consider classes of parsing algorithms expressible as restricted classes of pushdown automata (note: general pushdown automata are impractical to implement, unlike finite automata) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 9

10 Context-free Grammars M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 10

11 Example M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 11

12 Example of using the grammar M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 12

13 Another Example M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 13

14 Example of using the grammar M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 14

15 Finite languages M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 15

16 Notation M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 16

17 Notation M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 17

18 Types of rules in CFG There are three types of rules in CFG 1. Union Rule: S A B 2. Production Rule: S AB 3. Closure Rule: S AS ε M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 18

19 Examples M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 19

20 More Examples L = ε, a, b, bb, aa, aaa, bbb,, a n, b n n 0 L = ε, ab, aabb, aaabbb,, a n b n n 0 L = ε, ab, abab, ababab,, ab n n 0 L = a n b m n, m 0 L = a n b m c k : n, m, k 0 Well balances parentheses: (()(()))() M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 20

21 Regular Languages are Context-free... M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 21

22 Regular Grammar M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 22

23 An equivalent definition The above definition is used in Hopcroft and Ullman. The previous definition is used in Temblay and Sorenson. M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 23

24 Conversion to restricted form M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 24

25 Regular Grammar to NFA M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 25

26 How to prove the construction is valid M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 26

27 Example M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 27

28 Example of conversion from grammar to FA M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 28

29 Example of conversion from grammar to FA M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 29

30 Example of conversion from grammar to FA M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 30

31 Another Example M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 31

32 Example (contd.) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 32

33 Example: some arithmetic expressions M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 33

34 Ambiguity Parsing is the problem of: given a grammar and a string, find a derivation of the string using the grammar. (example on previous slide!) Given our claim that programming languages are often described using grammars, this is a key problem for compiler. We prefer unambiguous CFGs (ones where all derivations of a string are essentially the same, noting that variables of a CFG often correspond to specific structures of a program, e.g. arithmetic expression, procedure, statement, method etc. M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 34

35 Ambiguity (continued) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 35

36 Parse Trees (a.k.a. syntax trees, derivation trees M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 36

37 Parse Trees (continued) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 37

38 Rewriting a grammar M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 38

39 Rewriting a grammar (contd.) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 39

40 Leftmost/rightmost derivations M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 40

41 Leftmost/rightmost derivations (contd.) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 41

42 Another example of a simple ambiguous grammar M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 42

43 Example (contd.) M S Khan (Univ. of Liverpool) COMP218 Decision, Computation and Language 43

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