Discrete Math. and Logic II. Context-Free Grammars SFWR ENG 2FA3 Ryszard Janicki Winter 2014 Acknowledgments: Material partially based on Automata and Computability Ryszard Janicki by Dexter C. Kozen (Chapter 19). Discrete Math. and Logic II. Context-Free Grammars 1 / 16 Introduction An example of a context-free grammar Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 2 / 16 Introduction The objects hxxxi are called nonterminal symbols Each nonterminal symbol generates a set of strings over a nite alphabet Σ in a systematic way the nonterminal harith-expri in hassg-stmti ::= hvari := harith-expri generates the set of syntactically correct arithmetic expressions in this language The strings corresponding to the nonterminal hxxxi are generated using rules with hxxxi on the left-hand side Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 3 / 16 Introduction The alternatives on the righthand side, separated by vertical bars |, describe dierent ways strings corresponding to hxxxi can be generated These alternatives may involve other nonterminals hyyyi, which must be further eliminated by applying rules with hyyyi on the left-hand side while x ≤ y do begin x := (x + 1); y := y − 1 end is generated by the nonterminal hstmti Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 4 / 16 Introduction We obtain the while statement from hstmti through a sequence of expressions called sentential forms Each sentential form is derived from the previous by an application of one of the rules hstmti hwhile-stmti while hbool-expri do hstmti while harith-expri hcompare-opi harith-expri do hstmti while hvari hcompare-opi harith-expri do hstmti while hvari ≤ harith-expri do hstmti while hvari ≤ hvari do hstmti while x ≤ hvari do hstmti while x ≤ y do hstmti while x ≤ y do hbegin-stmti ... Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 5 / 16 Introduction Applying dierent rules will yield dierent results begin if z = (x + 3) then y := z else y := x end The set of all strings not containing any nonterminals generated by the grammar is called the language generated by the grammar In general, this set of strings may be innite, even if the set of rules is nite There may also be several dierent derivations of the same string A grammar is said to be unambiguous if a string cannot have more than one derivation Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 6 / 16 Introduction The language (subset of Σ∗ ) generated by the context-free grammar G is denoted L(G ) A subset of Σ∗ is called a context-free language (CFL) if it is L(G ) for some CFG G CFLs are good for describing innite sets of strings in a nite way They are particularly useful in computer science for describing the syntax of programming languages, well-formed arithmetic expressions, well-nested begin-end blocks, strings of balanced parentheses, All regular sets are CFLs, but not necessarily vice versa. Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 7 / 16 Pushdown Automata (PDAs): A Preview A pushdown automaton (PDA) is like a nite automaton, except it has a stack or pushdown store, which it can use to record a potentially unbounded amount of information Its input head is read-only and may only move right The machine can store information on the stack in a last-in-rst-out (LIFO) fashion It can push symbols onto the top of the stack or pop them o the top of the stack It may not read down into the stack without popping the top symbols o Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 8 / 16 Pushdown Automata (PDAs): A Preview a1 a2 a3 a4 a5 a6 a7 a8 an left to right, read only Q push/pop Finite Control Ryszard Janicki A B C B Stack Discrete Math. and Logic II. Context-Free Grammars 9 / 16 Formal Denition of CFGs and CFL Denition A context-free grammar (CFG) is a quadruple G = (N , Σ, P , S ), where N is a nite set (the nonterminal symbols), Σ is a nite set (the terminal symbols) disjoint from N, P is a nite subset of N × (N ∪ Σ)∗ (the productions), S ∈ N (the start symbol). Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 10 / 16 Formal Denition of CFGs and CFL We use capital letters A, B , C , · · · for nonterminals We use a, b, c , · · · for terminal symbols Strings in (N ∪ Σ)∗ are denoted α, β, γ, · · · Instead of writing productions as (A, α), we write A −→ α We often use the vertical bar | to abbreviate a set of productions with the same left-hand side Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 11 / 16 Formal Denition of CFGs and CFL Example Instead of writing A −→ α1 , we write A −→ α2 , A −→ α1 A −→ α3 , | α2 | α3 Denition If α, β ∈ (N ∪ Σ)∗ , we say that β is derivable from α in one step and write 1 α −→ β G if β can be obtained from α by replacing some occurrence of a nonterminal A in α with γ , where A −→ γ , is in P . Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 12 / 16 Formal Denition of CFGs and CFL Example We have A −→ γ a production in P α = α1 Aα2 , where α1 , α2 ∈ (N ∪ Σ)∗ β = α1 γα2 1 Then, we have α −→ β G Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 13 / 16 Formal Denition of CFGs and CFL Denition ∗ 1 Let −→ be the reexive transitive closure of the relation −→ ; that G G is, dene 0 α −→ α G n +1 n 1 α −→ β if there exists γ such that α −→ γ and γ −→ β G G G α −→ β if ∃(n | G ∗ n≥0 n : α −→ β ) G Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 14 / 16 Formal Denition of CFGs and CFL Terminology: A string in (N ∪ Σ)∗ derivable from the start symbol S is called a sentential form A sentential form is called a sentence if it consists only of terminal symbols The language generated by G , denoted L(G ), is the set of all sentences: L(G ) = {x ∈ Σ∗ | ∗ S −→ x} G A subset B ⊆ Σ∗ is a context-free language (CFL) if B = L(G ) for some context-free grammar G Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 15 / 16 Formal Denition of CFGs and CFL Example The nonregular set {an bn | n ≥ 0} is a CFL It is generated by the grammar G = (N , Σ, P , S ), where N = {S }, Σ = {a, b}, P = {S −→ aSb, S −→ } Here is a derivation of a3 b3 in G : 1 1 1 1 S −→ aSb −→ aaSbb −→ aaaSbbb −→ aaabbb G G G G n +1 n n S −→ ab G Ryszard Janicki Discrete Math. and Logic II. Context-Free Grammars 16 / 16