Copyright © by Curt Hill Grammar Types The Chomsky Hierarchy BNF and Derivation Trees

Introduction We are now familiar with the notion of a grammar and the language that it covers Next we wish to categorize grammars –This will be based on the forms that the productions take We will start with the simplest and work up Copyright © by Curt Hill

Chomsky Hierarchy Chomsky proposed an hierarchy of languages based on the strength of the rewriting rules There are four –Type 0 through Type 3 The hierarchy is based on the strength of the rewriting rules Type 0 is strongest, 3 is weakest Copyright © by Curt Hill

Type 3 - Regular Languages U  n or U  Wn U and W are non-terminals and n is a terminal A non-terminal may only be replaced by a terminal or non-terminal followed by a terminal Regular expressions are of this type –Do you know about regular expressions? Copyright © by Curt Hill

Regular (3) A  b | A  bC | A  Cd The production must have only one non-terminal on the left The right-hand side must be: – A terminal –A terminal followed by a non-terminal –A non-terminal followed by a terminal May not have a terminal non- terminal terminal on right –Terminal may lead or follow but not both Copyright © by Curt Hill

Type 2 - Context Free A  aNy Single non-terminal on left Any number or arrangement of non- terminals and terminals on the right Most programming languages are largely context free –The optional else in C is not Copyright © by Curt Hill

Type 1 - Context Sensitive xUy  xvy Where U is a non-terminal and v is any sequence of terminals and/or non-terminals –x, y are terminals U may be rewritten to v only in the context of x and y before and after We may have another rule aUb  aeb which is completely different replacement of U Copyright © by Curt Hill

Type 0 - Unrestricted u  v Unrestricted both sides of the production may have non-terminals or terminals, but u cannot be empty Unlike types 1-3 u could be a terminal Context is also important Very powerful, very little work done with it Copyright © by Curt Hill

Language Hierarchies Copyright © by Curt Hill Type 3 Regular Type 2 Context Free Type 1 Context Sensitive Type 0 Unrestricted

Languages and Automata Each of these languages corresponds to an automaton that can accept it The weakest is a regular language, which can be accepted by a regular expression or finite state automaton Later machines correspond to stronger languages We will consider these automatons later Copyright © by Curt Hill

Hierarchy Again Copyright © by Curt Hill TypeGrammarLanguageAutomata 3Finite StateRegularFinite 2Context Free Pushdown 1Context Sensitive Linear Bounded 0Recursively enumerable UnrestrictedTuring Machine

Again We use regular (type 3) languages are used for lexical analyzers –The lexical analyzer is typically the front-end of a compiler Most programing languages have a context-free grammar (type 2) –With a few ambiguities Efficient algorithms exist to implement parsers for both of these –This cannot be said for type 0 and 1 Copyright © by Curt Hill

Derivation or parse trees A multi-way tree where: –Each interior node is a non-terminal –Each leaf is a terminal –The start symbol is the root –Nested under each interior node is the RHS of the production, with the LHS being the node itself This is a handy data structure for compilers and the like Copyright © by Curt Hill

Example Parse Tree Copyright © by Curt Hill program stmts stmt varexpr = term = a b const var

Example Consider the following grammar V= {a,b,c,S} T = {a,b,c} P = { –S  abS –S  bcS –S  bbS –S  a –S  cb } Copyright © by Curt Hill

bcbba Copyright © by Curt Hill S b c b S b S a S  bcS S  bbS S  a

Audience Participation Lets try on the board bcabbbbbcb Bbbcbba Copyright © by Curt Hill

John Backus Principle designer of FORTRAN Substantial contributions to Algol60 Designed Backus Normal Form Eventually became a functional languages proponent Turing award winner Copyright © by Curt Hill

BNF John Backus defined FORTRAN with a notation similar to Context Free languages independent of Chomsky in 1959 Peter Naur extended it slightly in describing ALGOL Became known as BNF for Backus Normal Form or Backus Naur Form Meta-language is the language that describes another language Copyright © by Curt Hill

Simplest notation Form of productions: LHS ::= RHS Where: –LHS is a non-terminal (context free grammars) –RHS is any sequence of terminals and non-terminals, including empty There can be many productions with exactly the same LHS, these are alternatives If the RHS contains the LHS, the rule is recursive Copyright © by Curt Hill

Notation There is usually a simple way to distinguish terminals and non- terminals Rosen and others enclose non- terminals in angle brackets – ::= if ( ) – ::= if ( ) else Copyright © by Curt Hill

Simple extensions Some times there is an alternation symbol that allows us to only need one production with the same LHS, often the vertical bar – ::= + | - Some times things enclosed in [ and ] are optional, they may be present zero or one times Some times things enclosed in { and } may be present 1 or more times –Thus [{x}] allows zero or more x items Copyright © by Curt Hill

More The extensions are often called EBNF Syntax graphs are equivalent to EBNF These tend to be more easy to read Copyright © by Curt Hill

Syntax Graphs A circle represents a terminal –Reserved word or operator –No further definition A rectangle represents a non-terminal –For statement or expression –Must be defined else where An arrow represents the path between one item and another –The arrows may branch indicating alternatives Recursion is also allowed Copyright © by Curt Hill

Simple Expressions Copyright © by Curt Hill expression term + - factor * / constant ident ()expression

Parse tree example Trees are recursive Every sub-tree is a tree itself Consider the parse of: * ( ) –Using the previous syntax graph Copyright © by Curt Hill

Expression: * (3 – 4) Copyright © by Curt Hill term- factor 3 term factor 4 expression *factor 5 term + factor 2 expression factor ( )

BNF is generative A derivation is sentence generation Leftmost derivation –Only the leftmost non-terminal can be rewritten –This is usually the kind of derivation used by compilers –The previous derivation was leftmost There are also rightmost derivations The order of derivation does not affect the language defined Copyright © by Curt Hill

Example BNF productions Copyright © by Curt Hill ::= ::= | ; ::= = ::= a | b | c | d ::= + | - ::= | const

Example Derivation Copyright © by Curt Hill => => = => a = => a = + => a = b + => a = b + const

Exercises 13.1 b –1, 5, 13, 19, 25, 35 Copyright © by Curt Hill