LANGUAGE AND GRAMMARS COMP 319 University of Liverpool

LANGUAGE AND GRAMMARS COMP 319 © University of Liverpool slide 1

Contents • • • Languages and Grammars Formal languages Formal grammars Generative grammars Analytic grammars Context-free grammars LL parsers LR parsers Rewrite systems L-systems COMP 319 © University of Liverpool slide 2

Software Engineering Foundation Software engineering may be summarised by saying that it concerns the construction of programs to solve problems and that there are three parts: - Construction/engineering, and methods - Problems, and problem solving, and - Programs COMP 319 © University of Liverpool slide 3

Languages and grammar • Languages are spoken and written (linguistics) • To be effective they must be based on a shared set of rules – a grammar • Grammars are introspective they are based on and couched in language • Natural language grammars are constantly shifting and locally negotiated • A grammar is a formal language in which the rules of discourse are discussed and are the aim COMP 319 © University of Liverpool slide 4

Formal language concepts • The concept emerges because of the need to define rules (for language) • Formally, they are collections of words composed of smaller, atomic units • Issues of concern are - the number and nature of the atomic units, - the precision level required, - the completeness of the formalism COMP 319 © University of Liverpool slide 5

Examples of formal languages • The set of all words over {a, b} • The set {an : n is a prime number} • The set of syntactically correct programs in a given computer programming language • The set of inputs upon which a certain Turing machine halts COMP 319 © University of Liverpool slide 6

Formal language specification There are many ways in which a formal language can be specified e. g. • strings produced in a formal grammar • strings produced by regular expressions • the strings accepted by automata • logic and other formalisms COMP 319 © University of Liverpool slide 7

Language Production Operations • Concatenation of strings drawn from the two languages • Intersection or union of common strings in both languages • Complement of one language • Right quotient of one by the other • Kleene star operation on one language • Reverse of a language • Shuffle combination of languages COMP 319 © University of Liverpool slide 8

Formal Grammars • Noam Chomsky - Linguist, philosopher at MIT - 1956, papers on information and grammar • Types of formal grammar - Generative grammar - Analytical grammar COMP 319 © University of Liverpool slide 9

Generative formal grammars • Generative grammars: A set of rules by which all possible strings in a language to be described can be generated by successively rewriting strings starting from a designated start symbol. In effect it formalises an algorithm that generates strings in the language. COMP 319 © University of Liverpool slide 10

Analytic formal grammars • Analytic grammars: A set of rules that assumes an arbitrary string as input, and which successively reduces or analyses that string to yield a final boolean “yes/no” that indicates whether that string is a member of the language described by the grammar In effect a parser or recogniser for a language COMP 319 © University of Liverpool slide 11

Generative grammar components Chomsky’s definition – essentially for linguistics but perfect formal computing grammars; consists of the following components: - A finite set N of nonterminal symbols - A finite set of terminal symbols disjoint from N - A finite set P of production rules where a rule is of the form: string in ( N)* → string in ( N)* - A symbol S in N that is identified as the start symbol COMP 319 © University of Liverpool slide 12

Generative grammar definition • A language of a formal grammar: • G = (N, , P, S) • Is denoted by L(G) • And is defined as all those strings over � such that can be generated by starting from the symbol S and then applying P until no more nonterminal symbols are present COMP 319 © University of Liverpool slide 13

A generative formal grammar • Given the terminals {a, b}, nonterminals {S, A, B} where S is the special start symbol and • Productions: S → ABS S → (the empty string) BA → AB BS → b Bb → bb Ab → ab Aa → aa Defines all the words of the from anbn, (i. e. n copies of a followed by n copies of b) COMP 319 © University of Liverpool slide 14

Context Free Grammars • Theoretical basis of most programming languages. • Easy to generate a parser using a compiler. • Two main approaches exist: top-down parsing e. g. LL parsers, and bottom-up parsing e. g. LR parsers. COMP 319 © University of Liverpool slide 15

LL parser • Table based, top down parser for a subset of the context-free grammars (LL grammars). • Parsing is Left to right, and constructs a Leftmost derivation of the sentence. • LL(k) parsers use k tokens of look-ahead to parse the LL(k) grammar sentence. • LL(1) grammars are popular and fast because only the next token is considered in parsing decisions. COMP 319 © University of Liverpool slide 16

Table based LL parsing • Consider the grammar 1. S → F 2. S → ( S + F) 3. F → 1 • This has the parsing table ( ) 1 + $ S 2 - 1 - - F - - 3 - - e. g. 1 and S implies rule 1 i. e. Stack S is replaced with F and 1 is output Stack and Input same = delete Stack and Input different = error Architecture Input buffer: <null> | | +-------+ Stack | | S <---| Parser | --> Output $ | | +-------+ ^ | +------+ | Parsing | | table | +------+ • Example input (1+1)$ COMP 319 © University of Liverpool slide 17

Table based LL parsing • Consider the grammar 1. S → F 2. S → ( S + F) 3. F → 1 • This has the parsing table input stack action output ( S$ parse ( S : 2 2 ( (S + F)$ ( ( delete 2 1 S + F)$ parse 1 S : 1 21 ( ) 1 + $ 1 F + F)$ parse 1 F : 3 213 S 2 - 1 - - 1 1 + F)$ 1 1 delete 213 F - - 3 - - + + F)$ + + delete 213 1 F)$ parse 1 F : 3 2133 1 1)$ 1 1 delete 2133 ) )$ ) ) delete 2133 $ $ stop 2133 e. g. 1 and S implies rule 1 i. e. Stack S is replaced with F and 1 is output Stack and Input same = delete Stack and Input different = error • Example input (1+1)$ COMP 319 © University of Liverpool slide 18

Parse Tree

Left Right Parser • Bottom up parser for context-free grammars used by many program language compilers • Parsing is Left to right, and produces a Rightmost derivation. • LR(k) parsers uses k tokens of look-ahead. • LR(1) is the most common type of parser used by many programming languages. Usually always generated using a parser generator which constructs the parsing table; e. g. Simple LR parser (SLR), Look Ahead LR (LALR) e. g. Yacc, Canonical LR. COMP 319 © University of Liverpool slide 20

Left Right parser example. . • • • Rules. . . 1) E → E * B (2) E → E + B (3) E → B (4) B → 0 (5) B → 1 COMP 319 © University of Liverpool slide 21

Left Right parser example COMP 319 © University of Liverpool slide 22

Re-writing • Rewriting is a general process involving strings and alphabets. Classified according to what is rewritten e. g. strings, terms, graphs, etc. • A rewrite system is a set of equations that characterises a system of computation that provides one method of automating theorem proving and is based on use of rewrite rules. • Examples of practical systems that use this approach includes the software Mathematica. COMP 319 © University of Liverpool slide 23

Re-writing logic example • !!A=A // eliminate double negative • !(A AND B) = !A OR !B // de-morgan COMP 319 © University of Liverpool slide 24

Re-writing in Mathematica (Wolfram) COMP 319 © University of Liverpool slide 25

L-systems • Named after Aristid Lindenmeyer (1925 -1989) a Swedish theoretical biologist and botanist who worked at the University of Utrecht (Netherlands) • Are a formal grammar used to model the growth and morphology of plants and animals • In plant and animal modelling a special form, the parametric L-system is used – based on rewriting. • Because of their recursive, parallel, and unlimited nature they lead to concepts of selfsimilarity and fractional dimension and fractallike forms. COMP 319 © University of Liverpool slide 26

L-system structure • The basic system is identical to formal grammars: G = {V, S, Ω, P} • where G is the grammar defined V (the alphabet) a set of symbols that can be replaced by (variables) S is a set of symbols that remain fixed (constants) Ω(start, axiom or initiator) a string from V, the initial state P is a set of rules or productions defining the ways variables can be replaced by constants and other variables. Each rule, consists of a LHS (predecessor) and RHS (successor) COMP 319 © University of Liverpool slide 27

Slide 28 Example 1: Fibonacci numbers • • V: A B C: none Ω: A P: p 1: A → B p 2: B → AB COMP 319 N=0 A N=1 → B N=2 → AB N=3 → BAB N=4 → ABBAB N=5 → BABABBAB N=6 → ABBABBAB N=7 → BABABBAB. . . Counting lengths we get: 1, 1, 2, 3, 5, 8, 13, 21, . . . The Fibonacci numbers © University of Liverpool slide 28

Slide 29 Example 2: Algal growth • • V: A B C: none Ω: A P: p 1: A → AB p 2: B → A COMP 319 N=0 N=1 N=2 N=3 A → ABAABABA © University of Liverpool slide 29

Example 3: Koch snowflake • • V: F C: none Ω: F P: p 1: F → F+F-FF+F COMP 319 N=0 F N=1 → F+F-F-F+F N=2 → F+F-F-F+F+F. . . N=3 etc © University of Liverpool COMP 319 Software Engineering II slide 30

Example 4: 3 D Hilbert curve COMP 319 © University of Liverpool slide 31

Example 5: Branching COMP 319 © University of Liverpool slide 32