Context free grammars (CFG) and languages (CFL):
Aims of this section: CFGs and CFLs are models of computation which define the syntax of hierarchical formal notations as utilized in programming or mark-up languages. Recursion is an essential characteristic which distinguish CFGs and CFLs from FAs and regular languages. Properties, weakness and strengths of CFLs. Equivalence of the CFGs and NPDAs. Non-equivalence of the deterministic and non-deterministic PDAs, Parsing and Context sensitive grammars CSG.Context free grammars and languages (CFG, CFL):
Algol 60 pioneered the CFGs and CFLs to state the syntax of programming languages (Backus-Naur Form).
Example: Arithmetic expression E, term T, factor F, primary P, a-op A = {+, -}, m-op M = {•, /}, exp-op = ^.
E -> T | EAT | AT, T -> F | TMF, F -> P | F^P, P -> unsigned number |variable| function designator | (E) [Notice the recursion: E ->* (E)]
Example of Recursive data structures and their traversals:
Binary tree T, leaf L, node N: T -> L | N T T (prefix) or T -> L | T N T (infix) or T -> L | T T N (suffix).
Such definitions can be turned directly to the recursive traversal procedures, example: procedure traverse (p: ptr); begin when p ≠ nil then start visit(p); traverse(p.left); traverse(p.right); end; end;
Df CFG: G = (V, A, P, S)
V: non-terminal symbols, “variables”; A: terminal symbols; S ∈ V: start symbol, “sentence”;P: set of productions or rewriting rules of the form X -> w, Here X ∈ V, w ∈ (V ∪ A)*
Rewriting the step: for u, v, x, y, y’, z ∈ (V ∪ A)*: u -> v if u = xyz, v = xy’z and y -> y’ ∈ P.
Derivation: “->*” is the transitive, reflexive closure of “->”, that is, u ->* v iff ∃ w0, w1, .. , wk with k ≥ 0 and u = w0, wj-1 -> wj, wk = v.
L(G) context free language produce by G: L(G) = {w ∈ A* | S ->* w}.
Example Symmetric structures: L = {0n 1n | n ≥ 0}, or even palindromes L0 = {w wreversed | w ∈ {0, 1}*}
G(L) = ({S}, {0, 1}, { S -> 0S1, S -> ε}, S); G(L0) = ({S}, {0, 1}, { S -> 0S0, S -> 1S1, S -> ε}, S)
Palindromes (length even or odd): L1 = {w | w = wreversed}. G(L1): add all the rules: S -> 0, S -> 1 to G(L0).
Ex Parenthesis expressions: V = {S}, T = {(, ), [, ] }, P = { S -> ε, S -> (S), S -> [S], S -> SS }
Sample derivation: S -> SS -> SSS ->* ()[S][ ] -> ()[SS][ ] ->* ()[()[ ]][ ]
The rule S -> SS builds this grammar ambiguous. The ambiguity is undesirable in practice, as the syntactic structure is usually employed to convey the semantic information.
Ambiguous structures in natural languages:
‘Time flies like an arrow’ versus. ‘Fruit flies like a banana’.
Bad news: There exist CFLs which are inherently ambiguous, that is each and every grammar for them is ambiguous. Furthermore, the problem of deciding whether the given CFG G is ambiguous or not, is not decidable.
Good news: For practical purposes it is simple to design the unambiguous CFG’s.
Exercise:
i) For Algol 60 grammar G (simple arithmetic expressions) above, describe the aim of the rule E -> AT and show illustrations of its use. Prove or disprove: G is unambiguous.
ii) Build an unambiguous grammar for the language of above parenthesis expressions.
iii) The ambiguity of ‘dangling else’. Some programming languages (example: Pascal) assign to nested if-then[-else] statements an ambiguous structure. This is then left to the semantics of language to disambiguate.
Let E represents Boolean expression, S statement, and let consider the 2 rules:
S -> if E then S, and S -> if E then S else S. Explain the trouble with this grammar and fix it.
iv) Provide a CFG for L = {0i 1j 2k | i = j or j = k}. Try to verify: L is inherently ambiguous.
Latest technology based Theory of Computation Online Tutoring Assistance
Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Theory of Computation help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Theory of Computation, project ideas and tutorials. We provide email based Theory of Computation help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Theory of Computation. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Theory of Computation Homework help and assignment help services. They use their experience, as they have solved thousands of the Theory of Computation assignments, which may help you to solve your complex issues of Theory of Computation. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.
tutorsglobe.com processing of mrna molecules assignment help-homework help by online biosynthesis of rna tutors
first law of thermodynamics tutorial all along with the key concepts of mathematical representation of first law of thermodynamics, internal energy, internal energy as a function of state, isothermal expansion and isothermal reversible expansion
tutorsglobe.com reabsorption in proximal convoluted tubule assignment help-homework help by online mechanism of urine formation tutors
a wire’s length lying in the magnetic field and where an emf is induced is termed as a coil.
theory and lecture notes of mos transistor inverter all along with the key concepts of inverter dynamic characteristics and power dissipation analysis of cmos inverter. tutorsglobe offers homework help, assignment help and tutor’s assistance on theory of mos transistor inverter.
tutorsglobe.com turgor pressure assignment help-homework help by online plasmolysis tutors
Mass Transfer Process tutorial all along with the key concepts of Modes of Mass Transfer, Mass transfer operations, Properties of Mixtures, Concentration of Species, Velocities, Diffusion Flux
Buffering Capacity of Amino Acids tutorial all along with the key concepts of Ionization of Water, Calculation of Hydrogen ion Concentration, Dissociation of Weak Acids, Definition of pKa, Buffer Solutions
tutorsglobe.com transgenic microbes assignment help-homework help by online biotechnology tutors
Cost Accounting basically deals with collection, analysis of applicable of cost data for interpretation and presentation for several troubles of management.
In a species, differences are caused through the alterations in the environment or any alterations in the innate genetic setup of an organism or through the combination of both.
Thermal Effects of Electric Currents and Electric Power tutorial all along with the key concepts of Current Power, Power in Electric Circuits, Power Dissipation in a Resistor, Joule's Law, Electromotive Force, Incandescent Lamp and Electrical Equivalent of Heat
monopoly and its characteristics: a monopoly condition means an exclusive ownership of a market by a supplier of a product or a service for which there is no substitute or alternative available.
tutorsglobe.com market equilibrium assignment help-homework help by online demand and supply tutors
tutorsglobe.com somatic embryogenesis assignment help-homework help by online micropropagation tutors
1956384
Questions Asked
3689
Tutors
1443231
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!