Formal Language Exam

ambiguous

A context free grammar is called ambiguous if there exists more than one leftmost derivation or more than one rightmost derivation for a string which is generated by grammar. There will also be more than one derivation tree for a string in ambiguous grammar.

context-free grammar

A quadruple (NT, T, R S); i.e., non-terminals, terminals, rules and the initial symbol a grammar in which the left-hand side of each production consists of a single nonterminal symbol.

context-free grammar

A quadruple (NT, T, R S); i.e., non-terminals, terminals, rules and the initial symbol.

countable

A set is countable if it is finite or countably infinite

transitive

If a=b and b=c, then a=c

closed under addition

Two vectors, when added together, are still in the same subspace. To prove: have a vector *x* and *y*, and add them together (x1 + y1) etc. It has to satisfy requirements and equal well and everything. Mostly if you can make an *x*', it works.

deterministic finite automaton

a finite automaton that has at most one transition from a state for each input symbol and no empty transitions. Abbreviated DFA.

nondeterministic finite automaton

a finite automaton that has multiple state transitions from a single state for a given input symbol, or that has a null transition, not requiring an input symbol. Abbreviated NFA.

regular language

a language described by a regular grammar, or recognizable by a finite automaton, e.g. a simple item such as a variable name or a number in a programming language.

uncountable

a set is uncountable, if its cardinality is larger than that of natural numbers (All reals)

Regular Set

a set of variables that can be described by a regular language

reflexive

a=a

Symmetric

a=b, b=a

Pushdown Automata

can be formally described as a 7-tuple (Q, ∑, S, δ, q0, I, F) − Q is the finite number of states, ∑ is input alphabet, S is stack symbols, δ is the transition function, q0 is the initial state (q0 ∈ Q), I is the initial stack top symbol (I ∈ S), F is a set of accepting states (F ∈ Q)

show the it is a regular set

how to show that a set is closed

Pumping Lemma for Context Free Languages

s = uv(^i)xy(^i)z |xyz| <= p

Pumping Lemma for Regular Languages

s = xy^iz E Ai >= 0 |xy| <= p

Chomsky Normal Form

the Productions are in the following forms − A → a A → BC S → ε where A, B, and C are non-terminals and a is terminal.

distinguishable

x and y are ____________ by L if for some z either xz or yz is in L

indistinguishable

x and y are _____________ if for every x and y there exists a string z that whenever xz is present in L yz is present in L proof: it must be reflexive, transitive and symmetric, that transitive: "for all z, az is in L iff bs is in L and for all z bz is in L iff cz is in L" therefore "for all z, az is in L iff cz is in L"

Myhill-Nerode Theorem

x and y are indistinguishable by A iff for every string z in the set, either both xz and yz are in A or both xz and yz are not in A. We write x=y in this case