Theory of Automata

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How many Regular Languages

Countably infinite

Every CF language is in D

The set of context-free languages is a proper subset of D

A function is a special relation

True

Complexity

What makes some problems computationally hard and others easy? (easy vs hard)

NonDeterministic Turing accepts/rejects

M accepts w iff exists some path that accepts it, M rejects w iff all paths reject it

A path P can be infinite. For FSM, DPDA, or NDFSM and NDPDA without ε transitions

P always ends

Context Free Languages can be recognized by

Pushdown Automata

Reduction of SAT to TSP

SAT->3-SAT->Hamiltonian-circuit->TSP

Reduction of SAT to independent set

SAT->3-SAT->INDEPENDENT-SET

REDUCTION of SAT to vertex-cover

SAT->3-SAT->Vertex-cover

Does TM M accept epsilon?

SD/D

Does TM M accept w?

SD/D

Does TM M halt on the empty tape

SD/D

Does TM M halt on w?

SD/D

Is there any string on which TM M halts?

SD/D

Is there any string that TM M accepts ?

SD/D

NP-Complete Languages

Subset-SUM, Set-Partition, TSP-DECIDE, Hamiltonian Path. Hamiltonian circuit, KnapSack, Bin packing

P: contains languages that can be decided by a

TM in polynomial time

A relation is a set of ordered pairs

True

Backus Naur Form: a notation for writing practical context-free grammars

True

Every infinite language has a subsset that is not in D

True

For each NDFSM, there is an equivalent DFSM.

True

H(any) ={<M>: there exists at least one string on which TM M halts} is in SD?

True

If L is context-free , then L Union (not) L must be regular

True

If L is in SD and its complement is context-free, then L must be in D

True

If L1 and L2 are in D, then L1 - L2 must be in D

True

If L1 is in D and L2 is in SD then L1 intersect L2 must be in SD

True

If L1 is not in D and L2 is regular , then it is possible the L1 intersect L2 is regular

True

If L1 is reducible to L2 and L1 is not in D then L2 is not in D

True

If L1 is reducible to L2 and L1 is not in SD then L2 is not in SD

True

If L1 is reducible to L2 and L2 is in D then L1 is in D

True

If L1 is reducible to L2 and L2 is in SD then L1 is in SD

True

If not H where in D then every SD language would be in D

True

NDPDA = Context-free language > DPDA

True

PDA without epsilon-transitions Must Halt

True

The union of a finite number of regular Languages must be regular

True

The union of two context-free languages must be in D

True

There are uncountably many non-regular languages over (Epsilon)={a,b}

True

Every DFSM M, on input w, halts in at most |w| steps.

True (Theorem)

SD is closed under

Union , Intersection, Concatenation, Kleene star

Regular Language closure

Union, Concatenation, Kleene Star, Complement, Intersection, Difference, Reverse, Letter substitution

Context Free Language Closure Properties

Union, Concatenation, Kleene star Reverse , Letter Substitution

The set D is closed under

Union, Intersection, complement, set difference , concatenation, Kleene Star

Computatbility

What are the fundamental capabilities and limitations of computers? (Solvable and Unsolvable)

Is c++, perl a formal language

Yes

TM in complement of SD

accept all strings? ,Comparison of accepting languages, Not halt on any string, does not halt on its own description,

Decision procedure

answers a decision problem

NP

contains languages that can be decided by a NDTM, Can be verified in Polynomial time

PSPACE

contains languages that can be decided by a machine with polynomial space

Rice's Theorem

contains only even/odd number of strings, contains all strings that start with a, is infinite , is regular

The intersection of a regular language and a nonregular language must NOT be regular

False

The union of an infinite number of regular languages must NOT be regular

False

The union of an infinite number of regular languages must be regular

False

{<M> :L(M) is not context free} is in D

False

{<M>:L(M) is context free} is in D

False

Regular Languages Can be recognized

Finite State Machines

Formal Grammar Defines

Formal Language

Regular Grammar

Have left side non-terminal and right side lots

A sentence w is Valid

IF and only IF it is true in all interpertations

Using the pumping Theorem

If L is regular, then every long string in L is pumpable

A language is in D

If and only if both it and its complement are in SD

A sentence w is unsatisfiable

If and only if not w is valid

A language L is in D

If and only if there exists a TM M that halts on all inputs and accepts all strings in L and rejects all strings not in L

3-Sat is reducible to

Independent-set

H = {<M,w> : TM M halts on input string w

Is SemiDecidable NOT DECIDABLE

If l1 is reducible to L2 and L2 is not in SD then L1 is not in SD

False

Ambiguity

A grammar is ambiguous iff there is at least one string in L(G) for which G produces more than one parse tree.

Formal languages

A set of strings over a given alphabet

Languages in P

All regular languages and context-free languages

The intersection of a regular language and a nonregular Language must be regular

False

DFSM = NDFSM

Can be proven in Formal Theory

How to show a language is In P

DTM with an algorithm that runs on a regular computer

Languages that aren't in D

Does TM M halt on w? Does TM M not halt on w? Does TM M halt on the empty tape? Is there any string on which TM M halts? Does TM M accept all strings? Do TMs M(a) and M(b) accept that same languages? Is the language that TM M accepts regular?

If L is in SD then its complement must not be in D

False

If L1 and L2 are not in D, then L1-L2 cannot be regular

False

If L1 and L2 are not regular Languages , then L1(intersection)L2 is not regular

False

If L1 and L2 are regular Languages and L1 is a subset of L subset of L2 then L must be regular?

False

If L1 intersect L2 is in D then both L1 and L2 must be in D

False

If L1 is a proper subset of L2 and L2 is CF then L1 must be context-free

False

If L1 is in D and L2 is in SD then L1 intersect L2 must be in D

False

If L1 is reducible to L2 and L2 is not in D then L1 is not in D

False

If L1 and L2 are not in D, then L1 union L2 cannot be in D

false

How to show a language is in NP

find a polynomial verifier

A language L is in SD

if and only if there exists a TM M that accepts all strings in L and fails to accept every string not in L

Does TM M have an even number of states?

in D

Does TM M halt on all strings?

in NOT(SD)

not H = {<M,w>: TM M does not halt on input string w}

is NOT in SD

Decision problem is

simply a problem for which answer is yes or no

Automata theory

study of abstract machines and problems they are able to solve

Theorem: For Each DFSM M

there is an equivalent NDFSM M'

Theorem: For each NDFSM

there there is an equivalent DFSM

An inference rule is sound If and only If

whenever it is applied to a set A of axioms, any conclusion that it produces is entailed by A


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