Theory of Automata
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
