1. Automata, Grammars and Languages
CSC 473 Automata, Grammars and Languages
7/23/2018
Automata, Grammars and Languages
Discourse 07
Reduction x
C SC 473 Automata, Grammars and Languages

2. Reduction of One Problem to Another
CSC 473 Automata, Grammars and Languages
Reduction of One Problem to Another
7/23/2018
Often want to solve a new problem P similar to a problem Q that has already been solved.
One way of solving P is to transform each instance of P into an instance of the known problem Q, then solve the Q instance, and then use it to obtain a solution to the P instance.
The solution to P uses the solution to Q as a “subroutine”.
We often write P  Q for “P is reducible to Q”
Ex: Squaring  Multiplication:
Ex: Multiplication  Squaring:
Ex: DFA Equivalence  DFA Emptiness
C SC 473 Automata, Grammars and Languages

3. Using Reduction to Prove “Difficulty”
CSC 473 Automata, Grammars and Languages
Using Reduction to Prove “Difficulty”
7/23/2018
If P  Q and P is known to be “hard to solve”, then Q must be hard to solve too.
For example¶, if P  Q and P is undecidable, then Q must also be undecidable. For if Q is decidable, we can use the reduction P  Q to construct a decider for P; contradiction.
Ex: We will show by reduction that the problem
is reducible to the problem
The undecidability of will imply the undecidability of
_______________________________
¶Here  stands for many-one or mapping reduction denoted m . It will be defined precisely later.
C SC 473 Automata, Grammars and Languages

4. Undecidability via Reductions: Halting
CSC 473 Automata, Grammars and Languages
Undecidability via Reductions: Halting
7/23/2018
HALTING PROBLEM
ACCEPTANCE (MEMBERSHIP) PROBLEM
Thm 5.1: is undecidable.
Pf: We show that so that if we had a decider for we could build a decider for
This contradicts the undecidability of , and so
must be undecidable.
Assume, contrary to what is to be proved, that has a decider R. Following is a visual proof that is reducible to :
C SC 473 Automata, Grammars and Languages

5. Undecidability via Reductions (cont.)
CSC 473 Automata, Grammars and Languages
Undecidability via Reductions (cont.)
7/23/2018
Consider a “compiler” (algorithm) C that given  M  constructs a new TM C( M ) as follows:
Reduction: use this and R to build a decider for
acc
acc
rej
loop
acc
rej
So S is a decider for Contradiction  theorem. 
C SC 473 Automata, Grammars and Languages

6. Undecidability: Empty Tape Acceptance
CSC 473 Automata, Grammars and Languages
Undecidability: Empty Tape Acceptance
7/23/2018
Thm: The EMPTY-TAPE-ACCEPTANCE problem is undecidable:
Pf: We will show that Consider a “compiler” C that given M,w constructs TM C(M,w ):
acc
acc
rej
rej
Summarze here 3/2/04. Compare this reduction to next slide.
In fact:
C SC 473 Automata, Grammars and Languages

7. Empty Tape Acceptance (cont’d)
CSC 473 Automata, Grammars and Languages
Empty Tape Acceptance (cont’d)
7/23/2018
Reduction: assume a decider R for We construct a decider from it for
E is a decider for Contradiction. 
acc
acc
rej
rej
Summarze here 3/2/04. Compare this reduction to next slide.
C SC 473 Automata, Grammars and Languages

8. Undecidability: Empty Set Acceptance
CSC 473 Automata, Grammars and Languages
Undecidability: Empty Set Acceptance
7/23/2018
Thm: The EMPTY-SET-ACCEPTANCE problem is undecidable:
Pf: We will show that We reduce the complement of to this problem. Consider a “compiler” C that given M,w constructs TM C(M,w ):
rej
acc
Summarze here 3/2/04. Compare this reduction to next slide.
C SC 473 Automata, Grammars and Languages

9. Empty Set Acceptance (cont’d)
CSC 473 Automata, Grammars and Languages
Empty Set Acceptance (cont’d)
7/23/2018
Reduction: assume a decider R for We construct a decider from it for
E is a decider for Contradiction  theorem 
acc
acc
rej
rej
Summarze here 3/2/04. Compare this reduction to next slide.
C SC 473 Automata, Grammars and Languages

10. Undecidability: Regular Set Acceptance
CSC 473 Automata, Grammars and Languages
Undecidability: Regular Set Acceptance
7/23/2018
Thm 5.3: The REGULAR-SET-ACCEPTANCE problem is undecidable:
Pf: We will show that Consider a “compiler” C that given M,w constructs TM C(M,w ):
(2)
acc
acc
(1)
Summarze here 3/2/04. Compare this reduction to next slide.
rej
acc
rej
C SC 473 Automata, Grammars and Languages

11. Regular Set Acceptance (cont’d)
CSC 473 Automata, Grammars and Languages
Regular Set Acceptance (cont’d)
7/23/2018
Reduction: assume a decider R for We construct a decider from it for
E is a decider for Contradiction  theorem 
acc
acc
rej
rej
Summarze here 3/2/04. Compare this reduction to next slide.
C SC 473 Automata, Grammars and Languages

12. Mapping Reduction: Motivation
CSC 473 Automata, Grammars and Languages
Mapping Reduction: Motivation
7/23/2018
Halting Problem
Empty-Tape Acceptance Problem
Empty-Set Acceptance Problem
Regular-Set Acceptance Problem Y Y x N N
C is an algorithm in each case
C SC 473 Automata, Grammars and Languages

13. TMs can Act as Recognizers or Transducers
CSC 473 Automata, Grammars and Languages
TMs can Act as Recognizers or Transducers
7/23/2018
Defn 5.17: A function is a computable function† if  a TM transducer M such that on every input w, M halts with f(w) on its tape. Such a TM is called an algorithm.
Compare & contrast this definition with:
Defn 3.6: A language is a decidable language if  a TM recognizer M such that on every input w, if w L it halts with “accept” and if w L it halts with “reject” . Such a TM is called a decider.
A recognizer can be viewed as a special case of a transducer that prints only a 1 or 0
A language can be viewed as a special case of a function that returns a boolean value.
____________________________
†Also called total computable function. “Total” means “defined for all arguments w”.
C SC 473 Automata, Grammars and Languages

14. Function:Set :: Transducer:Recognizer
CSC 473 Automata, Grammars and Languages
Function:Set :: Transducer:Recognizer
7/23/2018
Functions
Sets (Languages)
Computable† f
 TM transducer M 
(w) M on w halts & prints M(w)
(w) M(w) = f(w)
M is called an algorithm
Decidable L
 TM recognizer M 
(w) M on w halts & prints 0,1
(w) M(w) = 1  w L
M is called a decider ‡
Partial† Computable f
(w) M on w halts  f(w) defined
M is called a procedure
Recognizable L
(w) M on w halts  w L
M is called a recognizer
Special case of
Special case of
Special case of
†A computable function is historically called a “total computable function”
†A partial computable function is sometimes called an “effective function” and a procedure an “effective procedure”. “Partial” signifies that the function f might not be defined for some arguments.
‡A decider is sometimes just called an “algorithm”
Special case of
C SC 473 Automata, Grammars and Languages

15. Mapping Reduction: Definition
CSC 473 Automata, Grammars and Languages
Mapping Reduction: Definition
7/23/2018
All have the same pattern
f must be a computable function
The “compiler” must be an algorithm
Defn 5.20: Language is mapping reducible to language iff  a computable function such that Function f is called the reduction of to Y Y x N N
C SC 473 Automata, Grammars and Languages

16. Mapping Reduction: Definition (cont’d)
CSC 473 Automata, Grammars and Languages
Mapping Reduction: Definition (cont’d)
7/23/2018
Picture: Y Y x N N
equivalent to ,
C SC 473 Automata, Grammars and Languages

17. CSC 473 Automata, Grammars and Languages
Reduction (cont.)
7/23/2018
Thms 5.22, 5.23,5.28, 5.29: Let Then
decidable  decidable
undecidable  undecidable
recognizable  recognizable
non-recognizable  non-recognizable
If you want to show a problem P is easier than
problem Q then reduce P to Q
If you want to show a problem P is harder than
problem Q then reduce Q to P
easiness
hardness P: ? Q Q: ? P
C SC 473 Automata, Grammars and Languages

18. CSC 473 Automata, Grammars and Languages
Reductions
7/23/2018
Thm:
Ex:
Reduction: major method for showing unsolvability or non-recognizability:
Goal: to show is not recognizable [not decidable]
Known: is not recognizable [not decidable]
Strategy: reduce to
Method: build computable “translator” f to accept assuming we have a recognizer [decider] for
yes w f
C SC 473 Automata, Grammars and Languages

19. Undecidable via Reductions
CSC 473 Automata, Grammars and Languages
Undecidable via Reductions
7/23/2018
Thm: The NONEMPTY-SET-ACCEPTANCE problem is TM-recognizable but not decidable:
Pf: It is easy to see that since
So cannot be decidable.
An acceptor for is the following nondeterministic TM
encode
yes
yes
†what if w is not a syntactically valid TM encoding? Which set is it in?
‡ a good illustration of the power of nondeterminism to describe processes
C SC 473 Automata, Grammars and Languages

20. Non-Recognizable via Reductions
CSC 473 Automata, Grammars and Languages
Non-Recognizable via Reductions
7/23/2018
Thm: The EQUIVALENCE problem is not recognizable:
Pf: We show that is not recognizable by showing that (which means that ).
The reducing function generates a pair of TMs with the following behaviors:
rej
†what if w is not a syntactically valid TM encoding? Which set is it in?
‡ a good illustration of the power of nondeterminism to describe processes
acc
C SC 473 Automata, Grammars and Languages

21. Non-recognizable via m (cont’d)
CSC 473 Automata, Grammars and Languages
Non-recognizable via m (cont’d)
7/23/2018
Reduction: assume a decider for We construct a decider from it for
S is a decider for So This implies that and so cannot be recognizable 
acc
acc
rej
rej
Summarze here 3/2/04. Compare this reduction to next slide.
C SC 473 Automata, Grammars and Languages

22. Non-Recognizable via Reductions (cont’d)
CSC 473 Automata, Grammars and Languages
Non-Recognizable via Reductions (cont’d)
7/23/2018
Exercise: The FINITE-SET-ACCEPTANCE problem is not TM-recognizable:
Proof: Show that
Exercise: The INFINITE-SET-ACCEPTANCE problem is not TM-recognizable:
C SC 473 Automata, Grammars and Languages

23. Equivalence and Completeness
CSC 473 Automata, Grammars and Languages
Equivalence and Completeness
7/23/2018
Definition: Let C be a class of sets. A set A is mapping reduction complete ( complete ) in C iff
Remark: “A is complete” says A is a “hardest problem in C”
A is mapping equivalent to B iff
Fact: all complete sets in C are mapping equivalent
Exercise:
Start here 3/3/04
C SC 473 Automata, Grammars and Languages

24. CSC 473 Automata, Grammars and Languages
Completeness
7/23/2018
Theorem: is complete in the class TM of Turing-recognizable sets.
Proof: is accepted by U, so is in TM.
To show completeness, let B be any Turing-recognizable set in TM. Then  a TM such that Define the computable function
Then
Since B was chosen arbitrarily in TM the result follows 
Exercise: Show the Halting Prob is complete in TM
C SC 473 Automata, Grammars and Languages