This document discusses reducibility and the halting problem. It introduces reducibility as a way to convert one problem into another problem such that solving the second problem solves the first. The document then discusses the halting problem (HALTTM) in detail, proving it is undecidable through a reduction from the acceptance problem (ATM). It introduces computation histories as a method to prove reductions and discusses their use in showing languages like ALBA, ELBA, and ALLCFG are decidable/undecidable.

1. EECS 2001: Introduction to the
Theory of Computation
Muhammad Umair Khan

2. Reducibility
EECS 2001 Introduction to Theory of Computation 2
 Converting one problem into another problem
 Aim: solution to the second problem can be used to solve the first one
 Reducibility can be used to prove that certain problems are
computationally unsolvable

3. Halting problem
EECS 2001 Introduction to Theory of Computation 3
 HALTTM
 Determine whether a TM stops (HALTS) on a given input
 It may accept or reject
 HALT does not mean reject
HALTTM = {⟨M,w⟩| M is a TM and M halts on input w}

4. Halting problem
EECS 2001 Introduction to Theory of Computation 4
 Prove HALTTM is undecidable (informal discussion)
 Assume it is decidable
 We use this assumption to show that ATM is also decidable
 We will try to show that ATM is reducible to HALTTM
 Suppose we have a TM R
 R decides HALTTM
 R can be used to test whether a TM M halts on w
 If R says the M does not halt at w, then reject as (M, w) will not be in ATM
 If R says the M halts on w, then we can be assured that the M will eventually
accept or reject w and we will have a decision on whether it is part of ATM or
not.
 What does this mean?

5. Halting problem
EECS 2001 Introduction to Theory of Computation 5
 HALTTM is undecidable (informal discussion)
 Assume it is decidable
 We use this assumption to show that ATM is also decidable
 We will try to show that ATM is reducible to HALTTM
 Suppose we have a TM R
 R decides HALTTM
 R can be used to test whether a TM M halts on w
 If R says the M does not halt at w, then reject as (M, w) will not be in ATM
 If R says the M halts on w, then we can be assured that the M will eventually
accept or reject w and we will have a decision on whether it is part of ATM or
not.
 What does this mean? That there is a decider for ATM
 This cannot be true because we have already proved that ATM is undecidable:
hence R cannot exist, hence HALTTM is undecidable

6. Halting problem
EECS 2001 Introduction to Theory of Computation 6
 Prove HALTTM is undecidable (Formal Proof -Algorithmic steps)
 Assume R decides HALTTM
 Construct S to decide ATM
 S works as follows:
 Input (M, w)
 Run R on (M, w)
 If R rejects, S rejects
 If R accepts, simulate M on w till we get a result
 If M accepts, S accept, if M rejects, S reject
 Hence, if R decides HALTTM, S decides ATM
 Contradiction: Nothing can decide ATM

7. Computation histories
EECS 2001 Introduction to Theory of Computation 7
 Method used to prove that ATM is reducible to certain languages
 Usually used for problems where we have to show the
existence of something
 Remember finding the contradictory example in pumping lemmas
 Example, finding the integral root of a polynomial
 Computation history: sequence of configurations of a TM when it is
provided an input i.e., the record of computation of this TM
 Remember the derivation in CFGs

8. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 8
 If M is a TM and v is a string,
 Then an accepting computation history of M on v is a sequence of
configurations C1, C2, …, Ck where Ck is the accepting configuration
of M on v
 And each Ci leads to Ci+1 through a legal transition
 A rejecting computation history is similar except that Ck is a
rejecting configuration
 Computation histories are finite sequences
 TM has to halt
 If it does not halt, then there is no history

9. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 9
 Deterministic TMs have exactly one computation history for a
given string
 Non-deterministic TMs may have more than one computation
history for a given string

10. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 10
 A linear bounded automaton (LBA) is a type of TM which
has some restriction on its tape
 The head can only move on the portion of the tape where there is input
 If it tries to move to the left of the leftmost input character or to the right of the
rightmost input character, it will stay where it was.
 Hence the amount of memory is limited
 What do you think about the power of an LBA?
 It can only solve those problems which can be solved using the memory that was
used to store the input.

11. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 11
 A linear bounded automaton (LBA) is a type of TM which
has some restriction on its tape
 The head can only move on the portion of the tape where there is input
 If it tries to move to the left of the leftmost input character or to the right of the
rightmost input character, it will stay where it was.
 Hence the amount of memory is limited
 What do you think about the power of an LBA?
 It can only solve those problems which can be solved using the memory that was
used to store the input.
 Deciders for ADFA, ACFG, EDFA, and ECFG are all LBAs

12. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 12
 The following language decidable
ALBA = {⟨M,w⟩| M is an LBA that accepts string w}
Compare with ATM = {⟨M,w⟩| M is a TM that accepts string w}

13. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 13
 Lemma 5.8
 Let M be an LBA with q states and g symbols in the tape alphabet.
There are exactly qngn distinct configurations of M for a tape of length
n.
 M has q states
 Number of tape symbols is g
 Tape length is n – hence the head can be in one of n many places
 Number of possible strings is gn
 The total number of different possible configurations is qngn

14. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 14
 Prove ALBA is decidable
 Informal discussion
 If M accepts or rejects, we accept or reject accordingly
 If M loops, it is bound to be in one of the same configurations it has been already.
 As M is an LBA, the tape is limited. According to Lemma 5.8, there can be a
limited number of configurations.
 Hence if M goes into a loop, it can be easily detected (simulate M for qngn steps
and if it does not halt it is in a loop)
 Formal proof
 Simulate M on w for qngn steps or until it halts
 If M has halted, accept if it has accepted, reject if it has rejected.
 If it is has not halted it must be in a loop, reject.

15. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 15
 ELBA is undecidable (language of a given LBA is empty)
 Idea: if ELBA is decidable then so will be ATM
 Informal discussion
 For a TM M and input string v, suppose we build an LBA C such that the
language of C comprises of all computation histories which make M end up in
the accepting state on v (this can be done in bounded memory)
 If M accepts v, then there is at least one computation history
 If M does not accept v, then C is empty
 Therefore, if we can determine that the language C is empty or not, then we can
say M accepts v or not
 Construction of C
 C gets a computation history, identifies the start configuration, finds an identical match for Ci
and Ci+1, and keeps moving on until it matches the accepting configuration

16. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 16
 ELBA is undecidable (language of a given LBA is empty)
 Idea: if ELBA is decidable then so will be ATM
 Formal proof
 The TM S gets as input ⟨M,v⟩ and works as follows:
 Build an LBA C as described above
 Run R on input ⟨C⟩ (R is the decider for ELBA)
 If R reject, accept. If R accepts, reject.
 If R has accepted then it means that the language of C is empty and M has no accepting computation
history for v. Hence S rejects
 If R rejects, then L(C) is non-empty, which is only possible when there is a computation history of v on
M. therefore M must accept v. Hence, S will accept.
 We have successfully made a decider for ATM using ELBA. Contradiction. Therefore there
cannot be an ELBA

17. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 17
 ALLCFG is undecidable (whether a CFG generates all possible strings)
ALLCFG = {⟨G⟩| G is a CFG and L(G) = Σ∗}.
 Same strategy assume ALLCFG is decidable and then show that this
makes ATM decidable
 Informal discussion:
 For a TM M and string v, build a CFG G
 If v is accepted by M then G will generate all strings
 If v is not accepted by M, then G does not generate one specific string
 Specific string = accepting computation history for v being accepted by M
 Other strings = all computation histories that do not allow M to accept v

18. Reduction via computation histories
EECS 2001 Introduction to Theory of Computation 18
 ALLCFG is undecidable (whether a CFG generates all possible strings)
ALLCFG = {⟨G⟩| G is a CFG and L(G) = Σ∗}.
 Informal discussion:
 Building G
 Remember a computation history is represented as a string
 Try to remember how we proved the uncountability of real number
 Similarly, we make G such that it generates all string (computation histories) except the
accepting one
 Starting configuration different
 Ending configuration different
 Some transition from Ci to Ci+1 does not match
 Details on page 226 – omit for the purposes of the test

19. References
EECS 2001 Introduction to Theory of Computation 19
Ideas, problems and their solutions in this lecture/tutorial have been taken from
• Prof. Jeffery Edmonds’ Lecture notes for EECS 2001 at York University
• Prof. Suprakash Datta’ Lecture notes for EECS 2001 at York University
• Introduction to the Theory of Computation (3rd edition) by Michael Sipser
• Introduction to Theory of Computation by Anil Maheshwari and Michiel Smid
• Wikipedia and other webpages of different professors/universities