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CS3331 – Assignment 3
due Nov. 30, 2021
2-day no-penalty extension until: Dec. 2, 11:55pm
(SRA’s cannot be used to extend further)
1. (20pt) Construct a deterministic Turing machine M that, given as input a binary string w, computes
the remainder of w modulo 4. M starts with the initial configuration (s,w) and halts with the
configuration (h,(w mod 4)2). It is assumed that the input, w, is a valid nonnegative number in
base 2, that is, w ∈ {0} ∪ 1{0, 1}∗.
Here are some examples of M ’s behaviour:
(s,0) `∗M (h,0); (s,1011) `∗M (h,11); (s,101) `∗M (h,1).
Describe M using the macro language (such as the ones in Examples 17.8-9, p. 275 of textbook).
2. (40pt) Prove that the following languages are in D (if you need to define TM’s, then clear English
description is sufficient):
(a) L = {|M has 3 states},
(b) L = {|M is a TM and L(M) ∈ SD},
(c) L = {| M halts on the input w in at most 3|w| steps},
(d) L = {| M uses only at most |w| tape cells before and after the input w (maximum
3|w| tape cells in total)}.
3. (20pt) Can you build a Turing machine that enumerates all encodings of Turing machines whose
language is not empty? Explain your answer.
4. (10pt) Given a language L that is not decidable (that is, L 6∈ D) and a context-free language C,
what can you say about the decidability of L ∪ C? Explain your answer.
5. (10pt) Consider the closure, SDc, of SD under complement, union, and intersection. Because SD
is not closed under complement, we have that SD ( SDc. We have obtained therefore a class of
languages, SDc, that is strictly larger than SD. Does this contradict Church’s thesis? Explain your
answer.
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