Computability and Complexity, CISC 462 - Assignment 1 (Fall 2014, K. Salomaa)
Due in lecture 4:30 PM, Tuesday September 30
1. We consider a formal proof system that should have the property that all true statements are provable.
Suppose that our domain in

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FORMAL DEFINITION OF REDUCTIONS:
MAPPING REDUCIBILITY
This material is covered in Section 5.3 of the textbook.
A central task in computability (and in complexity theory to be discussed in the second
half of the course) is to compare the diculty of compu

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THE CLASS NP
Many important algorithmic problems have the following property:
there is no known polynomial time algorithm to nd a solution to the problem;
if a solution to the problem is given, it is easy to verify (in polynomial time) that it is
inde

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COMPUTABILITY: Some history
David Hilbert (18621943) wanted to clarify the methods of mathematical reasoning by
creating a consistent and complete formal axiomatic proof system for all of mathematics.
A formal axiomatic proof system is a precisely dene

Computability and Complexity, CISC 462 - Assignment 5 (Fall 2014, K. Salomaa)
Due in lecture 4:30 PM, Tuesday November 25
1. Describe the error in the following fallacious proof purporting to show that P = NP:
We prove P = NP nonconstructively by showing

Computability and Complexity, CISC 462 - Assignment 4 (Fall 2014, K. Salomaa)
Due in lecture 4:30 PM, Tuesday November 11
1. (a) Exersice 7.1 a, b, e, f. (3rd edition page 322; 2nd edition page 294)
(b) Exercise 7.2 a, b, e, f.
2. Exercise 7.4
3. (a) Exer

Computability and Complexity, CISC 462 - Assignment 2 (Fall 2014, K. Salomaa)
Due in lecture 4:30 PM, Tuesday October 14
1. Give an implementation-level description of a deterministic single-tape Turing machine that decides
the language consisting of the

Computability and Complexity, CISC 462 - Assignment 3 (Fall 2014, K. Salomaa)
Due in lecture 4:30 PM, Tuesday October 28
1. A single-tape TM M is called infected if when started with the empty input string, I love you appears
somewhere on M s tape at some

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TURING MACHINES
Turing machines are one of the most commonly used models for general-purpose computers.
The formal denition of a TM is given in Denition 3.3, p. 168 in the text and we will go
through the denition in class. The precise details of TM deni

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SPACE COMPLEXITY
This material is covered in Chapter 8 of the textbook.
For simplicity, we dene the space used by a Turing machine computation only for
machines that halt on all inputs.
Denition. (Def. 8.1)
1. Let M be a deterministic Turing machine (DT

CISC-462 Practice Final Exam, December 2014
Page 1 of 12
Queens University, Faculty of Arts and Science, School of Computing
CISC-462 Final Exam, December 2007. Instructor: Kai Salomaa
INSTRUCTIONS
This is an open book exam: Candidates may use any writte

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FURTHER UNDECIDABLE PROBLEMS;
REDUCTIONS
This material is covered in Chapter 5 of the textbook.
The undecidability of the language ATM means that there is no algorithm that decides,
for an arbitrary given Turing machine M and input string w, whether or

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LOGARITHMIC SPACE
Whenever an input encoding is reasonable (non-redundant), all input symbols must be read.
Thus it is not customary to consider sublinear time bounds when dealing with the standard
Turing machine model (the situation is dierent with so

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SPACE AND TIME HIERARCHIES
This material is covered in section 9.1 in the textbook.
We have seen that comparing the power of deterministic and nondeterministic time
bounds is very challenging. For corresponding questions dealing with space bounds we hav

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DECIDABILITY AND UNDECIDABILITY
Decidable problems from language theory
For simple machine models, such as nite automata or pushdown automata, many decision
problems are solvable. In the case of deterministic nite automata, problems like equivalence
can

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CIRCUIT COMPLEXITY
Circuit complexity is discussed in section 9.3 of the textbook.
Above we have discussed complexity measures that limit the resources used by an algorithm (a Turing machine). The algorithmic approach has allowed us to obtain signicant

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TIME COMPLEXITY
This material is in Chapter 7 in the textbook.
Our previous discussion on computability gives absolute limits concerning which questions can, even in principle, be solved algorithmically. As we have seen, many important
problems related

The CISC462 FINAL EXAM is on
Friday December 12 at 9:00 AM.
Please consult the exam timetable for the room location.
- This is an open book exam. You can bring with you the textbook
and any written/printed material you wish.
*NO* laptops or electronic