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CS3052 Computational Complexity FULL COURSE
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CS3052 BibleNotes for CS3052: Computational Complexity
S2 - 2021/2022
Contents
1 Finite State Automata 3
1.1 Deterministic Finite State Automaton (DFA) . . . . . . . . . . . . . . . . . . . . . 3
1.2 Regular Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Non-deterministic Finite State Automaton (NFA) . . . . . . . . . . . . . . . . . . . 4
1.4 NFA to DFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Regular Expressions and Regular Languages . . . . . . . . . . . . . . . . . . . . . . 7
1.6 From RegEx to NFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 From NFAs to RegEx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 The Pumping Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Context-Free Languages and Pushdown Automata 10
2.1 Pushdown Automata (PDA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Context-free languages (CFL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 CFG to PDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Pumping Lemma for CFL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Turing Machines (TM) 14
3.1 TM Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Turing Recognisable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Variants of Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Multitape TM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Non-deterministic TM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.6 Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Limits of Computation 17
4.1 Church-Turing Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Turing-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3.1 Turing Decidable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 Acceptance Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.5 Halting Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.6 Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.7 Rice’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.8 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.9 Hierarchy of Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1
,5 Complexity 20
5.1 Running time/Time complexity/Complexity . . . . . . . . . . . . . . . . . . . . . . 20
5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.4 Complexity Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.5 Formal definitions of Poly and Expo . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Recursive Algorithms 22
6.1 The Master Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Non-deterministic Complexity Classes 23
7.1 Verifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.2 Polytime Verifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.3 PolyCheck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.4 NPoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7.5 PolyCheck, NPoly, Expo, Poly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8 Decision Problems 26
8.1 P vs NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.2 Polyreductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8.3 NP-Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.4 Polyequivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.4.1 UHC ≤P DHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.4.2 DHC ≤P UHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.5 NP-hard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8.6 Formal Definitions of Satisfiability Problems . . . . . . . . . . . . . . . . . . . . . . 29
8.6.1 CircuitSAT ≤P SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8.6.2 SAT ≤P CircuitSAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.6.3 3-SAT ≤P SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.6.4 SAT ≤P 3-SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.7 CircuitSAT is NP-complete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
8.7.1 Sketch of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
8.8 Proof Techniques for NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Summary 32
9.1 Space Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
10 Amortised Complexity 35
2
,1 Finite State Automata
Definitions:
• Alphabet: finite set A of symbols
• Elements of A are letters.
• A word (or string) over A is a finite sequence of letters, written w = a1 a2 ...an , where each
ai ∈ A.
• The length of a word is n (the number of letters).
• The empty word is written as ε (has length 0 and consists of no letters).
• Concatenation is writing one word after another to make a longer word.
– Note: εw = wε = w
• A∗ denotes the set of all words over A and includes ε. A language is a subset of A∗ .
1.1 Deterministic Finite State Automaton (DFA)
A deterministic finite state automaton (DFA) M is a 5-tuple consisting of:
1. Alphabet A (a finite set).
2. Finite set of states, S.
3. Transition function F : A × S → S with input: (letter, state), output: (state).
4. A state s0 ∈ S called the initial or start state.
5. A subset T ⊆ S containing the accept states or terminals.
Write M = (A, S, F, s0 , T ), which starts in state s0 and reads a word w = a1 a2 ...an (where w ∈ A∗ )
one letter at a time from left to right.
If M is in state s ∈ S and reads letter a, then it moves to state F (a, s).
M accepts the word w if after reading letter an it is in an accept state t (t ∈ T ).
M rejects the word w if after reading letter an it is in a state S \ T (a reject state).
Example
M = (A = {a, b}, S = {s0 , s1 , s2 }, F, s0 , T = {s2 }) where F is the state-transition function described
below:
F s0 s1 s2
a s1 s1 s1
b s2 s2 s1
3
, 1.2 Regular Language
Let M = (A, S, F, s0 , T ) be a DFA. The language L(M ) defined by M (also say accepted by M ) is
the set of all words from A∗ which bring M from s0 into an accept state .
A language L is regular if there exists a DFA M with L = L(M ).
Lemma: If L is a finite language, then L is regular.
1.3 Non-deterministic Finite State Automaton (NFA)
A non-deterministic finite state automaton (NFA) is a 5-tuple (A, S, F, s0 , T ) where A, S, s0 and T
are the same as for a DFA (Section 1.1), but now F is a function from (A ∪ {ε}) × S to P(S). The
power set of S is the set of all subsets of S.
It is allowed for an NFA to have:
• F (a, s) = ∅ to be the empty set, so no arrows are leaving the state s when a is read.
• F (a, s) = {s′ , s′′ } so from state s there can be more than one state transition when a is read.
• F (ε, s) = s′ empty transitions that do not require the NFA to read any symbols to move.
An NFA N accepts a word w if there exists a directed path from s0 to an accept state whose labels
concatenate to give w (i.e. the letters of w appear in order but there may be one or more ε before
or after each letter).
The language L(N ) defined by N is the set of all words that N accepts.
Can think of an NFA as computing in parallel on the word w, where more than one branch of com-
putation is ”alive” at the same time. It accepts if any of the branches of computations accepts .
Example:
N = (A = {a, b}, S = {s0 , s1 , s2 , s3 , s4 }, F, s0 , T = {s4 })
4
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