Prove that Clique problem is NP and cannot be reduced to P because the Deterministic Finite Automata of the Clique problem has exponential number of states. We can use the same concept to prove that NP is not equal to P using Turing Machine. We figured out a way to unify Mathematics. This proof is for those Theoretical Computing guys who do not know Boolean algebra but know Turing Machine. Kung fu computer science, Geometric complexity theory

1. NP vs P Proof using
Discrete Finite Automata
Using Clique Problem as example; How is it related to Turing Machine Time Complexity
Sing Kuang Tan
singkuangtan@gmail.com
11 November 2021

2. 4 Vertices 3-Clique Problem Boolean Algebra
The Boolean algebra for 4 vertices 3-Clique problem is
P=ACE+ADF+BCF+BDE
=A(CE+DF)+B(CF+DE)
Can be converted to a Regular Expression:
A(b|B)C(d|D)E(f|F)|A(b|B)(c|C)D(e|E)F|(a|A)BC(d|D)(e|E)F|(a|A)B(c|C)DE(f|F)

3. Plot the Discrete Finite Automata of the Boolean Algebra
P=ACE+ADF+BCF+BDE
=A(CE+DF)+B(CF+DE)
18 States !!!
3 branch out on first 3 alphabets
● 3 branch out result in 6 alphabet
combinations examined in state
10 to 15
● 6 is exponential with respect to
the number of vertices (4 in this
case) in this Clique problem

4. Simple Boolean Algebra
The Boolean algebra for a simple Boolean algebra is
Q=ACD+AEF+BCD+BEF
=(A+B)(CD+EF)
Can be converted to a Regular Expression:
A(b|B)CD(e|E)(f|F)|A(b|B)(c|C)(d|D)EF|(a|A)BCD(e|E)(f|F)|(a|A)B(c|C)(d|D)EF

5. Plot the Discrete Finite Automata of the Boolean Algebra
Q=ACD+AEF+BCD+BEF
=(A+B)(CD+EF)
11 States !!!
1 branch on first alphabet 1 branch on third alphabet
● 1 branch results in 2 alphabet
combinations examined in state 5 to 6
● 1 branch results in 2 alphabet
combinations examined in state 8 to 9

6. Conclusion
● Clique problem is more algorithmically complex than simple Boolean algebra
● Therefore Clique problem has many more states (18 vs 11)
○ Clique problem has exponential number of states and therefore NP is not equal to P
● No matter what representations we use (Boolean Algebra vs Discrete Finite Automata), we
get the same result that Clique problem has much more algorithmic complexity than a simple
Boolean algebra
Problem Type Boolean Algebra Number of States of Discrete Finite Automata
4 vertices 3-Clique problem P=ACE+ADF+BCF+BDE
=A(CE+DF)+B(CF+DE)
18
Simple Boolean Algebra Q=ACD+AEF+BCD+BEF
=(A+B)(CD+EF)
11

7. Same Result for Turing Machine
● If we convert the Discrete Finite Automata to Turing Machine, we will get the
same result
● That Clique problem has more algorithmic complexity than simple Boolean
algebra
Convert to
Discrete Finite Automata Turing Machine

8. Unify Mathematical Theories
Discrete Finite Automata Turing Machine
Convert to
Convert to
Convert to
Boolean Algebra
● We can unify Mathematical Theories
by converting it to standard form in
one of the theories
● This enables to solve all problems
using one theory

9. To All Theoretical Computing Guys Out There
● Not everyone in the NP vs P community are trained in digital electronics and
know Boolean Algebra
● I convert Boolean Algebra to Discrete Finite Automata so that those who are
not trained in Boolean Algebra can also understand
● I think Discrete Finite Automata and Turing Machine are very similar and use
of Turing Machine results in the same proof of NP is not equal P
● I sent my proofs to the Turing Machine guys and many are not able to
understand my Boolean Algebra proof
○ But I think my proof represented in Turing Machine is the same thing

10. I think the Problems with NP vs P Proofs in the internet
using Turing Machine
● Many of these proofs are hand-waving
● The proofs did not cover all the cases in the NP vs P problem
● Due to the convoluted change of values of the tape machine by Turing
Machine, an algorithm implemented in Turing Machine is difficult to
understand
○ I think it is easier to analyse the algorithm using Boolean algebra
○ The NP vs P proof is more succinct using Boolean Algebra than Turing Machine

11. Links to create Discrete Finite Automata (DFA)
● Link to create NFA, DFA and min DFA
○ https://cyberzhg.github.io/toolbox/nfa2dfa

12. Comments?
What do you think?
● If we proved NP is not equal to P using Boolean Algebra, it is the same when
we use Discrete Finite Automata and Turing Machine?
○ I think is Yes.
● Should we rewrite Mathematical Theories in Canonical form in one of the
Theories so that we can unify Mathematics?

13. Collaborations?
● We can work together to get the final proof for NP vs P. :D
○ Sing Kuang Tan singkuangtan@gmail.com
● Help me forward this email to anyone who is interested in this NP vs P
problem.
● Or tell me who are interested in this problem.

14. More about Clique Problem
● Step by step explanation of NP vs P in Clique problem
○ https://www.slideshare.net/SingKuangTan/clique-problem-stepbystep
● My Quora questions on simplification of Boolean algebras of Clique problems
○ https://www.quora.com/What-is-the-best-way-to-simplify-this-Boolean-algebra-expression-P-
ADB-DFE-FCB-AEC-This-is-the-4-Vertices-3-Clique-problem
○ https://www.quora.com/unanswered/Simplify-Boolean-algebra-P-sum_-1-le-i-j-k-l-leq-7-i-neq-j-
neq-k-neq-l-a_-i-j-a_-i-k-a_-i-l-a_-j-k-a_-j-l-a_-k-l-A-7-Vertices-4-Clique-problem-if-of-
operations-cannot-be-reduced-then-NP-is-not-equal-P-1

15. About Me
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■ Use Inductive or Deductive Logic to solve NP vs P?
■ Kung Fu Computer Science, Clique Problem: Step by Step
■ Beyond Shannon, Sipser and Razborov; Solve Clique Problem like an Electronic Engineer
■ A weird Soviet method to partially solve the Perebor Problems
■ 8 trends in Hang Seng Index
■ 4 types of Mathematical Proofs
■ How I prove NP vs P
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○ Prove Np not equal P using Markov Random Field and Boolean Algebra Simplification
○ https://vixra.org/abs/2105.0181
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