SLIDE 1
Direct Encodings of NP- Complete Problems into Horn Sequents of Multiplicative Linear Logic
AIST Satoshi Matsuoka
SLIDE 2 Motivation
- To solve NP-complete problems
- Success of SAT solvers to solve NP-
complete problems at a practical level
- Another Logical Viewpoint: Linear Logic
- Provability of Multiplicative Linear Logic
(MLL) is NP-complete
- Any NP-complete problem can be
encoded into MLL in principle
- No obvious existence of a direct encoding
- f a particular NP-complete problem
SLIDE 3 In this talk
- In the proceedings paper
- 1. Encodings of 3D MATCHING and
PARTITION into MLL
- 2. Their correctness proofs using MLL proof
nets
- In this talk
- 1. Encodings of these problems into HMLL
- 2. Only examples
- 3. Horn programs of these examples
SLIDE 4
The system IMLL
Formulas: Inference rules:
are multisets of IMLL formulas
SLIDE 5
Difference between IMLL and classical (or intuitionistic) logic
(C) (W) (W)
SLIDE 6 Difference between IMLL and classical (or intuitionistic) logic (Cont.)
cannot be proved in IMLL
- No contraction and weakening rules in IMLL
- IMLL is more resource sensitive than classical
(or intuitionistic) logic
SLIDE 7
The system HMLL
Simple Formulas: Horn Implications: Horn sequents: where is a multiset of Horn implications
SLIDE 8
The system HMLL (cont.)
Inference rules:
HMLL is a very restricted subsystem of IMLL
SLIDE 9
Multiplicative Horn Programs
Directed chains: vertices: simple formulas edges: Horn implications formulas
such that and are identified
SLIDE 10
Interpretation of HMLL into Horn programs
SLIDE 11
Interpretation of HMLL into Horn programs (Cont.)
SLIDE 12
Interpretation of HMLL into Horn programs (Cont.)
SLIDE 13
Interpretation of HMLL into Horn programs (Cont.)
SLIDE 14
Theorem (Kanovich)
Multiplicative Horn Programs
SLIDE 15
The ３D MATCHING Problem
Given where Find such that
SLIDE 16
The ３D MATCHING Problem (Example)
Given Find such that Solution:
SLIDE 17 The ３D MATCHING Problem (Example)
from from from from
SLIDE 18
The ３D MATCHING Problem (Example)
SLIDE 19 The ３D MATCHING Problem (Example)
So, we have obtained a Horn program for the sequent
SLIDE 20 The PARTITION problem
Given a finite set and a function
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Find a subset such that Example: A solution:
SLIDE 21 The PARTITION problem
from
from
SLIDE 22
The PARTITION problem
SLIDE 23
The PARTITION problem
SLIDE 24 The PARTITION problem
So, we have obtained a Horn program for the sequent
SLIDE 25 Summary
- Have obtained direct encodings of two NP-
complete problems into Horn programs
- A lot of work should be done:
– More encodings – First-order extensions – Implementations, etc.
