Knowledge Compilation

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Knowledge Compilation
Dr. Rolf Haenni Center for Junior Research
FellowsUniversity of Konstanz

• Introduction
• Negational Normal Forms
• Knowledge Compilation Map
• Conclusion

Inhalt
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1. Introduction

• Propositional languages are usefull tools to
  represent knowledge about binary variables
• propositional KB ? subset of n-dimensional
  state space
• On the basis of a propositional KB, different
  queries can answered

• The time to answer a query grows often
  exponentially with the size of the propositional
  KB

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• The more succinct a language is, the less queries
  can be answered in polytime
• The goal is thus to compile the original
  propositional KB into the most succinct language
  that supports (answer in polytime) all the
  necessary types of queries

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2. Negational Normal Forms

• Propositional sentences can be represented by
  rooted directed acyclic graphs propositional DAG

Leaves ? Propositions, true, false
Non-Leaves ? Connectors
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• A negational normal form (NNF) is a propositional
  DAG in which negations appear only at the bottom
• ? Leaves are literals

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Flatness

• A sentence is flat, if the depth
  of the DAG is smaller or equal to 2
• The corresponding language of all flat NNFs is
  denoted by
• Remark CNFs and DNFs are flat
• Example

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Decomposability

• A sentence is decomposable, if at
  every -node we have
• The corresponding language is denoted by
• Remark DNFs are decomposable
• Example

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Determinism

• A sentence is deterministic, if
  at every -node we have
• The corresponding language is denoted by
• Example

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Smoothness

• A sentence is smooth, if at every
  -nodewe have
• The corresponding language is denoted by
• Example

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Further Languages

• DNFs are sentences whose root is
  a -node and all other nodes are -nodes
• The corresponding language is denoted by
• CNFs are sentences whose root is
  a -node and all other nodes are -nodes
• The corresponding language is denoted by
• is the subset of whose sentences
  contain all prime implicates of
• is the subset of whose sentences
  contain all prime implicants of
• is the subset of whose
  sentences list all the models (conjunctions
  of maximal length) of

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Overview 1 ? Sub-Languages
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Decision

• A -node is a decision node, if it
  hasthe following form

This corresponds to a decision node in a binary
decision diagram (BDD)

• The language whose sentences consist only of
  decision nodes is denoted by
• Remark sentences are
  deterministic
• Free BDDs (each path from the root to a leaf
  contains a decision variable at most once)
• Ordered BDDs (all paths from the root to the
  leaves contain the decision variables in the same
  order)

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Overview 2 ? Sub-Languages
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3. Knowledge Compilation Map
Overview 3 ? Succinctness
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Succinctness
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Overview 4 Succinctness vs. Queries
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Overview 5 Succinctness vs. Transformations
CD, SFO, AND, BAND, OR, BOR, NOT
CD, FO, SFO, OR BOR
CD, SFO, AND, BAND, BOR
CD, NOT?
CD, FO, SFO, BAND, OR, BOR
CD, NOT?
CD, NOT?
CD, FO, SFO, BOR
CD, BAND
CD, SFO, BAND, BOR, NOT
CD, FO, SFO, BAND
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4. Conclusion

• Negational normal forms (NNFs) are interesting
  classes of propositional languages
• Special properties of NNFs are Flatness,
  Decomposability, Determinism, Smoothness, and
  Decision
• The more properties a language possesses the
  less succinct it is the more queries can be
  answered in polytime
• The idea of knowledge compilation is to find the
  most succinct language that supports all
  necessary queries
• An important open question is