### Abstract

Decomposition is an important property that we exploit in order to render problems more tractable. The decomposability of a problem implies the existence of some "independences" between relevant variables of the problem under consideration. In this paper we investigate the decomposability of functions which take values into an Abelian Group. Examples of such functions include: probability distributions, energy functions, value functions, fitness functions, and relations. For such problems we define a notion of conditional independence between subsets of the problem's variables. We prove a decomposition theorem that relates independences between subsets of the problem's variables with a factorization property of the respective function. As particular cases of this theorem we retrieve the Hammersley-Clifford theorem for probability distributions; an Additive Decomposition theorem for energy functions, value functions, fitness functions; and a relational algebra decomposition theorem.

Original language | English (US) |
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Title of host publication | 9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006 |

State | Published - 2006 |

Event | 9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006 - Fort Lauderdale, FL, United States Duration: Jan 4 2006 → Jan 6 2006 |

### Other

Other | 9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006 |
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Country | United States |

City | Fort Lauderdale, FL |

Period | 1/4/06 → 1/6/06 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Applied Mathematics

### Cite this

*9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006*

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*9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006.*9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006, Fort Lauderdale, FL, United States, 1/4/06.

**Independence, Decomposability and functions which take values into an Abelian Group.** / Silvescu, Adrian; Honavar, Vasant.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Independence, Decomposability and functions which take values into an Abelian Group

AU - Silvescu, Adrian

AU - Honavar, Vasant

PY - 2006

Y1 - 2006

N2 - Decomposition is an important property that we exploit in order to render problems more tractable. The decomposability of a problem implies the existence of some "independences" between relevant variables of the problem under consideration. In this paper we investigate the decomposability of functions which take values into an Abelian Group. Examples of such functions include: probability distributions, energy functions, value functions, fitness functions, and relations. For such problems we define a notion of conditional independence between subsets of the problem's variables. We prove a decomposition theorem that relates independences between subsets of the problem's variables with a factorization property of the respective function. As particular cases of this theorem we retrieve the Hammersley-Clifford theorem for probability distributions; an Additive Decomposition theorem for energy functions, value functions, fitness functions; and a relational algebra decomposition theorem.

AB - Decomposition is an important property that we exploit in order to render problems more tractable. The decomposability of a problem implies the existence of some "independences" between relevant variables of the problem under consideration. In this paper we investigate the decomposability of functions which take values into an Abelian Group. Examples of such functions include: probability distributions, energy functions, value functions, fitness functions, and relations. For such problems we define a notion of conditional independence between subsets of the problem's variables. We prove a decomposition theorem that relates independences between subsets of the problem's variables with a factorization property of the respective function. As particular cases of this theorem we retrieve the Hammersley-Clifford theorem for probability distributions; an Additive Decomposition theorem for energy functions, value functions, fitness functions; and a relational algebra decomposition theorem.

UR - http://www.scopus.com/inward/record.url?scp=84864544423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864544423&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84864544423

BT - 9th International Symposium on Artificial Intelligence and Mathematics, ISAIM 2006

ER -