# From 3-valued Semantics to Supported Model Computation for Logic Programs in Vector Spaces

### Taisuke Sato, Chiaki Sakama, Katsumi Inoue

#### Abstract

We propose a linear algebraic approach to computing 2-valued and 3-valued completion semantics of finite propositional normal logic programs in vector spaces. We first consider 3-valued completion semantics and construct the least 3-valued model of comp(DB), i.e. the iff (if-and-only-if) completion of a propositional normal logic program DB in Kleeneâ€™s 3-valued logic which has three truth values {t(true), f(false), âŠ¥(undefined)}. The construction is carried out in a vector space by matrix operations applied to the matricized version of a dualized logic program DBd of DB. DBd is a definite clause program compiled from DB and used to compute the success set Pâˆž as true atoms and finite failure set Nâˆž as false atoms that constitute the least 3-valued model Iâˆž = (Pâˆž,Nâˆž) of comp(DB). We then construct a supported model of DB, i.e. a 2-valued model of comp(DB) by carefully assigning t or f to undefined atoms not in Pâˆž or Nâˆž so that the resulting model is 2-valued and supported. The assigning process is guided by an atom dependency relation on undefined atoms. We implemented our proposal by matrix operations and conducted an experiment with random normal logic programs which demonstrated the effectiveness of our linear algebraic approach to computing completion semantics.

Download#### Paper Citation

#### in Harvard Style

Sato T., Sakama C. and Inoue K. (2020). **From 3-valued Semantics to Supported Model Computation for Logic Programs in Vector Spaces**.In *Proceedings of the 12th International Conference on Agents and Artificial Intelligence - Volume 2: ICAART,* ISBN 978-989-758-395-7, pages 758-765. DOI: 10.5220/0009093407580765

#### in Bibtex Style

@conference{icaart20,

author={Taisuke Sato and Chiaki Sakama and Katsumi Inoue},

title={From 3-valued Semantics to Supported Model Computation for Logic Programs in Vector Spaces},

booktitle={Proceedings of the 12th International Conference on Agents and Artificial Intelligence - Volume 2: ICAART,},

year={2020},

pages={758-765},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0009093407580765},

isbn={978-989-758-395-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 12th International Conference on Agents and Artificial Intelligence - Volume 2: ICAART,

TI - From 3-valued Semantics to Supported Model Computation for Logic Programs in Vector Spaces

SN - 978-989-758-395-7

AU - Sato T.

AU - Sakama C.

AU - Inoue K.

PY - 2020

SP - 758

EP - 765

DO - 10.5220/0009093407580765