Linear Algebraic Characterization of Logic Programs ⋆ Chiaki Sakama 1 , Katsumi Inoue 2 , and Taisuke Sato 3 1 Wakayama University, Japan sakama@sys.wakayama-u.ac.jp 2 National Institute of Informatics, Japan inoue@nii.ac.jp 3 AI research center AIST, Japan satou.taisuke@aist.go.jp Abstract. This paper introduces a novel approach for computing logic program- ming semantics based on multilinear algebra. First, a propositional Herbrand base is represented in a vector space and if-then rules in a program are encoded in a matrix. Then we provide methods of computing the least model of a Horn logic program, minimal models of a disjunctive logic program, and stable models of a normal logic program by algebraic manipulation of higher-order tensors. The result of this paper exploits a new connection between linear algebraic computa- tion and symbolic computation, which has potential to realize logical inference in huge scale of knowledge bases. 1 Introduction Logic programming (LP) provides languages for declarative problem solving and sym- bolic reasoning, while proof-theoretic computation like Prolog turns inefficient in real- world applications. Recent studies have developed efficient solvers for answer set pro- gramming (ASP)—LP under the stable model semantics [1]. In this paper, we take a different approach and introduce a new method of computing LP semantics in vec- tor spaces. There are several reasons for considering linear algebraic computation of LP. First, linear algebra is at the core of many applications of scientific computation, and integrating linear algebraic computation and symbolic computation is considered a challenging topic in AI [13]. Second, linear algebraic computation has potential to cope with Web scale symbolic data, and several studies develop scalable techniques to process huge relational knowledge bases [10, 11, 18]. Since relational KBs consist of ground atoms, the next challenge is applying linear algebraic techniques to LP and deductive DBs. Third, it would enable us to use efficient (parallel) algorithms of numer- ical linear algebra for computing LP. Moreover, matrix/tensor factorization techniques would be useful for approximation and optimization in LP. Several studies attempt to realize logical reasoning using linear algebra. Grefen- stette [5] introduces tensor-based predicate calculus in which elements of tensors rep- resent truth values of domain objects and logical operations are realized by third-order tensor contractions. Yang, et al. [18] introduce a method of mining Horn clauses from relational facts represented in a vector space. Serafini and Garcez [16] introduce logic tensor networks that integrate logical deductive reasoning and data-driven relational ⋆ This work is supported by NII Collaborative Research Program.