Structure and Complexity of Bag Consistency

Since the early days of relational databases, it was realized that acyclic hypergraphs give rise to database schemas with desirable structural and algorithmic properties. In a by-now classical paper, Beeri, Fagin, Maier, and Yannakakis established several different equivalent characterizations of acyclicity; in particular, they showed that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to-global consistency property for relations over that schema holds, which means that every collection of pairwise consistent relations over the schema is globally consistent. Even though real-life databases consist of bags (multisets), there has not been a study of the interplay between local consistency and global consistency for bags. We embark on such a study here and we first show that the sets of attributes of a schema form an acyclic hypergraph if and only if the local-to global consistency property for bags over that schema holds. After this, we explore algorithmic aspects of global consistency for bags by analyzing the computational complexity of the global consistency problem for bags: given a collection of bags, are these bags globally consistent? We show that this problem is in NP, even when the schema is part of the input. We then establish the following dichotomy theorem for fixed schemas: if the schema is acyclic, then the global consistency problem for bags is solvable in polynomial time, while if the schema is cyclic, then the global consistency problem for bags is NP-complete. The latter result contrasts sharply with the state of affairs for relations, where, for each fixed schema, the global consistency problem for relations is solvable in polynomial time

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions