We study a (first-order) spatial logic based on graphs of conjunctive queries for expressing (hyper-)graph languages. In this logic, each primitive positive (resp. existential positive) formula plays a role of an expression of a graph (resp. a finite language of graphs) modulo graph isomorphism. First, this paper presents a sound- and complete axiomatization for the equational theory of primitive/existential positive formulas under this spatial semantics. Second, we show Kleene theorems between this logic and hyperedge replacement grammars (HRGs), namely that over graphs, the class of existential positive first-order (resp. least fixpoint, transitive closure) formulas has the same expressive power as that of non-recursive (resp. all, linear) HRGs.
@InProceedings{nakamura:LIPIcs.CSL.2022.30, author = {Nakamura, Yoshiki}, title = {{Spatial Existential Positive Logics for Hyperedge Replacement Grammars}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {30:1--30:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.30}, URN = {urn:nbn:de:0030-drops-157504}, doi = {10.4230/LIPIcs.CSL.2022.30}, annote = {Keywords: Existential positive logic, spatial logic, Kleene theorem} }
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