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Foundations of Logic Programming in Hybridised Logics

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Book cover Recent Trends in Algebraic Development Techniques (WADT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9463))

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Abstract

The present paper sets the foundation of logic programming in hybridised logics. The basic logic programming semantic concepts such as query and solutions, and the fundamental results such as the existence of initial models and Herbrand’s theorem, are developed over a very general hybrid logical system. We employ the hybridisation process proposed by Diaconescu over an arbitrary logical system captured as an institution to define the logic programming framework.

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Notes

  1. 1.

    A category \(\mathcal{C}\) is a broad subcategory of \(\mathcal{C}'\) if \(\mathcal{C}\) is a subcategory of \(\mathcal{C}'\) and \(\mathcal{C}\) contains all objects of \(\mathcal{C}'\), i.e. \(|\mathcal{C}|=|\mathcal{C}'|\).

  2. 2.

    For all \(M'\in |{\mathbb Mod}(\varSigma ')|\) and \(M_1\in |Mod(\varSigma _1)|\) such that \(M'\!\upharpoonright \!_\chi =M_1\!\upharpoonright \!_\varphi \) there exists \(M_1'\in |{\mathbb Mod}(\varSigma _1')|\) such that \(M_1'\!\upharpoonright \!_{\varphi [\chi ]}=M'\) and \(M'_1\!\upharpoonright \!_{\chi (\varphi )}=M_1\).

  3. 3.

    \(h^{mod}\) is a |R|-indexed family of \(\varSigma \)-homomorphisms \(h^{mod}=\{h^{mod}_s:\mathcal{M}_s\rightarrow \mathcal{M}'_{h^{st}(s)}\}_{s\in |R|}\).

  4. 4.

    A category \(\mathcal{C}\) is a full subcategory of \(\mathcal{C}'\) if \(\mathcal{C}\) is a subcategory of \(\mathcal{C}'\) and for all objects \(A,B\in |C|\) we have \(\mathcal{C}(A,B)=\mathcal{C}'(A,B)\).

  5. 5.

    We denote by \(\rho [j\leftarrow k]\) the sentence \(\varphi (\rho )\), where the signature morphism \(\varphi :(\varSigma ,Nom\cup \{j\},\varLambda )\rightarrow (\varSigma ,Nom,\varLambda )\) is the canonical extension of the function \(\varphi _{\mathbb Nom}:\{j\}\rightarrow Nom\) defined by \(\varphi _{\mathbb Nom}(j)=k\).

  6. 6.

    This condition implies that there exists a basic model \((\mathcal{M}^B,R^B)\in |{\mathbb Mod}^{\mathtt {HI}}(\varDelta )|\), but it is also possible that \((\mathcal{M}^B,R^B)\not \in |{\mathbb Mod}^{\mathtt {CHI}}(\varDelta )|\).

  7. 7.

    The institution \({\mathbf {HFOL}}_2\) contains also sentences that are free of @. It follows that \({\mathbb Sen}^{\mathbf {HFOL}}_1(\varDelta )\subsetneq {\mathbb Sen}^{\mathbf {HFOL}}_2(\varDelta )\) for all \(\varDelta \in |{\mathbb Sig}^{\mathbf {HFOL}}|\).

  8. 8.

    Note that \((\forall X)\rho \) is an abbreviation for \((\forall \chi )\rho \), where \(\chi :(\varSigma ,Nom,\varLambda )\hookrightarrow (\varSigma [X],Nom,\varLambda )\in \mathcal{Q}^{\mathbf {HFOL}}\) is a signature extension with the finite set of first-order variables X and \(\rho \in {\mathbb Sen}^{\mathbf {IHFOL}}_2(\varSigma [X],Nom,\varLambda )\).

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Authors and Affiliations

  1. Research Center for Software Verification, Japan Advanced Institute of Science and Technology (JAIST), Nomi, Japan

    Daniel Găină

Authors
  1. Daniel Găină
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Correspondence to Daniel Găină .

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Editors and Affiliations

  1. Otto-von-Guericke-Universität Magedeburg , Magdeburg, Germany

    Mihai Codescu

  2. Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

    Răzvan Diaconescu

  3. Royal Holloway University of London, Egham, UK

    Ionuț Țuțu

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Găină, D. (2015). Foundations of Logic Programming in Hybridised Logics. In: Codescu, M., Diaconescu, R., Țuțu, I. (eds) Recent Trends in Algebraic Development Techniques. WADT 2015. Lecture Notes in Computer Science(), vol 9463. Springer, Cham. https://doi.org/10.1007/978-3-319-28114-8_5

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  • DOI: https://doi.org/10.1007/978-3-319-28114-8_5

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  • Online ISBN: 978-3-319-28114-8

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