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An Institutional Foundation for the \(\mathbb {K}\) Semantic Framework

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Recent Trends in Algebraic Development Techniques (WADT 2015)

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Abstract

We advance an institutional formalisation of the logical systems that underlie the \(\mathbb {K}\) semantic framework and are used to capture both structural properties of program configurations through pattern matching, and changes of configurations through reachability rules. By defining encodings of matching and reachability logic into the institution of first-order logic, we set the foundation for integrating \(\mathbb {K}\) into logic graphs of heterogeneous institution-based specification languages such as \(\textsc {HetCasl}\). This will further enable the use of the \(\mathbb {K}\) tool with other existing formal specification and verification tools associated with Hets.

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Notes

  1. 1.

    For simplicity, the formalism we used in this paper does not take into account the subsorting relation. We could further include subsorts following ideas developed for order-sorted equational logic [11].

  2. 2.

    Pgm is the sort defined in Listing 1.1 for capturing the syntax of an IMP program.

  3. 3.

    Technically, the quantification in \(\underline{\mathrm {ML}}^{+}\) is done only over variables that are interpreted in a deterministic manner. This means that every extension with variables over signature \(\mathrm {U}(\varSigma )\) (in \(\underline{\mathrm {ML}}\)) corresponds to a deterministic extension of \(\varSigma \) in \(\underline{\mathrm {ML}}^{+}\).

  4. 4.

    We recall from the definitions of the institutions of matching and first-order logic that from a technical point of view, variables are triples, consisting of name, sort, and signature over which they are defined. Consequently, the signature morphism \(\xi _\varSigma \) maps \( \langle x,t,\varSigma \rangle \) to \( \langle x,t,\varPhi (\varSigma ) \uplus \langle m,s,\varPhi (\varSigma ) \rangle \rangle \), and \( \langle m, s, \varPhi (\varSigma \uplus x) \rangle \) to \( \langle m,s,\varPhi (\varSigma ) \rangle \).

  5. 5.

    Note that, in this case, \(\mathrm {FOL}^{m \,\mathord {:}\,{s}}_{\varSigma }(\pi )\) is just a notation, and it should not be confused with the first-order sentence described in the previous section, for which we would need to instantiate \(\underline{M}\) with \(\underline{\mathrm {ML}}^{+}\).

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Chiriţă, C.E., Şerbănuţă, T.F. (2015). An Institutional Foundation for the \(\mathbb {K}\) Semantic Framework. 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_2

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