Abstract
In linear logic, formulas can be split into two sets: classical (those that can be used as many times as necessary) or linear (those that are consumed and no longer available after being used). Subexponentials generalize this notion by allowing the formulas to be split into many sets, each of which can then be specified to be classical or linear. This flexibility increases its expressiveness: we already have adequate encodings of a number of other proof systems, and for computational models such as concurrent constraint programming, in linear logic with subexponentials (
). Bigraphs were proposed by Milner in 2001 as a model for ubiquitous computing, subsuming models of computation such as CCS and the \(\pi \)-calculus and capable of modeling connectivity and locality at the same time. In this work we present an encoding of the bigraph structure in
, thus giving an indication of the expressive power of this logic, and at the same time providing a framework for reasoning and operating on bigraphs. Our encoding is adequate and therefore the operations of composition and juxtaposition can be performed on the logical level. Moreover, all the proof-theoretical tools of
become available for querying and proving properties of bigraph structures.
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Notes
- 1.
We posit that, given the way computer science is evolving, the lack of formal and mechanized reasoning capabilities for any formalisms can be fatal.
- 2.
As usual in the view of the sequent calculus as a proof search formalism, we read inference rules from conclusion to premises.
- 3.
If a or b is a natural number \(1 \le i \le m\) representing a root, than we map it to \(r_i\) in the subexponential signature.
- 4.
We always consider bigraphs to be equal up to the renaming of elements.
- 5.
t / s denotes the substitution of s by t.
- 6.
This is always possible due to renaming and \(\alpha \)-equivalence.
- 7.
Note that, by the definition of substitution, the \(r_j\) must be pairwise distinct. In contrast, \(\mathtt{v} _i\) can be repeated in case a node contains more than one site.
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Acknowledgment
This work was partially supported by the ERC Advanced Grant ProofCert.
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Chaudhuri, K., Reis, G. (2015). An Adequate Compositional Encoding of Bigraph Structure in Linear Logic with Subexponentials. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_11
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