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From Starvation Freedom to All-Path Reachability Problems in Constrained Rewriting

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Practical Aspects of Declarative Languages (PADL 2023)

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Abstract

An all-path reachability problem of a logically constrained term rewrite system is a pair of constrained terms representing state sets, and is demonically valid if every finite execution path from any state in the first set to a terminating state includes a state in the second set. We have proposed a framework to reduce the non-occurrence of specified error states in a transition system represented by a logically constrained term rewrite system to an all-path reachability problem of the system. In this paper, we extend the framework to verification of starvation freedom of asynchronous integer transition systems with shared variables such that some processes enter critical sections.

This work was partially supported by JSPS KAKENHI Grant Number 18K11160 and DENSO Corporation.

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Notes

  1. 1.

    When there exists a shared variable, we rename the variables in \(\ell \rightarrow r ~ [\psi ]\).

  2. 2.

    We also say that \(\langle t_\ell \mid \phi _\ell \rangle \Rightarrow \langle t_r \mid \phi _r \rangle \) is demonically valid w.r.t. \(\mathcal {R}\).

  3. 3.

    A constant-directed problem \(\langle t_\ell \mid \phi _\ell \rangle \Rightarrow \langle c\mid \phi _r\rangle \) is equivalent to the constant-directed problem \(\langle t_\ell \mid \phi _\ell \wedge (\exists \vec {x}.\ \phi _r)\rangle \Rightarrow \langle c\mid \textsf{true}\rangle \), where \(\{\vec {x}\}=\mathcal {V}ar(\phi _r)\setminus \mathcal {V}ar(t_\ell ,\phi _\ell )\).

  4. 4.

    If there exists a substitution \(\gamma \) such that \(\mathcal {R}an(\gamma |_{\mathcal {V}ar(\phi '_\ell )})\subseteq T(\varSigma _{ theory },\mathcal {V}ar(\phi _\ell ))\), \(t_\ell =t'_\ell \gamma \), and \(\phi _\ell \iff \phi '_\ell \gamma \) is valid, then \(\llbracket \langle t_\ell \mid \phi _\ell \rangle \rrbracket \subseteq \llbracket \langle t'_\ell \mid \phi '_\ell \rangle \rrbracket \) [15, Proposition 5.8].

  5. 5.

    To represent local variables \(y_1,\ldots ,y_k\) of process \(P_i\) with locations \(\ell _1,\ldots \), we use a ground term \(\ell _j(v_1,\ldots ,v_k)\) for the local state with location \(\ell _j\) and integers \(v_1,\ldots ,v_k\) that are assigned to \(y_1,\ldots ,y_k\), respectively.

  6. 6.

    For simplicity, in some examples, we use \(P_0\) instead of \(P_n\), i.e., use \(P_0,\ldots ,P_{n-1}\).

  7. 7.

    By definition, the sort of \(\langle s_0 \mid \phi _0 \rangle \) is \(\texttt{state}\).

  8. 8.

    https://www.trs.css.i.nagoya-u.ac.jp/~nishida/padl2023/.

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Authors and Affiliations

  1. Graduate School of Informatics, Nagoya University, Nagoya, 4648601, Japan

    Misaki Kojima & Naoki Nishida

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  1. Misaki Kojima
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  2. Naoki Nishida
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Correspondence to Misaki Kojima .

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  1. University of Kiel, Kiel, Germany

    Michael Hanus

  2. Miami University, Oxford, OH, USA

    Daniela Inclezan

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Kojima, M., Nishida, N. (2023). From Starvation Freedom to All-Path Reachability Problems in Constrained Rewriting. In: Hanus, M., Inclezan, D. (eds) Practical Aspects of Declarative Languages. PADL 2023. Lecture Notes in Computer Science, vol 13880. Springer, Cham. https://doi.org/10.1007/978-3-031-24841-2_11

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