Abstract
We propose a new deductive model checking methodology where narrowing-based logical model checking of symbolic states specified as disjunctions of constrained patterns is synergistically combined with inductive theorem proving to discharge inductive verification conditions. An obvious synergy is to use an inductive theorem prover in automated mode as an oracle to help logical model checking reach a fixpoint. But this is not the only possible synergy. In this paper we focus instead on a new deductive model checking methodology to verify invariants—including inductive invariants—of infinite-state systems, where logical model checking automates large parts of the verification effort with the help of an inductive theorem prover as an oracle, and the remaining tasks are left to the inductive theorem prover used in interactive mode. We demonstrate this methodology by means of Maude examples using two tools working in tandem: the DM-Check symbolic model checker, and the NuITP inductive theorem prover.
S Escobar, J. Sapiña and R. López-Rueda have been partially supported by the grant PID2021-122830OB-C42 funded by MCIN/AEI/10.13039/501100011033 and ERDF A way of making Europe, and by the grant CIPROM/2022/6 funded by Generalitat Valenciana. S Escobar and K. Bae have been partially supported by the NATO Science for Peace and Security Programme project SymSafe (grant number G6133). J. Meseguer, S Escobar, J. Sapiña and R. López-Rueda have been partially supported by INCIBE’s Chair funded by the EU-NextGenerationEU through the Spanish government’s Plan de Recuperación, Transformación y Resiliencia. K. Bae has been partially supported by the National Research Foundation of Korea (NRF) grants funded by the Korean government (2021R1A5A1021944 and RS-2023-00251577).
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Notes
- 1.
- 2.
A disjoint E-unifier of \(u=v\) is an E unifier of \(u'=v'\), with \(u'\) (resp. \(v'\)) a variable renaming of u (resp. v) such that \(u'\) and \(v'\) share no variables.
- 3.
A rewrite theory \(\mathcal {R}\) is topmost iff it has a sort \( State \) such that: (i) if \(f(t_{1},\ldots ,t_{n})\) has sort \( State \), none of the \(t_{i}\), \(1 \le i \le n\) has sort \( State \), and (ii) for any rewrite rule \(l \rightarrow r \; if \; \varphi \) in \(\mathcal {R}\), l and r have sort \( State \). Many common rewrite theories such as, e.g., actor systems, can be transformed into semantically equivalent topmost ones.
- 4.
\(\mathcal {R}= (\varSigma ,E \cup B,R)\) is admissible if \((\varSigma ,E \cup B)\) is ground confluent and the rules R are ground coherent w.r.t. the oriented equations \(\vec{E}\) modulo B, i.e., for each \(t \in T_{\varSigma , State }\) if \(t \rightarrow _{R,B} t'\), then there exists a \(u'\) such that \(t!_{\vec{E},B} \rightarrow _{R,B}u'\) and \(t'!_{\vec{E},B} =_{B} u'!_{\vec{E},B}\).
- 5.
Given a \(\varSigma \)-algebra \(\mathbb {A}\) and a subsignature \(\varSigma '\) with same poset of sorts, the reduct \(\mathbb {A}|_{\varSigma '}\) is the \(\varSigma '\)-algebra with same sorts as \(\mathbb {A}\) and same operations as \(\mathbb {A}\) for each \(f\in \varSigma '\).
- 6.
One case where this does not happen is deadlock-freedom, where a proof by the positive method is easier: see Sect. 4 for an example.
- 7.
At present, \(\sigma \) applies a collection of NuITP formula simplification rules.
- 8.
For example, in [25] a queue is defined using an empty list nil and a non-associative list constructor _::_, with auxiliary functions such as append defined recursively. Five predicates are declared in [25], and there are many lemmas about interactions between these predicates and the append function. We refer to [25] for further details.
- 9.
It is possible to simplify some of these rules; for example, drop-1a and drop-1b can be merged into a single rule. However, to ensure a fair comparison, we use the same rewrite rules as [25], except for the use of associative operators.
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Bae, K., Escobar, S., López-Rueda, R., Meseguer, J., Sapiña, J. (2024). Verifying Invariants by Deductive Model Checking. In: Ogata, K., Martí-Oliet, N. (eds) Rewriting Logic and Its Applications. WRLA 2024. Lecture Notes in Computer Science, vol 14953. Springer, Cham. https://doi.org/10.1007/978-3-031-65941-6_1
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