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A Proof System for a Unified Temporal Logic

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Computing and Combinatorics (COCOON 2019)

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Abstract

Theorem proving is a widely used approach to the verification of computer systems, and its theoretical basis is generally a proof system for formal derivation of logic formulas. In this paper, we propose a proof system for Propositional Projection Temporal Logic (PPTL) with indexed expressions, which is a unified temporal logic that subsumes the well-used Linear Temporal Logic (LTL). First, the syntax, semantics and logic laws of PPTL that allows indexed expressions are introduced, and the representation of LTL constructs by PPTL formulas is shown. Then, the proof system for the logic is presented which consists of axioms and inference rules for the derivation of both basic constructs and indexed expressions of PPTL. To show the capability of the proof system, several examples of formal proofs are provided. Finally, the soundness of the proof system is demonstrated.

This research is supported by the NSFC Grant Nos. 61751207, 61732013, 61672403, and 61572386.

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Correspondence to Xiaobing Wang or Xinfeng Shu .

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Appendix

Appendix

This appendix presents the proof of Theorem 1.

Proof

We only need to prove the soundness of axioms and inference rules in \(\varPi _I\). The soundness of axioms and inference rules in \(\varPi _B\) has been proved in [8].

(IST) For any interval \(\sigma \), we have

which indicates \(\sigma \models \bigvee _{i\in \mathbb {N}}Q^i \leftrightarrow Q^*\). Recall that \(Q^*= \varepsilon \vee Q^+\).

(INS) For any interval \(\sigma \), we have

$$\begin{aligned} \begin{array}{lll} &{} \sigma \models R[i] \\ \Longrightarrow &{} \sigma \models R[i] \text{ for } \text{ some } i\in \mathbb {N}\\ \Longleftrightarrow &{} \sigma \models \bigvee _{i\in \mathbb {N}}R[i] \\ \end{array} \end{aligned}$$

which indicates \(\sigma \models R[i] \rightarrow \bigvee _{i\in \mathbb {N}}R[i]\).

(INR) For any interval \(\sigma \), we have

$$\begin{aligned} \begin{array}{lll} &{} \sigma \models \bigvee _{i\in \mathbb {N}}R[i] \\ \Longleftrightarrow &{} \sigma \models R[i] \text{ for } \text{ some } i\in \mathbb {N}\\ \Longleftrightarrow &{} \sigma \models R[0] \text{ or } \sigma \models R[i+1] \text{ for } \text{ some } i\in \mathbb {N}\\ \Longleftrightarrow &{} \sigma \models R[0] \text{ or } \sigma \models \bigvee _{i\in \mathbb {N}}R[i+1] \\ \end{array} \end{aligned}$$

which indicates \(\sigma \models \bigvee _{i\in \mathbb {N}}R[i] \leftrightarrow R[0]\vee \bigvee _{i\in \mathbb {N}}R[i+1]\).

(INA) For any interval \(\sigma \), we have

$$\begin{aligned} \begin{array}{lll} &{} \sigma \models \bigvee _{i\in \mathbb {N}}P\wedge R[i] \\ \Longleftrightarrow &{} \sigma \models P\wedge R[i] \text{ for } \text{ some } i\in \mathbb {N}\\ \Longleftrightarrow &{} \sigma \models P \text{ and } \sigma \models R[i] \text{ for } \text{ some } i\in \mathbb {N}\\ \Longleftrightarrow &{} \sigma \models P \text{ and } \sigma \models \bigvee _{i\in \mathbb {N}}R[i] \\ \end{array} \end{aligned}$$

which indicates \(\sigma \models \bigvee _{i\in \mathbb {N}}P\wedge R[i] \leftrightarrow P\wedge \bigvee _{i\in \mathbb {N}}R[i]\). The proofs of (INO), (INN) and (INC) are similar.

(INM) Suppose \(\models R[i]\rightarrow R'[i]\). Then, for any interval \(\sigma \), \(\sigma \models R[i]\) implies \(\sigma \models R'[i]\). So, for any interval \(\sigma \), \(\sigma \models R[i]\) for some \(i\in \mathbb {N}\) implies \(\sigma \models R'[i]\) for some \(i\in \mathbb {N}\), which means \(\sigma \models \bigvee _{i\in \mathbb {N}}R[i]\) implies \(\sigma \models \bigvee _{i\in \mathbb {N}}R'[i]\), or equivalently \(\sigma \models \bigvee _{i\in \mathbb {N}}R[i]\rightarrow \bigvee _{i\in \mathbb {N}}R'[i]\).

(REF) Suppose \(\models R\leftrightarrow Q\vee P\wedge \bigcirc R\) and \(\models R\rightarrow \Diamond Q\). Then, \(R \equiv Q\vee P\wedge \bigcirc R\) and \(R \subset \Diamond Q\). According to Lemma 2, \(\bigvee _{i\in \mathbb {N}}P^{(i)} \wedge \bigcirc ^i Q \equiv R\), which means \(\models \bigvee _{i\in \mathbb {N}}P^{(i)} \wedge \bigcirc ^i Q \leftrightarrow R\). The proof of (REI) is similar. \(\square \)

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Zhao, L., Wang, X., Shu, X., Zhang, N. (2019). A Proof System for a Unified Temporal Logic. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_55

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  • DOI: https://doi.org/10.1007/978-3-030-26176-4_55

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