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Distinguishing and relating higher-order and first-order processes by expressiveness

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Abstract

This is a paper on distinguishing and relating two important kinds of calculi through expressiveness, settling some critical but long unanswered questions. The delimitation of higher-order and first-order process calculi is a basic and pivotal topic in the study of process theory. Particularly, expressiveness studies mutual encodability, which helps decide whether process-passing or name-passing is more fundamental, and the way they ought to be used in both theory and practice. In this paper, we contribute to such demarcation with three major results. Firstly \(\pi \) (first-order pi-calculus) can faithfully express \(\varPi \) (basic higher-order pi-calculus). The calculus \(\varPi \) has the elementary operators (input, output, composition and restriction). This actually is a corollary of a more general result, that \(\pi \) can encode \(\varPi ^r\) (\(\varPi \) enriched with the relabelling operator). Secondly \(\varPi \) cannot interpret \(\pi \) reasonably. This is of more significance since it separates \(\varPi \) and \(\pi \) by drawing a well-defined boundary. Thirdly an encoding from \(\pi \) to \(\varPi ^r\) is revisited and discussed, which not only implies how to make \(\varPi \) more useful but also stresses the importance of name-passing in \(\pi \).

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Notes

  1. An example: \(Q\stackrel{def}{=}(d)Q_1, Q_1\stackrel{def}{=}\overline{a}[d{.}0]{.}\overline{d}{.}0\). Then \(Q{\xrightarrow {(d)\overline{a}[d{.}0]}} \overline{d}{.}0\).

    Noticing \( {[}\!{[}Q_1 {]}\!{]}^{}_{}\equiv (m)(\overline{a}m{.} {[}\!{[}\overline{d}{.}0 {]}\!{]}^{}_{} |\,!m(Id) {.} {[}\!{[}d{.}0 {]}\!{]}^{}_{})\), we know

    $$\begin{aligned} \begin{array}{lll} {[}\!{[}Q {]}\!{]}^{}_{}&\equiv&(d) {[}\!{[}Q_1 {]}\!{]}^{}_{} \\&\equiv&(d)(m)(\overline{a}m {.} {[}\!{[}\overline{d}{.}0 {]}\!{]}^{}_{} |\,!m(Id){.} {[}\!{[}d{.}0 {]}\!{]}^{}_{}) \\&{\xrightarrow {\overline{a}(m)}}&(d)( {[}\!{[}\overline{d}{.}0 {]}\!{]}^{}_{} |\,!m(Id){.} {[}\!{[}d{.}0 {]}\!{]}^{}_{}) \end{array} \end{aligned}$$

  2. e.g., \(T{\xrightarrow {\tau }} _{j\times k} \; (m\widetilde{c})( {[}\!{[}P_1^{\prime } {]}\!{]}^{}_{\rho } |\, {[}\!{[}E {]}\!{]}^{}_{\rho }\{\epsilon /x\} |\,\prod ^j {[}\!{[}A {]}\!{]}^{}_{} |\,!m(\rho _1){.} {[}\!{[}A {]}\!{]}^{}_{})\) , where \( {[}\!{[}E {]}\!{]}^{}_{\rho }\{\epsilon /x\} \) naively means cancelling the prefix occurrences \(x\) in \( {[}\!{[}E {]}\!{]}^{}_{\rho }\), and \(\prod ^j {[}\!{[}A {]}\!{]}^{}_{}\) means the composition of \(j\) number of \( {[}\!{[}A {]}\!{]}^{}_{}\).

  3. http://basics.sjtu.edu.cn.

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Acknowledgments

We are particularly grateful to the anonymous referees for their instructive criticism and suggestions. We appreciate greatly the interest and advice from editor Ernst-Ruediger Olderog. We also thank Yuxi Fu, Huan Long, Qiang Yin and other members of the BASICS lab at SJTU for their helpful discussion.

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  1. Department of Computer Science and Technology, East China University of Science and Technology, 130 Mei Long Road, Shanghai, 200237, People’s Republic of China

    Xian Xu

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  1. Xian Xu
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Correspondence to Xian Xu.

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The work is supported by the National Natural Science Foundation of China (60903020, 60903189, 61173048) and the Doctoral Fund of Ministry of Education of China (20090073120024).

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Xu, X. Distinguishing and relating higher-order and first-order processes by expressiveness. Acta Informatica 49, 445–484 (2012). https://doi.org/10.1007/s00236-012-0168-9

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