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Using Well-Founded Relations for Proving Operational Termination

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Abstract

In this paper, we study operational termination, a proof theoretical notion for capturing the termination behavior of computational systems. We prove that operational termination can be characterized at different levels by means of well-founded relations on specific formulas which can be obtained from the considered system. We show how to obtain such well-founded relations from logical models which can be automatically generated using existing tools.

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Notes

  1. See http://maude.cs.illinois.edu/.

  2. The labels of the rules refer to such a system: \( SR \) stands for subject reduction, \( M1 \) and \( M2 \) for membership-1/-2, \( Rf \) for reflexivity, \( Rl \) for replacement, and \( T \) for transitivity.

  3. Some new labels referring to such a system are used now: \( M1 \) for membership-1 and \( C \) for congruence.

  4. Since the drawing of the tree in (1) suggests the back of a skeleton, we use ‘spine’ for the central part, or backbone, of the tree.

  5. In the following, iff means if and only if.

  6. Following [18, Sect. 1.1], these sets can be empty.

  7. As in [18], we use ‘structure’ and reserve the word ‘model’ to refer to those structures satisfying a given set of sentences (theory).

  8. We use ‘hook’ because this formula is intended to ‘catch’ the head of the next inference rule in the spine.

  9. This transformation keeps no information about the matrix formula \(F\) in the universally quantified formula \((\forall x:s)\,F\). This could compromise the success of its use with Theorem 4 below. More precise transformations could be obtained by considering constants \(c_{x,F}\) indexed not only by variables x but also by formulas \(F\) and envisaging appropriate conditions on such constants so that stability holds.

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Acknowledgements

I thank the anonymous referees for their comments and suggestions, leading to many improvements in the paper.

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Authors and Affiliations

  1. DSIC, Universitat Politècnica de València, Valencia, Spain

    Salvador Lucas

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  1. Salvador Lucas
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Correspondence to Salvador Lucas.

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Partially supported by the EU (FEDER), Projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/013.

Appendices

A. Infeasibility of Proof Jumps for \({\mathtt {INF}}\) Proved with AGES (Example 11)

Example 11 claims for infeasibility of rule \(( Rl )\) in Fig. 2, i.e., of

$$\begin{aligned} \begin{array}{c} \displaystyle {\mathtt {a}}:{\mathtt {S}}\\ \hline \displaystyle {\mathtt {a}}\rightarrow {\mathtt {b}} \end{array} \end{aligned}$$

We use AGES to find a model of \(\overline{{\mathtt {INF}}}\cup \{\lnot \,{\mathtt {a}}:{\mathtt {S}}\}\).

AGES specification.

figure g

AGES goal.

Note: universal quantification is implicit in AGES.

figure h

AGES   output.

figure i

Example 11 also claims infeasibility of \([ T ]^2\), i.e., of

$$\begin{aligned} x \rightarrow ^* z\!\mathrm {\;\; \Uparrow \;\;}\! x \rightarrow y,~ y \rightarrow ^* z \end{aligned}$$

We use AGES to find a model of \(\overline{{\mathtt {INF}}}\cup \{\lnot \,(\exists x)(\exists y)\,x \rightarrow y\}\).The specification is the same as for the previous example. The goal is also the same except for

figure j

AGES  output.

figure k

B. Infeasibility of Proof Jumps of \({\mathtt {3*NAT}}\) Proved with Mace4 (Example 12)

Example 12 claims for infeasibility of the proof jump \([( SR )_ Z ]^2\), i.e., of

$$\begin{aligned} x:{\mathtt {Zero}}\!\mathrm {\;\; \Uparrow \;\;}\! x \rightarrow y, ~ y:{\mathtt {Zero}} \end{aligned}$$

We use Mace4 to find a model of \(\overline{{\mathtt {3*NAT}}}\cup \{\lnot \,(\exists x,y)\,x \rightarrow y\}\). We use the following symbols:

figure l

Mace4  assumptions

figure m

Mace4  goal

figure n

Mace4  output

figure o

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Lucas, S. Using Well-Founded Relations for Proving Operational Termination. J Autom Reasoning 64, 167–195 (2020). https://doi.org/10.1007/s10817-019-09514-2

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