We describe a method for proving non-looping non-termination, that is, of
term rewriting systems that do not admit looping reductions. As certificates of
non-termination, we employ regular (tree) automata.Comment: Published at International Workshop on Termination 201

Endrullis, Jörg

Zantema, Hans

English

arXiv

We describe a method for proving non-termination of term rewriting systems that do not admit looping reductions. As certi¿cates of non-termination, we employ regular (tree) automata

Endrullis, J.

Zantema, H.

NARCIS 

Non-termination using regular languages

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 Non-termination using regular languagesCitation for published version (APA):Endrullis, J., & Zantema, H. (2014). Non-termination using regular languages. (arXiv; Vol. 1405.5662 [cs.LO]).s.n.Document status and date:Published: 01/01/2014Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. 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Nov. 2023Non-termination using Regular Languages∗— International Workshop on Termination —Jörg Endrullis1 and Hans Zantema21 VU University Amsterdam, The Netherlands2 Eindhoven University of Technology, The NetherlandsAbstractWe describe a method for proving non-termination of term rewriting systems that do not admitlooping reductions. As certificates of non-termination, we employ regular (tree) automata.1998 ACM Subject Classification D.1.1, D.3.1, F.4.1, F.4.2, I.1.1, I.1.3Keywords and phrases non-termination, finite automata, regular languages1 IntroductionWe describe a method for proving non-termination of term rewriting systems that do notadmit looping reductions, that is, reductions from a term t to a term C[tσ] containing asubstitution instance of t. For this purpose, we employ tree automata as certificates ofnon-termination. For proving non-termination of a term rewriting system R, we search atree automaton A whose language L(A) is not empty, weakly closed under rewriting andevery term of the language contains a redex occurrence. We have fully automated the searchfor these certificates employing SAT-solvers.All automata that we use as example in this paper have been found automatically; thisconcerns in particular fully automated proofs of non-termination for the following two rewritesystems.I Example 1. We consider the following string rewriting system:zL→ Lz Rz → zR bL→ bR Rb→ LzbThis rewrite system admits no reductions of the form s→∗ `sr.I Example 2. We consider the S-rule from combinatory logic:ap(ap(ap(S , x), y), z)→ ap(ap(x, z), ap(y, z))For the S-rule it is known that there are no reductions t→∗ C[t] for ground terms t, see [15].For open terms t the existence of reductions t→∗ C[tσ] is open.It turns out that the method can be fruitfully applied to obtain non-termination proofsof several string rewriting systems that have remained unsolved in the last full run of thetermination competition.∗ Published at the International Workshop on Termination 2014.© J. Endrullis and H. ZantemaLeibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, GermanyarXiv:1405.5662v1  [cs.LO]  22 May 20142 Non-termination using Regular LanguagesRelated WorkThe paper [11] investigates necessary conditions for the existence of loops. The work [17]employs SAT solvers to find loops, [18] uses forward closures to find loops efficiently, and thewook [16] introduces ‘compressed loops’ to find certain forms of (possibly very long) loops.Non-termination beyond loops has been investigated in [14] and [2]; we note that Example 2cannot be handled by these techniques.Here we prove non-looping non-termination on regular languages. The converse, localtermination on regular languages, has been investigated in [3]. Regular (tree) automata havebeen fruitfully applied to a wide rage of properties of term rewriting systems: for provingtermination [10, 8, 12], for infinitary normalization [4], for proving liveness [13], and foranalysing reachability and deciding the existance of common reducts [9, 5].2 Non-termination and Weakly Closed LanguagesI Definition 3. Let L ⊆ T (Σ,∅) a language and R a TRS over Σ. Then L is called:closed under rewriting if for every t ∈ L and s such that t→ s, one has s ∈ L, andweakly closed under rewriting if for every t ∈ L that is not in normal form, there existss ∈ L such that t→R s.The following theorem describes the basic idea that we employ for proving non-termination.I Theorem 4. A term rewriting system R over Σ is non-terminating if and only if thereexists a non-empty language L ⊆ T (Σ,X ) such that(i) every t ∈ L contains a redex (that is, t→ s for some term s), and(ii) L is weakly closed under rewriting. JA language fulfilling the properties of Theorem 4 is also called a recurrence set, see [1].To automate this method, we need to restrict to a certain family of languages. In thispaper, we consider regular tree languages. To guarantee that the language of a tree automatonis weakly closed under rewriting, we check that the language is not empty and that theautomaton is a quasi-model (see Definition 13) for the rewrite system. The latter conditionis actually too strict; it implies that the languages is not only weakly closed, but also closedunder rewriting. In future, we plan to relieve this restriction.3 Tree AutomataI Definition 5. A (nondeterministic finite) tree automaton A over a signature Σ is a tupleA = 〈Q,Σ, F, δ 〉 where(i) Q is a finite set of states,(ii) F ⊆ Q is a set of accepting states, and(iii) {δf}f∈Σ is a family of transition relations such that for every f ∈ Σ:δf ⊆ Qn ×Qwhere n is the arity of f .In examples, we often write the transition relation δf as →f .J. Endrullis and H. Zantema 3I Example 6. The following is a tree automaton for the signature in Example 1. We considerstring rewriting systems as term rewriting systems by interpreting all symbols as unary andadding a special constant ε to denote the end of the word. Let ALR = 〈Q,Σ, F,→〉 whereQ = {0, 1, 2, 3}, Σ = {b, L,R, 0, ε}, F = {3} and→ε 0 1→z 1 0→b 1 1→R 2 1→L 22→z 2 2→b 3The transition relation for ε can be thought of as defining the initial states (here 0) of a wordautomaton.I Example 7. The following is a tree automaton for Example 2. Let AS = 〈Q,Σ, F,→〉where Q = {0, 1, 2, 3, 4}, Σ = {ap,S}, F = {4} and→S 0 (0, 0)→ap 1 (1, 0)→ap 2 (2, 2)→ap 3 (3, 3)→ap 3(0, 2)→ap 2 (2, 3)→ap 3 (3, 3)→ap 4(0, 3)→ap 2In Example 12 we show that this automaton accepts the term SSS(SSS)(SSS(SSS)).I Definition 8. Let A = 〈Q,Σ, F, δ 〉 be a tree automaton over Σ. For terms t ∈ T (Σ,X )and assignments α : X → P(Q) we define the interpretation [t, α]A by:[x, α]A = α(x)[f(t1, . . . , tn), α]A = {q | (q1, . . . , qn) ∈ [t1, α]A × . . .× [tn, α]A, 〈 (q1, . . . , qn), q 〉 ∈ δf}Whenever A is clear from the context, we write [t, α] as shorthand for [t, α]A. For groundterms t, the interpretation is independent of α, allowing is to write [t]A or [t] for short.I Example 9. We use the automaton AS from Example 7. Let α(x) = {2}, then we have:[S , α] = {0} [ap(S ,S), α] = {1} [ap(ap(S ,S),S), α] = {2}[ap(x, x), α] = {3} [ap(ap(x, x), ap(x, x)), α] = {3, 4}I Definition 10. Let A = 〈Q,Σ, F, δ 〉 be a tree automaton over Σ. The language L(A)accepted by A is the set L(A) = {t | t ∈ T (Σ,∅), [t]A ∩ F 6= ∅} of ground terms.I Example 11. The automaton in Example 6 accepts all words of the form b z∗ (L|R) z∗ b,that is, all words that start with b, end with b, contain one L or R and otherwise only z.I Example 12. We continue Example 9:[ap(ap(S ,S),S)] = {2} [ap( ap(ap(S ,S),S) , ap(ap(S ,S),S) )] = {3}[ap( ap(ap(ap(S ,S),S), ap(ap(S ,S),S)) , ap(ap(ap(S ,S),S), ap(ap(S ,S),S)) )] = {3, 4}Thus F ∩[SSS(SSS)(SSS(SSS))] = {4} 6= ∅ and hence the term is accepted by the automaton.4 Closure under RewritingI Definition 13. A tree automaton A = 〈Q,Σ, F, δ 〉 is a quasi-model for a term rewritingsystem R over Σ if [`, α]A ⊆ [r, α]A for every `→ r ∈ R and α : X → P(Q).4 Non-termination using Regular LanguagesActually, it suffices to check the property [`, α]A ⊆ [r, α]A for assignments α : X → P(Q)that map variables to singleton sets.I Lemma 14. A tree automaton A = 〈Q,Σ, F, δ 〉 is a quasi-model for a term rewritingsystem R over Σ iff [`, α]A ⊆ [r, α]A for every `→ r ∈ R and α : X → {{q} | q ∈ Q}.I Example 15. It is not difficult to check that the automaton ALR from Example 6 is aquasi-model for rewrite system in Example 1.I Example 16. We consider the automaton AS from Example 7. We write (a, b, c)→ d ifd ∈ [`, α] when α(x) = {a}, α(y) = {b}, α(z) = {c}. Then for [`, α] we have:(0, 0, 2)→ 1 (2, 2, 3)→ 3 (2, 3, 3)→ 3 (3, 2, 3)→ 3 (3, 3, 3)→ 3(0, 0, 3)→ 1 (2, 2, 3)→ 4 (2, 3, 3)→ 4 (3, 2, 3)→ 4 (3, 3, 3)→ 4The interpretation [r, α] has all the above and additionally:(0, 2, 2)→ 3 (1, 1, 0)→ 3 (2, 2, 2)→ 3(0, 2, 3)→ 3 (2, 2, 2)→ 4(0, 3, 3)→ 3As a consequence AS is a quasi-model for the S-rule.The following theorem is immediate:I Theorem 17. Let A = 〈Q,Σ, F, δ 〉 be a tree automaton and R a term rewriting systemover Σ. If A is a quasi-model for R then the language of A is closed under rewriting. J5 Ensuring Redex OccurrencesNext, we want to guarantee that every term in the language L(A) of an automaton A containsa redex with respect to the term rewriting system R. For left-linear systems R, this problemcan be reduced to deciding the inclusion of regular languages.Let R be a left-linear term rewriting system. Then the set of ground terms containinga redex is a regular tree language. A deterministic automaton B for this language can beconstructed using the overlap-closure of subterms of left-hand sides, see further [6, 7].I Example 18. The following tree automaton C = 〈Q,Σ, F,→〉 accepts the language ofground terms that contain a redex occurrence with respect to the S-rule. Here Q = {0, 1, 2, 3},Σ = {ap,S}, F = {3} and→S 0 (0, q)→ap 1 (1, q)→ap 2 (2, q)→ap 3 (3, q)→ap 3 (q, 3)→ap 3for all q ∈ {0, 1, 2}.As a consequence the problem of checking whether every term in L(A) contains a redexboils down to checking that L(A) ⊆ L(B). For non-deterministic A and deterministic B,this property can be decided by constructing the product automaton and considering thereachable states.I Definition 19. The product A·B of tree automata A = 〈Q,Σ, F, δ 〉 and B = 〈Q′,Σ, F ′, δ′ 〉is the automaton C = 〈Q × Q′,Σ,∅, γ 〉 where for every f ∈ Σ of arity n, we define thetransition relation γf ⊆ (Q×Q′)n × (Q×Q′) by〈 (q1, p1), . . . , (qn, pn) 〉 γ (q′, p′) ⇐⇒ 〈 q1, . . . , qn 〉 δf q′ ∧ 〈 p1, . . . , pn 〉 δ′f p′J. Endrullis and H. Zantema 5I Definition 20. The set of reachable states of a tree automaton A = 〈Q,Σ, F, δ 〉 is thesmallest set S ⊆ Q such that q ∈ S whenever 〈 q1, . . . , qn 〉 δf q for some q1, . . . , qn ∈ S andf ∈ Σ with arity n.The following theorem gives a method for checking L(A) ⊆ L(B) without the need fordeterminising A (only B needs to be deterministic).I Theorem 21. Let A = 〈Q,Σ, F, δ 〉 and B = 〈Q′,Σ, F ′, δ′ 〉 be tree automata such that Bis deterministic. Let S be the set of reachable states of the product automaton A ·B. ThenL(A) ⊆ L(B) if and only if for all (q, p) ∈ S it holds that q ∈ F =⇒ p ∈ F ′.I Example 22. The reachable states of product automaton AS ·C of the automata AS fromExample 7 and C from Example 18 are (0, 0), (1, 1), (2, 2), (2, 1), (3, 3), (3, 2), (2, 3), (4, 3). Theonly state (q, q′) such that q is accepting in AS is (4, 3) and 3 is an accepting state of C.Thus the conditions of Theorem 21 are fulfilled and hence L(AS) ⊆ L(C). Thus every termaccepted by AS contains a redex.6 Future WorkWe plan to investigate whether the method described in this paper can be fruitfully extendedfrom regular automata to pushdown automata, that is, context-free languages. For thispurpose, it is important that it is decidable whether a context-free language is a subset of aregular language (the language of terms containing left-linear redex occurrences). However,it remains to be investigated whether context-free certificates can be found efficiently.References1 B. Cook. Priciples of program termination. http://research.microsoft.com/en-us/um/cambridge/projects/terminator/principles.pdf.2 F. Emmes, T. Enger, and J. Giesl. Proving Non-looping Non-termination Automatically.In International Joint Conference on Automated Reasoning (IJCAR 2012), volume 7364 ofLecture Notes in Computer Science, pages 225–240. Springer, 2012.3 J. Endrullis, R.C. de Vrijer, and J. Waldmann. Local Termination: Theory and Practice.Logical Methods in Computer Science, 6(3), 2010.4 J. Endrullis, C. Grabmayer, D. Hendriks, J.W. Klop, and R.C. de Vrijer. Proving InfinitaryNormalization. In Postproc. Int. Workshop on Types for Proofs and Programs (TYPES2008), volume 5497 of Lecture Notes in Computer Science, pages 64–82. Springer, 2009.5 J. Endrullis, C. Grabmayer, J.W. Klop, and V. van Oostrom. On Equal µ-Terms. Theor-etical Computer Science, 412(28):3175–3202, 2011.6 J. Endrullis and D. Hendriks. Transforming Outermost into Context-Sensitive Rewriting.Logical Methods in Computer Science, 6(2), 2010.7 J. Endrullis and D. Hendriks. Lazy Productivity via Termination. Theoretical ComputerScience, 412(28):3203–3225, 2011.8 J. Endrullis, D. Hofbauer, and J. Waldmann. Decomposing Terminating Rewrite Relations.In Proc. Workshop on Termination (WST ’06), pages 39–43, 2006.9 B. Felgenhauer and R. Thiemann. Reachability Analysis with State-Compatible Automata.In Proc. Conf. on Language and Automata Theory and Applications (LATA 2014), volume8370 of Lecture Notes in Computer Science, pages 347–359. Springer, 2014.10 A. Geser, D. Hofbauer, J. Waldmann, and H. Zantema. On tree automata that certifytermination of left-linear term rewriting systems. Inf. Comput., 205(4):512–534, 2007.11 A. Geser and H. Zantema. Non-looping string rewriting. ITA, 33(3):279–302, 1999.6 Non-termination using Regular Languages12 M. Korp and A. Middeldorp. Match-bounds revisited. Inf. Comput., 207(11):1259–1283,2009.13 M. Mousazadeh, B. T. Ladani, and H. Zantema. Liveness verification in trss using treeautomata and termination analysis. Computing and Informatics, 29(3):407–426, 2010.14 M. Oppelt. Automatische Erkennung von Ableitungsmustern in nichtterminierenden Wort-ersetzungssystemen. Technical report, HTWK Leipzig, Germany, 2008. Diploma Thesis.15 J. Waldmann. The Combinator S. Information Computation, 159(1–2):2–21, 2000.16 Johannes Waldmann. Compressed loops (draft).17 H. Zankl and A. Middeldorp. Nontermination of String Rewriting using SAT. 2007.18 Harald Zankl, Christian Sternagel, Dieter Hofbauer, and Aart Middeldorp. Finding andcertifying loops. In Proc. Conf. on Theory and Practice of Computer Science (SOFSEM2010), volume 5901 of Lecture Notes in Computer Science, pages 755–766. Springer, 2010.

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Non-termination using Regular Languages

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