Innermost Termination of Context-Sensitive Rewriting

Innermost context-sensitive rewriting (CSR) has been proved useful for modeling the computational behavior of programs of algebraic languages like Maude, OBJ, etc, which incorporate an innermost strategy which is used to break down the nondeterminism which is inherent to reduction relations. Furthermore, innermost termination of rewriting is often easier to prove than termination. Thus, under appropriate conditions, a useful strategy for proving termination of rewriting is trying to prove termination of innermost rewriting. This phenomenon has also been investigated for context-sensitive rewriting. Up to now, only few transformation-based methods have been proposed and used to (specifically) prove termination of innermost CSR. Powerful and e cient techniques for proving (innermost) termination of (unrestricted) rewriting like the dependency pair framework have not been considered yet. In this work, we investigate the adaptation of the dependency pair framework to innermost CSR. We provide a suitable notion of innermost context-sensitive dependency pair and show how to extend and adapt the main notions which conform the framework (chain, termination problem, processor, etc.). Thanks to the innermost context-sensitive dependency pairs, we can now use powerful techniques for proving termination of innermost CSR. This is made clear by means of some benchmarks showing that our techniques dramatically improve over previously existing transformational techniques, thus establishing the new state-of-the-art in the area. We have implemented them as part of the termination tool MU-TERM.Alarcón Jiménez, B.; Lucas, S. (2011). Innermost Termination of Context-Sensitive Rewriting. http://hdl.handle.net/10251/1079

Function Calls at Frozen Positions in Termination of Context-Sensitive Rewriting

Context-sensitive rewriting (CSR) is a variant of rewriting where only selected arguments of function symbols can be rewritten. Consequently, the subterm positions of a term are classified as either active, i.e., positions of subterms that can be rewritten; or frozen, i.e., positions that cannot. Frozen positions can be used to denote subexpressions whose evaluation is delayed or just forbidden. A typical example is the if-then-else operator whose second and third arguments are not evaluated until the evaluation of the first argument yields either true or false. Imposing replacement restrictions can improve the termination behavior of rewriting-based computational systems. Termination of CSR has been investigated by several authors and a number of automatic tools are able to prove it. In this paper, we analyze how frozen subterms affect termination of CSR. This analysis helps us to improve our context-sensitive dependency pair (CS-DP) framework for automatically proving termination of CSR. We have implemented these improvements in our tool MU-TERM. The experiments show the power of the improvements in practice.Gutiérrez Gil, R.; Lucas Alba, S. (2015). Function Calls at Frozen Positions in Termination of Context-Sensitive Rewriting. http://hdl.handle.net/10251/5075

MU-TERM: Verify Termination Properties Automatically (System Description)

[EN] We report on the new version of mu-term, a tool for proving termination properties of variants of rewrite systems, including conditional, context-sensitive, equational, and order-sorted rewrite systems. We follow a unified logic-based approach to describe rewriting computations. The automatic generation of logical models for suitable first-order theories and formulas provide a common basis to implement the proofs.Supported by EU (FEDER), and projects RTI2018-094403-B-C32,PROMETEO/ 2019/098, and SP20180225. Also by INCIBE program "Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad" (Raul Gutiérrez).Gutiérrez Gil, R.; Lucas Alba, S. (2020). MU-TERM: Verify Termination Properties Automatically (System Description). Springer Nature. 436-447. https://doi.org/10.1007/978-3-030-51054-1_28S436447Alarcón, B., et al.: Improving context-sensitive dependency pairs. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 636–651. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89439-1_44Alarcón, B., Gutiérrez, R., Lucas, S.: Context-sensitive dependency pairs. Inf. Comput. 208(8), 922–968 (2010). https://doi.org/10.1016/j.ic.2010.03.003Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201–208. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-17796-5_12Alarcón, B., Lucas, S., Meseguer, J.: A dependency pair framework for $${A} \vee {C}$$-termination. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 35–51. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16310-4_4Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000). https://doi.org/10.1016/S0304-3975(99)00207-8Clavel, M., et al.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71999-1Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reasoning 40(2–3), 195–220 (2008). https://doi.org/10.1007/s10817-007-9087-9Giesl, J., Arts, T.: Verification of erlang processes by dependency pairs. Appl. Algebra Eng. Commun. Comput. 12(1/2), 39–72 (2001). https://doi.org/10.1007/s002000100063Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and disproving termination of higher-order functions. In: Gramlich, B. (ed.) FroCoS 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005). https://doi.org/10.1007/11559306_12Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006). https://doi.org/10.1007/s10817-006-9057-7Goguen, J.A., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105(2), 217–273 (1992). https://doi.org/10.1016/0304-3975(92)90302-VGutiérrez, R., Lucas, S.: Function calls at frozen positions in termination of context-sensitive rewriting. In: Martí-Oliet, N., Ölveczky, P.C., Talcott, C. (eds.) Logic, Rewriting, and Concurrency. LNCS, vol. 9200, pp. 311–330. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23165-5_15Gutiérrez, R., Lucas, S.: Proving termination in the context-sensitive dependency pair framework. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 18–34. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16310-4_3Gutiérrez, R., Lucas, S.: Automatic generation of logical models with AGES. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 287–299. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29436-6_17Gutiérrez, R., Lucas, S.: Automatically proving and disproving feasibility conditions. In: Peltier, N., Sofronie-Stokkermans, V. (eds.) IJCAR 2020. LNAI, vol. 12167, pp. 416–435. Springer, Heidelberg (2020)Lucas, S.: Context-sensitive computations in functional and functional logic programs. J. Funct. Log. Program. 1998(1), 1–61 (1998). http://danae.uni-muenster.de/lehre/kuchen/JFLP/articles/1998/A98-01/A98-01.htmlLucas, S.: Context-sensitive rewriting strategies. Inf. Comput. 178(1), 294–343 (2002). https://doi.org/10.1006/inco.2002.3176Lucas, S.: Proving semantic properties as first-order satisfiability. Artif. Intell. 277 (2019). https://doi.org/10.1016/j.artint.2019.103174Lucas, S., Gutiérrez, R.: Automatic synthesis of logical models for order-sorted first-order theories. J. Autom. Reasoning 60(4), 465–501 (2017). https://doi.org/10.1007/s10817-017-9419-3Lucas, S., Gutiérrez, R.: Use of logical models for proving infeasibility in term rewriting. Inf. Process. Lett. 136, 90–95 (2018). https://doi.org/10.1016/j.ipl.2018.04.002Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95(4), 446–453 (2005). https://doi.org/10.1016/j.ipl.2005.05.002Lucas, S., Meseguer, J.: Order-sorted dependency pairs. In: Antoy, S., Albert, E. (eds.) Proceedings of the 10th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming, 15–17 July 2008, Valencia, Spain, pp. 108–119. ACM (2008). https://doi.org/10.1145/1389449.1389463Lucas, S., Meseguer, J.: Dependency pairs for proving termination properties of conditional term rewriting systems. J. Log. Algebraic Methods Program. 86(1), 236–268 (2017). https://doi.org/10.1016/j.jlamp.2016.03.003Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems. Part I: Definition and basic processors. J. Comput. Syst. Sci. 96, 74–106 (2018). https://doi.org/10.1016/j.jcss.2018.04.002Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems—part II: advanced processors and implementation techniques. J. Autom. Reasoning (2020). https://doi.org/10.1007/s10817-020-09542-3McCune, W.: Prover9 & Mace4. Technical report (2005–2010). http://www.cs.unm.edu/~mccune/prover9/Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer (2002). https://doi.org/10.1007/978-1-4757-3661-8 . http://www.springer.com/computer/swe/book/978-0-387-95250-5Ölveczky, P.C., Lysne, O.: Order-sorted termination: the unsorted way. In: Hanus, M., Rodríguez-Artalejo, M. (eds.) ALP 1996. LNCS, vol. 1139, pp. 92–106. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61735-3_6Zantema, H.: Termination of term rewriting: interpretation and type elimination. J. Symb. Comput. 17(1), 23–50 (1994). https://doi.org/10.1006/jsco.1994.1003Zantema, H.: Termination of context-sensitive rewriting. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 172–186. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62950-5_6

Termination of canonical context-sensitive rewriting and productivity of rewrite systems

[EN] Termination of programs, i.e., the absence of infinite computations, ensures the existence of normal forms for all initial expressions, thus providing an essential ingredient for the definition of a normalization semantics for functional programs. In lazy functional languages, though, infinite data structures are often delivered as the outcome of computations. For instance, the list of all prime numbers can be returned as a neverending stream of numerical expressions or data structures. If such streams are allowed, requiring termination is hopeless. In this setting, the notion of productivity can be used to provide an account of computations with infinite data structures, as it "captures the idea of computability, of progress of infinite-list programs" (B.A. Sijtsma, On the Productivity of Recursive List Definitions, ACM Transactions on Programming Languages and Systems 11(4):633-649, 1989). However, in the realm of Term Rewriting Systems, which can be seen as (first-order, untyped, unconditional) functional programs, termination of Context-Sensitive Rewriting (CSR) has been showed equivalent to productivity of rewrite systems through appropriate transformations. In this way, tools for proving termination of CSR can be used to prove productivity. In term rewriting, CSR is the restriction of rewriting that arises when reductions are allowed on selected arguments of function symbols only. In this paper we show that well-known results about the computational power of CSR are useful to better understand the existing connections between productivity of rewrite systems and termination of CSR, and also to obtain more powerful techniques to prove productivity of rewrite systems.Partially supported by the EU (FEDER), Spanish MINECO TIN 2013-45732-C4-1-P, and GV PROMETEOII/2015/013.Lucas Alba, S. (2015). Termination of canonical context-sensitive rewriting and productivity of rewrite systems. Electronic Proceedings in Theoretical Computer Science. 200:18-31. https://doi.org/10.4204/EPTCS.200.2S183120

Automatically Proving and Disproving Feasibility Conditions

[EN] In the realm of term rewriting, given terms s and t, a reachability condition s>>t is called feasible if there is a substitution O such that O(s) rewrites into O(t) in zero or more steps; otherwise, it is called infeasible. Checking infeasibility of (sequences of) reachability conditions is important in the analysis of computational properties of rewrite systems like confluence or (operational) termination. In this paper, we generalize this notion of feasibility to arbitrary n-ary relations on terms defined by first-order theories. In this way, properties of computational systems whose operational semantics can be given as a first-order theory can be investigated. We introduce a framework for proving feasibility/infeasibility, and a new tool, infChecker, which implements it.Supported by EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098, and SP20180225. Also by INCIBE program "Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad" (Raul Gutiérrez).Gutiérrez Gil, R.; Lucas Alba, S. (2020). Automatically Proving and Disproving Feasibility Conditions. Springer Nature. 416-435. https://doi.org/10.1007/978-3-030-51054-1_27S416435Andrianarivelo, N., Réty, P.: Over-approximating terms reachable by context-sensitive rewriting. In: Bojańczyk, M., Lasota, S., Potapov, I. (eds.) RP 2015. LNCS, vol. 9328, pp. 128–139. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24537-9_12Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3(1/2), 69–116 (1987). https://doi.org/10.1016/S0747-7171(87)80022-6Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006). https://doi.org/10.1007/s10817-006-9057-7Goguen, J.A., Meseguer, J.: Models and equality for logical programming. In: Ehrig, H., Kowalski, R., Levi, G., Montanari, U. (eds.) TAPSOFT 1987. LNCS, vol. 250, pp. 1–22. Springer, Heidelberg (1987). https://doi.org/10.1007/BFb0014969Gutiérrez, R., Lucas, S.: Automatic generation of logical models with AGES. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 287–299. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29436-6_17Kojima, Y., Sakai, M.: Innermost reachability and context sensitive reachability properties are decidable for linear right-shallow term rewriting systems. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 187–201. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70590-1_13Kojima, Y., Sakai, M., Nishida, N., Kusakari, K., Sakabe, T.: Context-sensitive innermost reachability is decidable for linear right-shallow term rewriting systems. Inf. Media Technol. 4(4), 802–814 (2009)Kojima, Y., Sakai, M., Nishida, N., Kusakari, K., Sakabe, T.: Decidability of reachability for right-shallow context-sensitive term rewriting systems. IPSJ Online Trans. 4, 192–216 (2011)Lucas, S.: Analysis of rewriting-based systems as first-order theories. In: Fioravanti, F., Gallagher, J.P. (eds.) LOPSTR 2017. LNCS, vol. 10855, pp. 180–197. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94460-9_11Lucas, S.: Context-sensitive computations in functional and functional logic programs. J. Funct. Logic Program. 1998(1) (1998). http://danae.uni-muenster.de/lehre/kuchen/JFLP/articles/1998/A98-01/A98-01.htmlLucas, S.: Proving semantic properties as first-order satisfiability. Artif. Intell. 277 (2019). https://doi.org/10.1016/j.artint.2019.103174Lucas, S.: Using well-founded relations for proving operational termination. J. Autom. Reasoning 64(2), 167–195 (2019). https://doi.org/10.1007/s10817-019-09514-2Lucas, S., Gutiérrez, R.: Use of logical models for proving infeasibility in term rewriting. Inf. Process. Lett. 136, 90–95 (2018). https://doi.org/10.1016/j.ipl.2018.04.002Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95(4), 446–453 (2005). https://doi.org/10.1016/j.ipl.2005.05.002Lucas, S., Meseguer, J.: Proving operational termination of declarative programs in general logics. In: Chitil, O., King, A., Danvy, O. (eds.) Proceedings of the 16th International Symposium on Principles and Practice of Declarative Programming, Kent, Canterbury, United Kingdom, 8–10 September 2014, pp. 111–122. ACM (2014). https://doi.org/10.1145/2643135.2643152Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems. Part I: definition and basic processors. J. Comput. Syst. Sci. 96, 74–106 (2018). https://doi.org/10.1016/j.jcss.2018.04.002Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems—Part II: advanced processors and implementation techniques. J. Autom. Reasoning (2020, in press)McCune, W.: Prover9 and Mace4. https://www.cs.unm.edu/~mccune/mace4/Meßner, F., Sternagel, C.: nonreach – a tool for nonreachability analysis. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 337–343. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_19Middeldorp, A., Nagele, J., Shintani, K.: Confluence competition 2019. In: Beyer, D., Huisman, M., Kordon, F., Steffen, B. (eds.) TACAS 2019. LNCS, vol. 11429, pp. 25–40. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17502-3_2Nishida, N., Maeda, Y.: Narrowing trees for syntactically deterministic conditional term rewriting systems. In: Kirchner, H. (ed.) Proceedings of the 3rd International Conference on Formal Structures for Computation and Deduction. FSCD 2018. Leibniz International Proceedings in Informatics (LIPIcs), vol. 108, pp. 26:1–26:20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018). https://doi.org/10.4230/LIPIcs.FSCD.2018.26Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Heidelberg (2002). http://www.springer.com/computer/swe/book/978-0-387-95250-5Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Dover, New York (2006)Sternagel, C., Sternagel, T., Middeldorp, A.: CoCo 2018 Participant: ConCon 1.5. In: Felgenhauer, B., Simonsen, J. (eds.) Proceedings of the 7th International Workshop on Confluence. IWC 2018, p. 66 (2018). http://cl-informatik.uibk.ac.at/events/iwc-2018/Sternagel, C., Yamada, A.: Reachability analysis for termination and confluence of rewriting. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 262–278. 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Using Well-Founded Relations for Proving Operational Termination

[EN] 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.Partially supported by the EU (FEDER), Projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/013.Lucas Alba, S. (2020). Using Well-Founded Relations for Proving Operational Termination. Journal of Automated Reasoning. 64(2):167-195. https://doi.org/10.1007/s10817-019-09514-2S167195642Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. 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Sci. 248, 93–113 (2009)Durán, F., Lucas, S., Marché, C., Meseguer, J., Urbain, X.: Proving operational termination of membership equational programs. High. Order Symb. Comput. 21(1–2), 59–88 (2008)Falke, S., Kapur, D.: Operational termination of conditional rewriting with built-in numbers and semantic data structures. Electron. Notes Theor. Comput. Sci. 237, 75–90 (2009)Floyd, R.W.: Assigning meanings to programs. Math. Asp. Comput. Sci. 19, 19–32 (1967)Giesl, J., Arts, T.: Verification of Erlang processes by dependency pairs. Appl. Algebra Eng. Commun. Comput. 12, 39–72 (2001)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reason. 37(3), 155–203 (2006)Giesl, J., Thiemann, R., Schneider-Kamp, P.: The dependency pair framework: combining techniques for automated termination proofs. In: Proceedings of LPAR’04, LNAI, vol. 3452, pp. 301–331 (2004)Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: automatic termination proofs in the dependency pair framework. In: Proceedings of IJCAR’06, LNAI, vol. 4130, pp. 281–286 (2006)Goguen, J., Meseguer, J.: Models and equality for logical programming. In: Proceedings of TAPSOFT’87, LNCS, vol. 250, pp. 1–22 (1987)Goguen, J., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105, 217–273 (1992)Gutiérrez, R., Lucas, S., Reinoso, P.: A tool for the automatic generation of logical models of order-sorted first-order theories. In: Proceedings of PROLE’16, pp. 215–230 (2016)Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: Proceedings of RTA 2009, LNCS, vol. 5595, pp. 295–304 (2009)Lalement, R.: Computation as Logic. 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Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

Termination of Rewriting with and Automated Synthesis of Forbidden Patterns

We introduce a modified version of the well-known dependency pair framework that is suitable for the termination analysis of rewriting under forbidden pattern restrictions. By attaching contexts to dependency pairs that represent the calling contexts of the corresponding recursive function calls, it is possible to incorporate the forbidden pattern restrictions in the (adapted) notion of dependency pair chains, thus yielding a sound and complete approach to termination analysis. Building upon this contextual dependency pair framework we introduce a dependency pair processor that simplifies problems by analyzing the contextual information of the dependency pairs. Moreover, we show how this processor can be used to synthesize forbidden patterns suitable for a given term rewriting system on-the-fly during the termination analysis.Comment: In Proceedings IWS 2010, arXiv:1012.533

Loops under Strategies ... Continued

While there are many approaches for automatically proving termination of term rewrite systems, up to now there exist only few techniques to disprove their termination automatically. Almost all of these techniques try to find loops, where the existence of a loop implies non-termination of the rewrite system. However, most programming languages use specific evaluation strategies, whereas loop detection techniques usually do not take strategies into account. So even if a rewrite system has a loop, it may still be terminating under certain strategies. Therefore, our goal is to develop decision procedures which can determine whether a given loop is also a loop under the respective evaluation strategy. In earlier work, such procedures were presented for the strategies of innermost, outermost, and context-sensitive evaluation. In the current paper, we build upon this work and develop such decision procedures for important strategies like leftmost-innermost, leftmost-outermost, (max-)parallel-innermost, (max-)parallel-outermost, and forbidden patterns (which generalize innermost, outermost, and context-sensitive strategies). In this way, we obtain the first approach to disprove termination under these strategies automatically.Comment: In Proceedings IWS 2010, arXiv:1012.533