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Proving Termination of Context-Sensitive Rewriting with MU-TERM

Beatriz Alarcón, Raúl Gutiérrez, José Iborra, Salvador Lucas
2007 Electronical Notes in Theoretical Computer Science  
Furthermore, we show how to combine them to furnish MU-TERM with an expert which is able to automatically perform the termination proofs.  ...  Context-sensitive rewriting (CSR) is a restriction of rewriting which forbids reductions on selected arguments of functions.  ...  A direct and automatic proof of μ-termination of R is possible with CSDPs by using an auxiliary polynomial ordering generated by a linear polynomial interpretation (computed by MU-TERM, see Figure 4 )  ... 
doi:10.1016/j.entcs.2007.05.041 fatcat:cgu24egybnhezf4mw5zwtqfldy

Complexity Analysis of Term Rewriting Based on Matrix and Context Dependent Interpretations

Georg Moser, Andreas Schnabl, Johannes Waldmann, Marc Herbstritt
2008 Foundations of Software Technology and Theoretical Computer Science  
For a given (terminating) term rewriting system one can often estimate its derivational complexity indirectly by looking at the proof method that established termination.  ...  of context dependent interpretations and restricted triangular matrix interpretations.  ...  (A direct proof of this result, i.e., a proof that argues only about context dependent interpretations, can be found in [15] .) COROLLARY 17.  ... 
doi:10.4230/lipics.fsttcs.2008.1762 dblp:conf/fsttcs/MoserSW08 fatcat:ntpolclienacfphg7qdc2hptya

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

2011 Mathematical Structures in Computer Science  
We also present its application to the automated verification of termination certificates, as produced by termination tools.  ...  Termination is an important property of programs; notably required for programs formulated in proof assistants.  ...  Contexts Contexts are defined as terms with a unique hole in a similar way (module AContext): Thanks to the use of dependent types, contexts are well-formed by construction and replacing a hole by some  ... 
doi:10.1017/s0960129511000120 fatcat:by6ksdib7nfpdjj54sscojozei

Proving Termination Properties with mu-term [chapter]

Beatriz Alarcón, Raúl Gutiérrez, Salvador Lucas, Rafael Navarro-Marset
2011 Lecture Notes in Computer Science  
of order-sorted rewriting, termination of context-sensitive rewriting, termination of innermost context-sensitive rewriting and termination of rewriting modulo specific axioms.  ...  mu-term is a tool which can be used to verify a number of termination properties of (variants of) Term Rewriting Systems (TRSs): termination of rewriting, termination of innermost rewriting, termination  ...  The most important one has been the development of the context-sensitive dependency pair framework (CSDP framework, [3, 20] ), for mechanizing proofs of termination of CSR.  ... 
doi:10.1007/978-3-642-17796-5_12 fatcat:5333qvps3bfgnlfywktz2wqch4

mu-term: Verify Termination Properties Automatically (System Description) [chapter]

Raúl Gutiérrez, Salvador Lucas
2020 Lecture Notes in Computer Science  
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  ...  The automatic generation of logical models for suitable firstorder theories and formulas provides a common basis to implement the proofs.  ...  Infeasibility in Termination Proofs.  ... 
doi:10.1007/978-3-030-51054-1_28 fatcat:hfw3ljwsajdm7nzubf4ok6s4de

mu-term: A Tool for Proving Termination of Context-Sensitive Rewriting [chapter]

Salvador Lucas
2004 Lecture Notes in Computer Science  
The tool implements the generation of the appropriate orderings for proving termination of CSR by means of polynomial interpretations over the rational numbers.  ...  In fact, mu-term is the first termination tool which generates term orderings based on such polynomial interpretations.  ...  I thank Albert Rubio and Cristina Borralleras for providing most of the proofs (and disproofs) of termination of CSR with CS-RPO summarized in the aforementioned WWW site.  ... 
doi:10.1007/978-3-540-25979-4_14 fatcat:v7qxcoyarzgt3a5pzcnjp2g55y

Cdiprover3: A Tool for Proving Derivational Complexities of Term Rewriting Systems [chapter]

Andreas Schnabl
2010 Lecture Notes in Computer Science  
This paper describes cdiprover3 a tool for proving termination of term rewrite systems by polynomial interpretations and context dependent interpretations.  ...  The methods used by cdiprover3 induce small bounds on the derivational complexity of the considered system. We explain the tool in detail, and give an overview of the employed proof methods.  ...  In Section 3., we briefly describe polynomial and context dependent interpretations, which are used by cdiprover3.  ... 
doi:10.1007/978-3-642-14729-6_11 fatcat:k5rt4ximf5gl7d2ceujmc5n5mm

CeTA - A Tool for Certified Termination Analysis [article]

Christian Sternagel and René Thiemann and Sarah Winkler and Harald Zankl
2012 arXiv   pre-print
Since the first termination competition in 2004 it is of great interest, whether a proof that has been automatically generated by a termination tool, is indeed correct.  ...  Hence the interest in automated certification of termination proofs. This led to the general approach of using proof assistants (like Coq and Isabelle) for certification.  ...  For dpTrans , the dependency pairs are provided by dps .  ... 
arXiv:1208.1591v1 fatcat:m4ozzjbfx5cddic2riokjvmm4m

Typed syntactic meta-programming

Dominique Devriese, Frank Piessens
2013 SIGPLAN notices  
and proof tactics.  ...  Our system supports typesafe construction and analysis of terms, types and typing contexts.  ...  The first reduction rule interprets an empty context by the unit type .  ... 
doi:10.1145/2544174.2500575 fatcat:7bi6pljz3rcutnwdi2xnz6zrdi

Typed syntactic meta-programming

Dominique Devriese, Frank Piessens
2013 Proceedings of the 18th ACM SIGPLAN international conference on Functional programming - ICFP '13  
and proof tactics.  ...  Our system supports typesafe construction and analysis of terms, types and typing contexts.  ...  The first reduction rule interprets an empty context by the unit type .  ... 
doi:10.1145/2500365.2500575 dblp:conf/icfp/DevrieseP13 fatcat:v3cztsz4inc2dbx3tig6gc5vdm

A Mechanised Semantics for HOL with Ad-hoc Overloading [article]

Johannes Åman Pohjola, Arve Gengelbach
2020 arXiv   pre-print
In this paper, we present a mechanised proof that HOL with ad-hoc overloading is consistent. All our results have been formalised in the HOL4 theorem prover.  ...  This termination proof is quite laborious, at 495 lines of proof script.  ...  Our termination criterion on contexts, however, is not executable as expressed.  ... 
arXiv:2002.10212v2 fatcat:uz7qmnbbc5d2lbv5d2dzc7g5ne

Termination proofs for higher-order rewrite systems [chapter]

Jaco Pol
1994 Lecture Notes in Computer Science  
The result is a proof technique for the termination of a HRS, similar to the proof technique \Termination by interpretation in a wellfounded monotone algebra", described in 8, 19] .  ...  This paper deals with termination proofs for Higher-Order Rewrite Systems (HRSs), introduced in 12].  ...  qy:(qy) 2qy:2(qy) The Model of Hereditarily Monotonic Functionals We try to apply the general idea of the proof technique \termination by interpretation" for TRSs in 8, 19] to HRSs.  ... 
doi:10.1007/3-540-58233-9_14 fatcat:w3fhd2cpafcthiepevrsix7xj4

Proof Theory at Work: Complexity Analysis of Term Rewrite Systems [article]

Georg Moser
2009 arXiv   pre-print
Abstract In this paper we study context dependent interpretations, a semantic termination method extending interpretations over the natural numbers, introduced by Hofbauer.  ...  Derivational Complexities Induced by Polynomial Context Dependent Interpretations Proof. We proceed by induction on t. As t ∈ T (F), the evaluation is independent of the assignment.  ...  In interpreting defined and dependency pair symbols, we restrict to polynomials whose coefficients are in the range {0, 1, . . . , 5}.  ... 
arXiv:0907.5527v2 fatcat:jxonbatpw5hdlabp5z7swfrgdy

Towards the Certification of Complexity Proofs [article]

René Thiemann
2012 arXiv   pre-print
We report on our formalization of matrix-interpretation in Isabelle/HOL. Matrices are required to certify termination proofs and we wish to utilize them for complexity proofs, too.  ...  For the latter aim, only basic methods have already been integrated, and we discuss some upcoming problems which arise when formalizing more complicated results on matrix-interpretations, which are based  ...  Our formalization of matrix interpretations for termination proofs in presented in Sec. 3. We discuss the extension to complexity proofs in Sec. 4 where we also discuss some open problems.  ... 
arXiv:1208.1609v1 fatcat:wvir2f2emrdvbd73ydlnqmblye

Termination by Abstraction [chapter]

Nachum Dershowitz
2004 Lecture Notes in Computer Science  
Proof. Harmony(→ ∩ , ) follows from Mono( ) and (3,5,7). Turning to the dependency methods: Lemma 9 ([1]). Main ( ) ⇒ Basic ( ). Proof. Follows from (14). Lemma 10.  ...  A small number of firms have decided to terminate their independent abstraction schemes. Lemma 7. Right ( ) ⇒ Subterm ( ). Proof.  ...  If is such a flat context and t a term (or context), then by [t] we denote the term (or context) f (t 1 , . . . , t i−1 , t, t i+1 , . . . , t n ).  ... 
doi:10.1007/978-3-540-27775-0_1 fatcat:jh6omiqxzrarfl2eret5lopr3i
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