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Levels of undecidability in rewriting

Jörg Endrullis, Herman Geuvers, Jakob Grue Simonsen, Hans Zantema
2011 Information and Computation  
Undecidability of various properties of first-order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it.  ...  In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies.  ...  Finally, we believe it interesting to study restricted classes of rewriting systems where the properties we consider are undecidable, yet at strictly lower levels in the hierarchies than proved in this  ... 
doi:10.1016/j.ic.2010.09.003 fatcat:4ctb23nn7ndeney7iipn4476ky

Levels of Undecidability in Infinitary Rewriting: Normalization and Reachability [article]

Joerg Endrullis
2010 arXiv   pre-print
In [EGZ09] it has been shown that infinitary strong normalization (SNi) is Pi-1-1-complete.  ...  Suprisingly, it turns out that infinitary weak normalization (WNi) is a harder problem, being Pi-1-2-complete, and thereby strictly higher in the analytical hierarchy.  ...  The rewrite rules of R M are: Theorem 8 . 8 Infinitary weak normalization for single terms, WN ∞ R (s), and reachability in infinitary rewriting are Σ 1 1 -complete.  ... 
arXiv:1003.1057v1 fatcat:3q4opm3hqvfkfpqffwg5j4bpki

The Reachability Problem over Infinite Graphs [chapter]

Wolfgang Thomas
2009 Lecture Notes in Computer Science  
The problem has a history of 100 years, and it is central for automatic verification of infinitestate systems. Our focus is on graphs that are presented in terms of word or tree rewriting systems.  ...  The first undecidability results (in fact, showing also undecidability of Thue's problem) follow from the fundamental work of Turing and Post in the 1930's and 1940's.  ...  Since it is well-known that the MSO-theory of the infinite grid is undecidable, the MSO model-checking problem over ground tree rewriting graphs is in general undecidable.  ... 
doi:10.1007/978-3-642-03351-3_2 fatcat:ocehdjg765ejvbzjlb3qshwxve

Page 4865 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews  
in term rewriting.  ...  Summary: “Ground reachability, ground joinability and conflu- ence are shown undecidable for flat term rewriting systems, i.e., systems in which all left and right members of rule have depth at most one  ... 

Page 7770 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
Summary: “Level-confluence is an important property of condi- tional term rewriting systems that allow extra variables in the rewrite rules because it guarantees the completeness of narrowing for such  ...  In this paper we present a syntactic condition ensuring level-confluence for orthogonal, not necessarily terminat- ing, conditional term rewriting systems that have extra variables in the right-hand sides  ... 

Primitive Rewriting [chapter]

Nachum Dershowitz
2005 Lecture Notes in Computer Science  
Early undecidability results in string and term rewriting were proved by reduction from undecidable problems of Turing machines (e.g. [9, 25] ).  ...  Undecidability results in rewriting have usually been proved by reduction from undecidable problems of Turing machines or, more recently, from Post's Correspondence Problem.  ...  Theorem 5 Matching of primitive rewriting is undecidable. The proof of undecidability of matching for terminating confluent systems in [22, Cor. 3 .11] uses a non-erasing, but overlapping system.  ... 
doi:10.1007/11601548_10 fatcat:ndwwizgbe5gzzj2quk743cq3yy

Page 6989 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
Summary: “We present a collection of results on regular tree lan- guages and rewrite systems. Moreover we prove the undecidability of the preservation of regularity by rewrite systems.  ...  When used at the proof level, rewriting pro- vides a very powerful methodology for proving completeness re- sults, a technique that is illustrated here.  ... 

Metamorphism, Formal Grammars And Undecidable Code Mutation

Eric Filiol
2007 Zenodo  
As an illustration we then present, on a formal basis, a proof-of-concept metamorphic mutation engine denoted PB MOT, whose detection has been proven to be undecidable.  ...  This paper presents a formalisation of the different existing code mutation techniques (polymorphism and metamorphism) by means of formal grammars.  ...  at the same level; • the whole code must be organised in terms of procedures (or blocks of codes) even if the coding itself is not structured in this way.  ... 
doi:10.5281/zenodo.1329887 fatcat:fz3k3gafk5eqnidtq5wyk65yw4

Page 5456 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
of rewriting.  ...  The main theorem in this paper states that the universal unification problem for C, is undecidable.  ... 

An on-line problem database [chapter]

Nachum Dershowitz, Ralf Treinen
1998 Lecture Notes in Computer Science  
The RTA list seeks to summarize open problems and subsequent solutions in elds of interest to this conference.  ...  languages We continue to solicit electronic contributions of new problems, progress reports, solutions, and comments.  ...  Termination of a single term-rewriting rule was proved undecidable in 16,38 .  ... 
doi:10.1007/bfb0052380 fatcat:x57lyotlxrh5bdan42az6tileq

Relative undecidability in the termination hierarchy of single rewrite rules [chapter]

Alfons Geser, Aart Middeldorp, Enno Ohlebusch, Hans Zantema
1997 Lecture Notes in Computer Science  
For a hierarchy of properties of term rewriting systems, related to termination, we prove relative undecidability even in the case of single rewrite rules: for implications X ~ Y in the hierarchy the property  ...  X is uudecidable for rewrite rules satisfying Y.  ...  In this paper we address the question of relativ@ undecidability for TRSs consisting of a single rewrite rule.  ... 
doi:10.1007/bfb0030600 fatcat:4uluxiav4zdrfhhf536qld4dxq

Problems in rewriting III [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1995 Lecture Notes in Computer Science  
Termination of string-rewriting systems is known to be undecidable [49] . Termination of a single term-rewriting rule was proved undecidable in [32; 64] .  ...  If the termination of a finite rewrite system over a finite signature can be proved using a simplification ordering, then the derivation lengths are bounded by a Hardy function of ordinal level less than  ... 
doi:10.1007/3-540-59200-8_82 fatcat:in5aspimbrbvngnrhzcm7tfwei

Page 7114 of Mathematical Reviews Vol. , Issue 97K [page]

1997 Mathematical Reviews  
The procedural semantics of metalogic programs, built on the basis of procedu- ral semantics of logic programs, uses a standard resolution rule and a meta-level to object-level reflection rule.  ...  A theory of graph rewriting systems is developed which describes how to rewrite graphs of very general types.  ... 

Page 4401 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
The paper gives a proof of the undecidability of the loop invariants for those examples essentially based on the idea of recursive inseparability—a concept introduced by Trakhtenbrot in 1953.  ...  As an ex- tended example they deal with a concurrent system and describe labelled transition specifications both at the requirements level and at the level of implementation of labelled transition specifications  ... 

Some Perspectives of Infinite-State Verification [chapter]

Wolfgang Thomas
2005 Lecture Notes in Computer Science  
We report on recent progress in the study of infinite transition systems for which interesting properties (like reachability of designated states) can be checked algorithmically.  ...  by regular sets of words or trees.  ...  If k levels of stacks occur, we speak of a level-k pushdown automaton.  ... 
doi:10.1007/11562948_3 fatcat:z6i5hzxbvndrvj3rq52gqym5ju
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