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Levels of undecidability in rewriting

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 ...

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

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. ...

##
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The Reachability Problem over Infinite Graphs
[chapter]

2009
*
Lecture Notes in Computer Science
*

The problem has a history

doi:10.1007/978-3-642-03351-3_2
fatcat:ocehdjg765ejvbzjlb3qshwxve
*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*. ...##
###
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]

2005
*
Lecture Notes in Computer Science
*

Early

doi:10.1007/11601548_10
fatcat:ndwwizgbe5gzzj2quk743cq3yy
*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. ...##
###
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

2007
*
Zenodo
*

As an illustration we then present, on a formal basis, a proof-

doi:10.5281/zenodo.1329887
fatcat:fz3k3gafk5eqnidtq5wyk65yw4
*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. ...##
###
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]

1998
*
Lecture Notes in Computer Science
*

The RTA list seeks to summarize open problems and subsequent solutions

doi:10.1007/bfb0052380
fatcat:x57lyotlxrh5bdan42az6tileq
*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 . ...##
###
Relative undecidability in the termination hierarchy of single rewrite rules
[chapter]

1997
*
Lecture Notes in Computer Science
*

For a hierarchy

doi:10.1007/bfb0030600
fatcat:4uluxiav4zdrfhhf536qld4dxq
*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. ...##
###
Problems in rewriting III
[chapter]

1995
*
Lecture Notes in Computer Science
*

Termination

doi:10.1007/3-540-59200-8_82
fatcat:in5aspimbrbvngnrhzcm7tfwei
*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 ...##
###
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]

2005
*
Lecture Notes in Computer Science
*

We report on recent progress

doi:10.1007/11562948_3
fatcat:z6i5hzxbvndrvj3rq52gqym5ju
*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. ...
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