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### Problems in rewriting III [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1995 Lecture Notes in Computer Science
Related results include the following: The existential fragment of the subterm ordering is decidable, but its ~Us (3"V'3") fragment is not [102] .  ...  The first-order theory of encompassment (the instance-of-subterm relation) is decidable [19] .  ...

### A new method for undecidability proofs of first order theories

Ralf Treinen
1992 Journal of symbolic computation
The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering  ...  Furthermore we are interested in showing not only undecidability of the whole theory of 1, but of a smallest possible fragment of this theory.  ...  ID is the algebra of FDground terms where < is interpreted as the subterm relation . The representation of strings has been given in Example (1) .  ...

### The decision problem for equational bases of algebras

George F. McNulty
1976 Annals of Mathematical Logic
Eqo is the set ofequaticns of type o. if e is the equation ~o ~ ~" then e I is ~0 and er is ~. Definition 1.3; Le~ o be a similarity type and r c Eq~,. tr = ~ei: e e r }u {e,: ee r}.  ...  Tarski announced that a certaiv finitely based equational theory connected with relation algebras is essentially undecidable in [34] .  ...  Tarski {34] found a finitely based undecidvble equational theory of relation algebras.  ...

### A new method for undecidability proofs of first order theories [chapter]

Ralf Treinen
1990 Lecture Notes in Computer Science
The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering  ...  Furthermore we are interested in showing not only undecidability of the whole theory of 1, but of a smallest possible fragment of this theory.  ...  ID is the algebra of FDground terms where < is interpreted as the subterm relation . The representation of strings has been given in Example (1) .  ...

### Relational analysis of algebraic datatypes

Viktor Kuncak, Daniel Jackson
2005 Proceedings of the 10th European software engineering conference held jointly with 13th ACM SIGSOFT international symposium on Foundations of software engineering - ESEC/FSE-13
Such formulas arise when using the language of sets and relations to reason about structured values such as algebraic datatypes.  ...  The key idea of our technique is to identify a natural syntactic class of formulas in relational logic for which reasoning about infinite structures can be reduced to reasoning about finite structures.  ...  Indeed, [39] shows that even the satisfiability of formulas with bounded universal quantifiers is undecidable (although the satisfiability of the purely existential fragment with a subterm relation is  ...

### Relational analysis of algebraic datatypes

Viktor Kuncak, Daniel Jackson
2005 Software engineering notes
Such formulas arise when using the language of sets and relations to reason about structured values such as algebraic datatypes.  ...  The key idea of our technique is to identify a natural syntactic class of formulas in relational logic for which reasoning about infinite structures can be reduced to reasoning about finite structures.  ...  Indeed, [39] shows that even the satisfiability of formulas with bounded universal quantifiers is undecidable (although the satisfiability of the purely existential fragment with a subterm relation is  ...

### Page 3634 of Mathematical Reviews Vol. , Issue 95f [page]

1995 Mathematical Reviews
Summary: “In this paper we give a new proof of undecidability results about term algebras with subterm relation.  ...  with subterm relation.  ...

### Coming to terms with quantified reasoning

Laura Kovács, Simon Robillard, Andrei Voronkov
2017 Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages - POPL 2017
The theory of finite term algebras provides a natural framework to describe the semantics of functional languages.  ...  However, as the theory of finite term algebras is not finitely axiomatizable, reasoning about quantified properties over term algebras is challenging.  ...  Moreover, one can combine T + F T with other theories, going even to undecidable fragments of the combined theory of term algebras and other theories.  ...

### Coming to Terms with Quantified Reasoning [article]

Laura Kovacs, Simon Robillard, Andrei Voronkov
2016 arXiv   pre-print
The theory of finite term algebras provides a natural framework to describe the semantics of functional languages.  ...  However, as the theory of finite term algebras is not finitely axiomatizable, reasoning about quantified properties over term algebras is challenging.  ...  Moreover, one can combine T + F T with other theories, going even to undecidable fragments of the combined theory of term algebras and other theories.  ...

### A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus

A. J. Kfoury, J. B. Wells
1994 Proceedings of the 1994 ACM conference on LISP and functional programming - LFP '94
We s h o w that typability is an undecidable problem at every rank k 3 of this strati cation.  ...  We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML.  ...  The two problems are undecidable for every fragment o f F of rank 3 and are decidable for rank 2.  ...

### A direct algorithm for type inference in the rank-2 fragment of the second-order λ-calculus

A. J. Kfoury, J. B. Wells
1994 ACM SIGPLAN Lisp Pointers
We s h o w that typability is an undecidable problem at every rank k 3 of this strati cation.  ...  We study the problem of type inference for a family of polymorphic type disciplines containing the power of Core-ML.  ...  The two problems are undecidable for every fragment o f F of rank 3 and are decidable for rank 2.  ...

### Sort inheritance for order-sorted equational presentations [chapter]

Claus Hintermeier, Claude Kirchner, Hélène Kirchner
1995 Lecture Notes in Computer Science
We focus in this work on testing sort inheritance, an undecidable property of speci cations, needed for uni cation in this context.  ...  In an algebraic framework, where equational, membership and existence formulas can be expressed, decorated terms and rewriting provide operational semantics and decision procedures for these formulas.  ...  We currently investigate relations with uni ed algebras Mos89] or many-sorted algebras with semantic sorts and sort operations Mei92].  ...

### More problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1993 Lecture Notes in Computer Science
Any rewrite relation comnmtes with the strict-subterm relation; hence, the union of the latter with an arbitrary terminating rewrite relation is terminating, and atso "fully htvariant" (closed under instantiation  ...  with linear constant restrictions is undecidable?  ...

### Ordered fragments of first-order logic [article]

Reijo Jaakkola
2021 arXiv   pre-print
Using a recently introduced algebraic framework for the classification of fragments of first-order logic, we study the complexity of the satisfiability problem for several ordered fragments of first-order  ...  logic, which are obtained from the ordered logic and the fluted logic by modifying some of their syntactical restrictions.  ...  The author also wishes to thank Antti Kuusisto for many helpful discussions on fragments of first-order logic.  ...

### Reachability of Hennessy-Milner Properties for Weakly Extended PRS [chapter]

Mojmír Křetínský, Vojtěch Řehák, Jan Strejček
2005 Lecture Notes in Computer Science
We also strengthen some related undecidability results on some PRS subclasses.  ...  As a corollary we observe that the problem of strong bisimilarity between wPRS and finite-state systems is decidable.  ...  p ∈ M (∆) and each term t over C with all idle subterms in IdleTerms, there is a rule of Rules(h) applicable to p t.D.  ...
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