Grid Structure and Undecidable Constraint Theories

We express conditions for a term to be a finite grid-like structure. Together with definitions of term properties by excluding “forbidden patterns” we obtain three new undecidability results in three areas: the ∃*∀*-fragment of the theory of one-step rewriting for linear and shallow rewrite systems, the emptiness for automata with equality tests between first cousins (i.e. only tests at depth 2 below the current node are available), and the ∃*∀*-fragment of the theory of set constraints.

[1]  Detlef Seese,et al.  The Structure of Models of Decidable Monadic Theories of Graphs , 1991, Ann. Pure Appl. Log..

[2]  M. A. Taitslin,et al.  Elementary lattice theories for ideals in polynomial rings , 1968 .

[3]  Joachim Niehren,et al.  On Equality Up-to Constraints over Finite Trees, Context Unification, and One-Step Rewriting , 1997, CADE.

[4]  Jerzy Marcinkowski.  Undecidability of the First Order Theory of One-Step Right Ground Rewriting , 1997, RTA.

[5]  Sergei G. Vorobyov,et al.  The First-Order Theory of One Step Rewriting in Linear Noetherian Systems is Undecidable , 1997, RTA.

[6]  Sophie Tison,et al.  Some new Decidability Results on Positive and Negative Set Constraints , 1994, CCL.

[7]  Hans Läuchli,et al.  Monadic second order definable relations on the binary tree , 1987, The Journal of Symbolic Logic.

[8]  Sophie Tison,et al.  Solving systems of set constraints with negated subset relationships , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[9]  Ralf Treinen,et al.  The First-Order Theory of One-Step Rewriting is Undecidable , 1996, RTA.

[10]  Witold Charatonik.  An Undecidable Fragment of the Theory of Set Constraints , 1998, Inf. Process. Lett..

[11]  Jean-Pierre Jouannaud,et al.  Syntacticness, Cycle-Syntacticness, and Shallow Theories , 1994, Inf. Comput..

[12]  Hubert Comon,et al.  Tree automata techniques and applications , 1997 .

[13]  Sophie Tison,et al.  The Recognizability Problem for Tree Automata with Comparisons between Brothers , 1999, FoSSaCS.

[14]  Florent Jacquemard,et al.  Pumping, Cleaning and Symbolic Constraints Solving , 1994, ICALP.

[15]  Dexter Kozen.  Set Constraints and Logic Programming , 1994, CCL.

[16]  Wolfgang Thomas.  On Logics, Tilings, and Automata , 1991, ICALP.

[17]  Christos H. Papadimitriou,et al.  Elements of the Theory of Computation , 1997, SIGA.

[18]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs IV: Definability Properties of Equational Graphs , 1990, Ann. Pure Appl. Log..

[19]  Ralf Treinen.  The First-Order Theory of Linear One-Step Rewriting is Undecidable , 1998, Theor. Comput. Sci..

[20]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[21]  Sophie Tison,et al.  Set Constraints and Automata , 1999, Inf. Comput..

[22]  M. Rabin.  Decidability of second-order theories and automata on infinite trees , 1968 .

[23]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[24]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[25]  Joxan Jaffar,et al.  A decision procedure for a class of set constraints , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[26]  Sophie Tison,et al.  Deciding the Satisfiability of Quantifier free Formulae on One-Step Rewriting , 1999, RTA.

[27]  Alexander Aiken,et al.  Decidability of Systems of Set Constraints with Negative Constraints , 1994, Inf. Comput..

[28]  Dexter Kozen.  Logical Aspects of Set Constraints , 1993, CSL.

[29]  Sophie Tison,et al.  Equality and Disequality Constraints on Direct Subterms in Tree Automata , 1992, STACS.

[30]  Witold Charatonik,et al.  Set constraints with projections are in NEXPTIME , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[31]  Nevin Charles Heintze,et al.  Set based program analysis , 1992 .

[32]  Joxan Jaffar,et al.  A finite presentation theorem for approximating logic programs , 1989, POPL '90.