Negative set constraints with equality

Systems of set constraints describe relations between sets of ground terms. They have been successfully used in program analysis and type inference. So far two proofs of decidability of mixed set constraints have been given: by R. Gilleron, S. Tison and M. Tommasi (1993) and A. Aiken, D. Kozen, and E.L. Wimmers (1993). However, both these proofs are long, involved and do not seem to extend to more general set constraints. Our approach is based on a reduction of set constraints to the monadic class given in a paper by L. Bachmair, H. Ganzinger, and U. Waldmann (1993). We first give a new proof of decidability of systems of mixed (positive and negative) set constraints. We explicitly describe a very simple algorithm working in NEXPTIME and we give in all detail a relatively easy proof of its correctness. Then, we sketch how our technique can be applied to get various extensions of this result. In particular we prove that the problem of consistency of mixed set constraints with restricted projections and unrestricted diagonalization is in NEXPTIME.<<ETX>>

[1]  Wilhelm Ackermann,et al.  Solvable Cases Of The Decision Problem , 1954 .

[2]  A. Ehrenfeucht.  An application of games to the completeness problem for formalized theories , 1961 .

[3]  John C. Reynolds,et al.  Automatic computation of data set definitions , 1968, IFIP Congress.

[4]  B. Dreben,et al.  The decision problem: Solvable classes of quantificational formulas , 1979 .

[5]  Neil D. Jones,et al.  Flow analysis and optimization of LISP-like structures , 1979, POPL.

[6]  Harry R. Lewis,et al.  Complexity Results for Classes of Quantificational Formulas , 1980, J. Comput. Syst. Sci..

[7]  Prateek Mishra,et al.  Declaration-free type checking , 1985, POPL.

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

[9]  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.

[10]  Alexander Aiken,et al.  Static type inference in a dynamically typed language , 1991, POPL '91.

[11]  Alexander Aiken,et al.  Implementing Regular Tree Expressions , 1991, FPCA.

[12]  Alex Aiken,et al.  Solving Systems of Set Constraints (Extended Abstract) , 1992, LICS 1992.

[13]  Alexander Aiken,et al.  Solving systems of set constraints , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

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

[15]  Harald Ganzinger,et al.  Set constraints are the monadic class , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

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

[17]  Alexander Aiken,et al.  The Complexity of Set Constraints , 1993, CSL.

[18]  Sophie Tison,et al.  Solving Systems of Set Constraints using Tree Automata , 1993, STACS.

[19]  K. Stefansson,et al.  Systems of set constraints with negative constraints are NEXPTIME-complete , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[20]  A. Aiken,et al.  Decidability of Systems of Set Constraints with Negative Constraints , 1994 .