## The Complexity of Set Constraints (1993)

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Citations: | 67 - 11 self |

### BibTeX

@MISC{Aiken93thecomplexity,

author = {Alexander Aiken and Dexter Kozen and Moshe Vardi and Ed Wimmers},

title = {The Complexity of Set Constraints},

year = {1993}

}

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### Abstract

. Set constraints are relations between sets of terms. They have been used extensively in various applications in program analysis and type inference. We present several results on the computational complexity of solving systems of set constraints. The systems we study form a natural complexity hierarchy depending on the form of the constraint language. 1 Introduction Systems of set constraints have received considerable attention as a formalism for expressing algorithms in program analysis and type inference. Many algorithms based on set constraints have been proposed and implemented, but very little is known about the computational complexity of solving systems of set constraints. In this paper we present complexity results for a natural hierarchy of decision problems involving set constraints. Set constraints are formal inclusions between expressions representing subsets of T \Sigma , the set of ground terms over a finite ranked alphabet \Sigma . A positive set constraint is an in...

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Citation Context ... 2 Applications and Related Work The greatest interest in set constraints stems from the area of program analysis, where set constraints have been used for a number of years in many di erent settings =-=[18, 14, 16, 17, 20, 13, 2, 3]-=-. In these applications, set constraints are generated from the program text and then solved to obtain useful information about the program (e.g., whether it is well-typed). Representing basic data st... |

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Citation Context ... 2 Applications and Related Work The greatest interest in set constraints stems from the area of program analysis, where set constraints have been used for a number of years in many di erent settings =-=[18, 14, 16, 17, 20, 13, 2, 3]-=-. In these applications, set constraints are generated from the program text and then solved to obtain useful information about the program (e.g., whether it is well-typed). Representing basic data st... |

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Citation Context ...atis able if there is a valuation that satis es all constraints in S simultaneously. An algorithm for determining the satis ability of general systems of positive set constraints was rst presented in =-=[4]-=-. In this paper, we extend the results of [4] in two ways. In Section 5, we giveanew characterization of the satis ability problem that may be of independent interest. We showthat deciding whether S i... |

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Citation Context ...ur knowledge, these are the rst upper and lower bound results for set constraint problems other than the NEXPTIME-completeness result for the general problem, which has been obtained independently in =-=[6]-=-. Our complexity results are summarized in the following table. number of elements in arity 0 arity 1 arity 2+ of complexity of the satis ability problem 0 0+ 0+ trivial 1+ 0 0 NP-complete 1+ 1 0 PSPA... |

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Citation Context ...by an ad hoc technique for checking the satis ability of systems of negative set constraints E 6 F . The satis ability of systems of positive and negative constraints 2has been shown to be decidable =-=[1, 11]-=-, and Stefansson [19] and independently Charatonik and Pacholski [9] have recently shown that the problem is NEXPTIME-complete, thus has the same complexity as positive constraints alone. Special case... |

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Citation Context ... 2 Applications and Related Work The greatest interest in set constraints stems from the area of program analysis, where set constraints have been used for a number of years in many dierent settings =-=[17, 13, 15, 16, 19, 12, 3, 4]-=-. In these applications, set constraints are generated from the program text and then solved to obtain useful information about the program (e.g., whether it is well-typed). Representing basic data st... |

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Citation Context ...by an ad hoc technique for checking the satis ability of systems of negative set constraints E 6 F . The satis ability of systems of positive and negative constraints 2has been shown to be decidable =-=[1, 11]-=-, and Stefansson [19] and independently Charatonik and Pacholski [9] have recently shown that the problem is NEXPTIME-complete, thus has the same complexity as positive constraints alone. Special case... |

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Citation Context ...tive set constraints E 6 F . The satis ability of systems of positive and negative constraints 2has been shown to be decidable [1, 11], and Stefansson [19] and independently Charatonik and Pacholski =-=[9]-=- have recently shown that the problem is NEXPTIME-complete, thus has the same complexity as positive constraints alone. Special cases of set constraints have also arisen naturally in the study of - ni... |

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Citation Context ...ults on solving negative constraints in arbitrary Boolean algebras. Finally, set constraints have been studied for their own sake and several algorithms for solving set constraints have been proposed =-=[12, 4, 10]-=-. Our results di er from these in that we areinterested primarily in the complexity ofthe satis ability problem for set constraints. 3 Set Expressions and Set Constraints Let be a nite ranked alphabet... |

21 |
Systems of set constraints with negative constraints are NEXPTIME-complete
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Citation Context ...or checking the satis ability of systems of negative set constraints E 6 F . The satis ability of systems of positive and negative constraints 2has been shown to be decidable [1, 11], and Stefansson =-=[19]-=- and independently Charatonik and Pacholski [9] have recently shown that the problem is NEXPTIME-complete, thus has the same complexity as positive constraints alone. Special cases of set constraints ... |

19 | Systems of negative boolean constraints
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Citation Context ...ding the satis ability of equations between regular languages with free variables is in EXPTIME. Set constraints with only nullary symbols correspond to Boolean algebras over a nite set of atoms. See =-=[15]-=- for more general results on solving negative constraints in arbitrary Boolean algebras. Finally, set constraints have been studied for their own sake and several algorithms for solving set constraint... |

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Automatic computation of data set de
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Citation Context ... 2 Applications and Related Work The greatest interest in set constraints stems from the area of program analysis, where set constraints have been used for a number of years in many dierent settings =-=[17, 13, 15, 16, 19, 12, 3, 4]-=-. In these applications, set constraints are generated from the program text and then solved to obtain useful information about the program (e.g., whether it is well-typed). Representing basic data st... |

7 |
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4 |
On equations for regular languages, automata, and sequential networks
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Citation Context ...ts alone. Special cases of set constraints have also arisen naturally in the study ofs- nite automata. An example is an algorithm for solving equations between regular 2 languages with free variables =-=[7]-=-. In [7], no complexity analysis is given. There is a simple linear-time reduction of regular expressions to systems of set constraints over unary and nullary symbols. Thus, our results show that deci... |

2 |
On equations for regular languages, nite automata, and sequential networks, Theor
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Citation Context ...ints alone. Special cases of set constraints have also arisen naturally in the study of - nite automata. An example is an algorithm for solving equations between regular languages with free variables =-=[7]-=-. In [7], no complexity analysis is given. There is a simple linear-time reduction of regular expressions to systems of set constraints over unary and nullary symbols. Thus, our results show that deci... |

1 |
The complexity of set constraints. Cornell University tech
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Citation Context ...is a truth assignmenttoX satisfying B. For each term t0 of the form ft1 :::tn, extend (t0) to a truth assignment to X [ Zf as follows: De ne a set assignment (t0)(z f ij )= by (ti)(xj) �z f ij 2 Zf : =-=(5)-=- (xj) =ft j (t)(xj) =1g (z f ij (z f 0j )= (f 1 | :::1 xj {z } 1 | :::1 {z }) i;1 n;i )= (f 1 | :::1 {z }\xj) n for f 2 n, 1 i n, and1 j m. Weshowthatsatis es S. It is immediate from the de nition of ... |