## Tarskian Set Constraints (1996)

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Venue: | IN PROCEEDINGS, 11 TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE |

Citations: | 27 - 0 self |

### BibTeX

@INPROCEEDINGS{Givan96tarskianset,

author = {Robert Givan and Dexter Kozen and David McAllester and Carl Witty},

title = {Tarskian Set Constraints},

booktitle = {IN PROCEEDINGS, 11 TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE},

year = {1996},

pages = {138--147},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

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### Abstract

We investigate set constraints over set expressions with Tarskian functional and relational operations. Unlike the Herbrand constructor symbols used in recent set constraint formalisms, the meaning of a Tarskian function symbol is interpreted in an arbitrary first order structure. We show that satisfiability of Tarskian set constraints is decidable in nondeterministic doubly exponential time. We also consider various extensions of the basic language and show that: satisfiability of Tarskian set constraints with recursion (µ-sets) is undecidable but satisfiability for Tarskian set constraints with µ-sets but without function symbols is linear time equivalent to satisfiability in the propositional µ-calculus and is therefore decidable in deterministic exponential time.

### Citations

612 | An overview of the KL-ONE knowledge representation system - Brachman, Schmolze - 1985 |

371 | Propositional dynamic logic of regular programs - Fischer, Ladner |

267 | Results on the propositional -calculus - Kozen - 1983 |

230 | The complexity of concept languages - Donini, Lenzerini, et al. - 1991 |

222 | Semantical considerations on Floyd-Hoare logic - Pratt |

196 | Boolean algebras with operators - Jónsson, Tarski - 1951 |

191 | Soft typing with conditional types - Aiken, Wimmers, et al. - 1994 |

98 | Logic programs as types for logic programs - Frühwirth, Shapiro, et al. - 1991 |

96 | A finite presentation theorem for approximating logic programs - Heintze, Jaffar - 1990 |

87 | Boosting the correspondence between description logics and propositional dynamic logics (extended abstract - Giacomo, Lenzerini - 1994 |

86 |
Linear time algorithms for testing the satisfiability of propositional Horn formulae
- Dowling, Gallier
- 1984
(Show Context)
Citation Context ... The positive entailment satisfiability problems for and can both be reduced in linear time to the satisfiability problem for propositional Horn clauses which is known to be decidable in linear time (=-=Downing and Gallier, 1984-=-), as follows. Any constraint set with no negative constraints is trivially satisfiable by the empty model. If there is one negative constraint, we can focus on a single domain object d witnessing the... |

72 | Set constraints are the monadic class - Bachmair, Ganzinger, et al. - 1993 |

67 | The complexity of set constraints - Aiken, Kozen, et al. - 1993 |

65 |
Varieties of complex algebras
- Goldblatt
- 1989
(Show Context)
Citation Context ...tion. They did not study decision theoretic complexity issues. Representation theorems for subclasses of Boolean algebras with operations have recently been studied in a general setting by Goldblatt (=-=Goldblatt, 1989-=-). Kozen (Kozen, 1993) has recently obtained a Stone duality in the context of Herbrand set constraints between the algebra of set constraints and the topological term automata of (Kozen, et. al., 199... |

54 | Efficient Inference of Partial Types - Kozen, Palsberg, et al. - 1992 |

53 | A decision procedure for a class of set constraints - Heintze, Jaffar - 1991 |

52 | Decidability of systems of set constraints with negative constraints - Aiken, Kozen, et al. - 1993 |

51 | Efficient recursive subtyping
- Kozen, Palsberg, et al.
- 1995
(Show Context)
Citation Context ...decision theoretic complexity issues. Representation theorems for subclasses of Boolean algebras with operations have recently been studied in a general setting by Goldblatt (Goldblatt, 1989). Kozen (=-=Kozen, 1993-=-) has recently obtained a Stone duality in the context of Herbrand set constraints between the algebra of set constraints and the topological term automata of (Kozen, et. al., 1993) and (Kozen, et. al... |

48 |
A Near-Optimal Method for Reasoning about Action
- Pratt
- 1980
(Show Context)
Citation Context ...an be taken to denote the set of worlds in which it is true. Since modal formulas denote sets, modal logics can be viewed as set calculi. Propositional dynamic logic (PDL) (Fischer and Ladner, 1979) (=-=Pratt, 1980-=-) and the modal -calculus (Kozen, 1983) are particularly significant modal logics. If R is a binary relation symbol and C is a set expression then in both these logics the set expression [ R]C denotes... |

47 |
Complexity results for classes of quantificational formulas
- Lewis
- 1980
(Show Context)
Citation Context ...o those used for PDL (Pratt, 1980) (see Sections 3.1, 4.2, and 4.3). The lower bound in line three is proved using techniques similar to those used in proving NEXPTIME hardness for the monadic class (=-=Lewis, 1980-=-) (see Section 3.2). The upper bound in line three is proved by a filtration-like argument (see Section 4.4). Standard techniques fail for the fourth line upper bound, the case of nonrecursive constra... |

38 | Concept language with number restrictions and fixpoints, and its relationship with p-calculus - Giacomo, Lenzerini - 1994 |

36 | Taxonomic syntax for first order inference
- McAllester, Givan
- 1993
(Show Context)
Citation Context ... finite sets of constraints of the form C1 C2 or C1 / C2. In spite of the apparent naturality of Tarskian set constraints, their computational properties have not been widely studied. It is shown in (=-=McAllester and Givan, 1993-=-) that satisfiability of nonrecursive Tarskian set constraints not involving Boolean operations is decidable in cubic time (assuming unit time hash table operations). It is shown in (Givan and McAlles... |

33 | Negative set constraints with equality - Charatonik, Pacholski - 1994 |

32 | Set constraints with projections are in NEXPTIME - Charatonik, Pacholski - 1994 |

30 | Boolean algebras with operators, part I - Jónsson, Tarski - 1951 |

28 | New results on local inference relations - Givan, McAllester - 1992 |

25 | A Unified Framework for Class-based Representation Formalisms - Calvanese, Lenzerini, et al. - 1992 |

25 | Logical aspects of set constraints - Kozen - 1993 |

21 |
Attributive concept descriptions with complements
- Schmidt-SchauB, Smolka
- 1991
(Show Context)
Citation Context ...nential time (Street and Emerson, 1989) (Emerson and Jutla, 1988) (Safra, 1988). AI concept languages have been developed for knowledge representation in expert systems (Brachman and Schmolze, 1985) (=-=Schmidt-Schaub and Smalka, 1991-=-). The set expressions of concept languages are constructed from set variables and relation variables using a variety of compositional mechanisms. For example, the expression R.C where R is a relation... |

21 | Boosting the correspondence between description logics and propositional dynamic logics - Lenzerini - 1994 |

17 | Natural language syntax and first order inference
- McAllester, Givan
- 1992
(Show Context)
Citation Context ...}. This provides a natural meaning for English verb phrases such as “contains every prime number.” One simple but expressive Montagovian concept language has a polynomial time satisfiability problem (=-=McAllester and Givan, 1992-=-). Tarskian set expressions have been studied by Jònnson and Tarski in the framework of Boolean algebra with operations (Jònnson and Tarski, 1951) (Jònnson and Tarski, 1952). In the work of Jònnson an... |

10 | Boolean algebras with operators, part II - Jónsson, Tarski - 1952 |

8 |
Complexity of tree automata and modal logics of programs
- EMERSON, JUTLA
- 1988
(Show Context)
Citation Context ...Tarskian Set Constraints, July 6, 1999 2many applications. Satisfiability for both PDL and the modal -calculus are known to be complete for deterministic exponential time (Street and Emerson, 1989) (=-=Emerson and Jutla, 1988-=-) (Safra, 1988). AI concept languages have been developed for knowledge representation in expert systems (Brachman and Schmolze, 1985) (Schmidt-Schaub and Smalka, 1991). The set expressions of concept... |

2 | An automaton theoretic decision procedure for the propositional -calculus - Street, Emerson, et al. - 1989 |

1 | Tree automata, -calculus, and determinancy - Emerson, Jutla - 1991 |

1 |
Safra. On the complexity of -automaton
- Safra
- 1988
(Show Context)
Citation Context ...July 6, 1999 2many applications. Satisfiability for both PDL and the modal -calculus are known to be complete for deterministic exponential time (Street and Emerson, 1989) (Emerson and Jutla, 1988) (=-=Safra, 1988-=-). AI concept languages have been developed for knowledge representation in expert systems (Brachman and Schmolze, 1985) (Schmidt-Schaub and Smalka, 1991). The set expressions of concept languages are... |

1 | Natural languagesyntax and first order inference - McAllester, Givan - 1992 |

1 | Tarskian Set Constraints, July 6 - A, Jutla - 1999 |

1 | Jònnson [20 - Math - 1951 |

1 |
21] D. Kozen. Results on the propositional -calculus
- Kozen
- 1983
(Show Context)
Citation Context ...s in which it is true. Since modal formulas denote sets, modal logics can be viewed as set calculi. Propositional dynamic logic (PDL) (Fischer and Ladner, 1979) (Pratt, 1980) and the modal -calculus (=-=Kozen, 1983-=-) are particularly significant modal logics. If R is a binary relation symbol and C is a set expression then in both these logics the set expression [ R]C denotes the set { x : y Rx ( , y) y C} . The ... |