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## Simplification by cooperating decision procedures (1979)

Venue: | ACM Transactions on Programming Languages and Systems |

Citations: | 450 - 2 self |

### Citations

758 |
A Decision Method for Elementary Algebra and Geometry, 2nd Edition,
- Tarski
- 1951
(Show Context)
Citation Context ...rs under addition and ACM Transactions on Programming Languages and Systems, Vol. 1, No. 2, October 1979.sSimplification by Cooperating Decision Procedures 249 multiplication is decidable (see Tarski =-=[14]-=-), but we have not tried to implement a decision procedure for this theory. The theory of arrays, d, has the nonlogical symbols store and select and the axioms VvVeViVj select(store(v, i, e),j) = if/=... |

236 | A basis for a mathematical theory of computation
- McCarthy
- 1963
(Show Context)
Citation Context ... A C are equivalent, but it is not easy to decide which is simpler. When an expression does not simplify to a constant such as true, our simplifier returns a variant of cond normal form (see McCarthy =-=[6]-=-), although this is not necessarily the most satisfactory form. We assume in this section that the expression given to the simplifier is Boolean valued. The simplifier first replaces A, V, -% and D by... |

129 |
Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory.
- Craig
- 1957
(Show Context)
Citation Context ...on Lemma). If F and G are formulas such that F entails G, then there exists a formula H such that F entails H and H entails G, and each parameter of H is a parameter of both F and G. PROOF. See Craig =-=[1]-=- or Shoenfield [10]. LEMMA 3. If F is any formula, then there exists a simple formula Res(F), the residue ofF, which is the strongest simple formula that F entails; that is, if H is any simple formula... |

116 |
Variations on the common subexpression problem.
- Downey, Sethi, et al.
- 1980
(Show Context)
Citation Context ...d ~ and prove that 8 and are convex. Both these satisfiability procedures take time O(n 2) to determine the satisfiability of, and the equalities entailed by, a conjunction of length n. Downey et al. =-=[3]-=- have improved the underlying algorithm to O(n log 2 n). Oppen [9] describes a satisfiability procedure for ~q~ which runs in linear time if list structure is assumed to be acyclic. The satisfiability... |

70 | A practical decision procedure for arithmetic with function symbols
- Shostak
- 1979
(Show Context)
Citation Context ...here has been some research on decision procedures for specific quantifierfree theories with mixed terms: Kaplan [4] gives a decision procedure for the theory of arrays with constant indices, Shostak =-=[11]-=- for Presburger arithmetic with uninterpreted function symbols, and Suzuki and Jefferson [12] for Presburger arithmetic with arrays and uninterpreted function symbols. In this paper we give a general ... |

44 |
Reasoning about Recursively Defined Data Structures
- Oppen
- 1980
(Show Context)
Citation Context ...cedures take time O(n 2) to determine the satisfiability of, and the equalities entailed by, a conjunction of length n. Downey et al. [3] have improved the underlying algorithm to O(n log 2 n). Oppen =-=[9]-=- describes a satisfiability procedure for ~q~ which runs in linear time if list structure is assumed to be acyclic. The satisfiability problem for conjunctions of d-literals is NP complete [2]. The al... |

26 |
Verification decidability of Presburger array programs
- Suzuki, Jefferson
- 1980
(Show Context)
Citation Context ...mixed terms: Kaplan [4] gives a decision procedure for the theory of arrays with constant indices, Shostak [11] for Presburger arithmetic with uninterpreted function symbols, and Suzuki and Jefferson =-=[12]-=- for Presburger arithmetic with arrays and uninterpreted function symbols. In this paper we give a general method for combining decision procedures for two quantifier-free theories into a single decis... |

23 |
Mathematical Logic
- Schoenfield
- 1967
(Show Context)
Citation Context ...d G are formulas such that F entails G, then there exists a formula H such that F entails H and H entails G, and each parameter of H is a parameter of both F and G. PROOF. See Craig [1] or Shoenfield =-=[10]-=-. LEMMA 3. If F is any formula, then there exists a simple formula Res(F), the residue ofF, which is the strongest simple formula that F entails; that is, if H is any simple formula entailed by F, the... |

17 |
Assignment commands with array references
- Downey, Sethi
(Show Context)
Citation Context .... Oppen [9] describes a satisfiability procedure for ~q~ which runs in linear time if list structure is assumed to be acyclic. The satisfiability problem for conjunctions of d-literals is NP complete =-=[2]-=-. The algorithm for ~ยข is not difficult (it just does the obvious case splits) but can be very costly. Most of the theories we would like to add to our simplifier are nonconvex. The theory of the real... |

14 |
Fast Decision Algorithms Based on Congruence Closure
- Nelson, Oppen
- 1980
(Show Context)
Citation Context ... set; the solution set of a disjunction of equalities is a finite union of hyperplanes; and a convex set cannot be contained in a finite union of hyperplanes unless it is contained in one of them. In =-=[7]-=- we describe satisfiability procedures for ~ and ~ and prove that 8 and are convex. Both these satisfiability procedures take time O(n 2) to determine the satisfiability of, and the equalities entaile... |

7 |
personal communication
- Takeuchi
- 2012
(Show Context)
Citation Context ... (In the last example, the first two literals imply that d __ 0, the third that d _ 0. Hence d = 0 and x = y.) The following is a somewhat artificial example involving a larger formula. Ikuo Takeuchi =-=[13]-=- defined the function: tak(x, y, z) = if x _< y then y else tak(tak(x - 1, y, z), tak(y - 1, z, x), tak(z - 1, x, y)). ACM Transactions on Programming Languages and Systems, Vol. 1, No. 2, October 197... |

3 |
Some Completeness Results in the Mathematical Theory of Computation
- Kaplan
- 1968
(Show Context)
Citation Context ...hmetic function +, the array function store, and the uninterpreted function symbol f. There has been some research on decision procedures for specific quantifierfree theories with mixed terms: Kaplan =-=[4]-=- gives a decision procedure for the theory of arrays with constant indices, Shostak [11] for Presburger arithmetic with uninterpreted function symbols, and Suzuki and Jefferson [12] for Presburger ari... |

1 |
Diophantine representation ofrecursively enumerable predicates
- MATIYASEVICH
- 1970
(Show Context)
Citation Context ...stants is handled. To some extent this incompleteness is inherent, since no decision procedure can exist for the quantifier-free theory of integers under addition and multiplication (see Matiyasevich =-=[5]-=-). The theory of the real numbers under addition and ACM Transactions on Programming Languages and Systems, Vol. 1, No. 2, October 1979.sSimplification by Cooperating Decision Procedures 249 multiplic... |

1 |
Convexity, complexity, and combinations of theories
- OPPEN
(Show Context)
Citation Context ... splits {unless there are 100 variables equal to 1, 2, .... 100, respectively!). The theory of sets is wantonly nonconvex; for example, {a, b, c} N {c, d, e} ~ { } forces a nine-way case split. Oppen =-=[8]-=- shows how the complexity of a combination of theories is determined ACM Transactions on Programming Languages and Systems, Vol. 1, No. 2, October 1979.sSimplification by Cooperating Decision Procedur... |