Nelson, G. and Oppen, D.C., "Fast decision procedures based on congruence closure," JACM 27(2) pp. 356-364 (April 1980).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....uninterpreted predicates and functions are intensively used, but also interpreted predicates, and more particularly linear arithmetic. Our method combines classical BDD handling with usual rst order satis ability procedures, based on the Nelson Oppen combination framework [19, 24] and algorithm [20]. This will not only allow to deal with equality and non interpreted functions, but also with interpreted terms for some important decidable theories (e.g. linear arithmetic) This technique relies on two ecient procedures to nd very general constraints on variables in the BDD. One applies to ....

....logic with equality is decidable, but did not give a practical algorithm. It is only in the late seventies that this problem has been better understood and that a usable decision procedure has been found by Nelson, Oppen, Downey, Sethi, and Tarjan. It is known as the Nelson Oppen algorithm (See [20] for the algorithm and early references) This decision procedure and those mentioned below are restricted to conjunctive sets of literals. General quanti er free formulas have rst to be put in conjunctive normal forms. In the meantime, Nelson and Oppen also managed [19] to combine (some) ....

G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356-364, 1980.

....[ The fact that any GLSP can be interfaced to the separated inference rules is a fortunate situation because there appear to be a large number of useful GLSPs. The argument supporting this claim has three parts. First, a significant number of GLSPs have been identified and published [N O 79] N O 80] Cyrluk 96] Second, other work reports on techniques for extending some GLSPs that have been identified. Third, there are techniques that enable GLSPs to be combined. Nelson Oppen in [N O 80] show how to extend a GLSP for the theory of equality with uninterpreted function symbols to a theory ....

....has three parts. First, a significant number of GLSPs have been identified and published [N O 79] N O 80] Cyrluk 96] Second, other work reports on techniques for extending some GLSPs that have been identified. Third, there are techniques that enable GLSPs to be combined. Nelson Oppen in [N O 80] show how to extend a GLSP for the theory of equality with uninterpreted function symbols to a theory of LISP list structure, i.e. a theory in which the function symbols HEAD, TAIL, CONS and NIL are interpreted. Their procedure can be interfaced to our system and used to check satisfiability of ....

Nelson, G., and Oppen, D., "Fast decision procedures based on congruence closure," Journal of the ACM, 27, 2, pp. 356-364, 1980.

....programming for arithmetic over integers or reals, congruence closure algorithms to deal with uninterpreted functions (i.e. second order variables) and the Fourier Motzkin transformation [7] for dealing with linear inequalities. This research was initiated by Shostak [26] and Nelson and Oppen [18]. See also [23, 14] Current research is devoted to combining decision procedures for di#erent theories [25] The DNF method has a clear bottleneck, because the transformation to disjunctive normal is not feasible: the resulting formula may be exponentially bigger than the original. This is ....

G. Nelson and D.C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356--364, 1980.

....and = and 3. detects inconsistencies by exploiting the strategy suggested above. The above procedure can be readily generalized to a SAT based procedure for the quanti er free fragment of FOL with uninterpreted function symbols by using a standard congruence closure algorithm (see, e.g. Nelson and Oppen, 1980 ] to perform the consistency checks. 4.2 Linear Constraints over the Reals In [ Armando et al. 1999 ] the logic admits the function constant and the domain of interpretation is xed to the set of the real numbers. Formally, a temporal constraint is a linear inequality of the form x y ....

Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356-364, 1980.

....from disjoint Shostak theories such as linear arithmetic and lists. The congruence closure procedure sets up the template for the extended procedure in Section 5. The congruence closure decision procedure for pure equality has been studied by Kozen [Koz77] Shostak [Sho78] Nelson and Oppen [NO80], Downey, Sethi, and Tarjan [DST80] and, more recently, by Kapur [Kap97] We present the congruence closure algorithm in a Shostak style, i.e. as an online algorithm for computing and using canonical forms by successively processing the input equations from the set T . For ease of presentation, ....

G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356--364, 1980.

....Shostak s decision procedure by presenting a correct version of the algorithm along with detailed and rigorous proofs for its correctness. If the terms in a conjecture of the form T a = b are constructed solely from variables and uninterpreted function symbols, then congruence closure [NO80, Sho78, DST80, CLS96, Kap97, BRRT99] can be used to partition the subterms into equivalence classes respecting T and congruence. For example, when congruence closure is applied to (x) f(x) f (x) f(x) the equivalence classes generated by the antecedent equality are fxg; ff(x) f (x)g; and ff 4 (x)g. This ....

G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356-364, 1980. 1, 7

....the ideas underlying Shostak s decision procedure by presenting a correct version of the algorithm along with rigorous proofs for its correctness. If the terms in a conjecture of the form T a = b are constructed solely from variables and uninterpreted function symbols, then congruence closure [NO80, Sho78, DST80, CLS96, Kap97, BRRT99] can be used to partition the subterms into equivalence classes respecting T and congruence. For example, when congruence closure is applied to (x) f(x) f (x) f(x) the equivalence classes generated by the antecedent equality are fxg; ff(x) f (x)g; and ff 4 (x)g. This ....

G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356-364, 1980.

.... , it can search for an answer such that : is unsatis able. Now, for some theories there exist more ecient algorithms for computing answers to given queries. A prime example is the free theory over a signature consisting of uninterpreted functions, where the congruence closure algorithm [9, 1] can process the input query and change its state appropriately so that new equations between variables can be directly seen from it. Shostak made an important discovery that a similar inference pattern is possible for many other theories [14] Roughly speaking, the theory module maintains a ....

G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. JACM, 27(2):356-364, 1980.

....structural base formula can be written as a quantifier free formula. 2.4. Conversion to Structural Base Formulas The conversion to structural base formulas builds on the conversion to disjunctions of well defined conjunctions of unnested literals [25, Section 2. 3] congruence closure algorithms [33], and the equality (1) Proposition 13 (Quantifier Free to Structural Base) Every well defined quantifier free formula # is equivalent on to true, false, or a disjunction of structural base formulas. Proof Sketch. We outline an algorithm for converting # into a disjunction of structural base ....

G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM (JACM), 27(2):356--364, 1980.

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Nelson, G. and Oppen, D.C., "Fast decision procedures based on congruence closure," JACM 27(2) pp. 356-364 (April 1980).

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. J. ACM, 27(2):356--364, 1980.

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Nelson, G. and D. C. Oppen, Fast decision procedures based on congruence closure, Journal of the ACM (JACM) 27 (1980), pp. 356--364.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. J. ACM, 27(2):356-364, April 1980.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of ACM, 27(2):356-364, April 1980.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356-364, 1980.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356364, 1980.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356-364, 1980.

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G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM (JACM), 27(2):356--364, 1980.

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G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. Journal of the Association for Computing Machinery, 27(2):356-364, 1980.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of the Association for Computing Machinery, 27(2):356--364, 1980.

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G. Nelson and D. C. Oppen. Fast decision procedures based on congruence closure. JACM, 27(2):356-364, 1980.

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Nelson, G., and Oppen, D., "Fast decision procedures based on congruence closure," Journal of the ACM, 27, 2, pp. 356-364, 1980.

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Greg Nelson and Derek C. Oppen. Fast decision procedures based on congruence closure. Journal of the ACM, 27(2):356--364, April 1980.

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G. Nelson, and D. C. Oppen, \Fast decision procedures based on the congruence closure," J. ACM, Vol. 27, No. 2 (1980), pp. 356-364.

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G. Nelson and D. C. Oppen. Fast Decision Procedures Based on Congruence Closure. In J. Association for Computing Machinery 27(2), 356-364, 1980.

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