R. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21:583--585, July 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....include linear and integer 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 ....

R.E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-- 585, 1978.

....include linear and integer 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 ....

R.E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-- 585, 1978.

....to handle quantifier instantiation separately by heuristic or interactive means. Algorithms for deciding unquantified combinations of suitable theories were introduced by Shostak [28] and by Nelson and Oppen [19] equality with uninterpreted functions is decided by the method of congruence closure [26], while linear arithmetic can be decided by several methods, including loop residue [27] and SUP INF [25] PVS uses methods based on those of Shostak [8] Rewriting is another capability that is crucial to effective verification: it can be used to expand definitions and automatically to locate ....

Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583--585, July 1978.

....automatically through the use of ground decision procedures for equality and arithmetic inequality. Ground decision procedures decide quantifer free statements, namely those that contain free variables that are implicitly universally quanti ed at the outermost level. Congruence closure [Koz77,NO80,Sho78,DST80] can be used as a decision procedure for equality for terms built from variables and uninterpreted functions. Most proof obligations contain a combination of uninterpreted function symbols and function symbols that are interpreted within a theory such as linear arithmetic. Nelson and Oppen ....

R. Shostak. An algorithm for reasoning about equality. Comm. ACM, 21:583-585, July 1978.

....by Kapur [76, 124] Bachmair [9] and Dershowitz [51] Larry Wos and his group incorporated their own rewriting techniques into their resolution provers [122] These methods are called demodulation and paramodulation. Another method for handling equality is called the congruence closure technique [85, 92, 105]. This method is complete for ground equality. That is to say that for any set of equalities without variables, and for any theorem that follows from those equalities, this method will prove the theorem using the equalities. In tableau based methods, some of these techniques do not work the same ....

....4.2.4. The method mentioned there has not yet been implemented in IPR but has proved useful in examples proved by hand. 4.1. 2 Congruence Closure At an early stage in the development of IPR, it also used an incomplete E unification algorithm incorporating the ground congruence closure technique [85, 92, 105]. This part of the code has not been kept current. To use this technique, a grammar is built from the positive equalities on a branch and the 70 simple equalities in the knowledge base. That grammar is used any time unification is performed between clashing terms. The congruence closure method ....

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Robert E. Shostak. An algorithm for reasoning about equality. Comm. of the ACM, 21(2):583-- 585, 1978.

....restricted so that it can only generate assertions involving input terms. This algorithm terminates in O(N 3 ) time (dominated by the transitivity rule for equality) where N is the number of input terms. It is possible to prove that running the congruence rule on only the input terms suffices [22]. Now we consider the congruence closure algorithm given in figure 7. These rules compute the same equivalence relation on the terms in the input as do the rules in figure 6. In particular, if hu 1 ; u 2 i and hw 1 ; w 2 i are both input terms C1 EQUAL (x; y) INPUT(x) INPUT(y) C2 EQUAL (x; y) ....

R. Shostak. An algorithm for reasoning about equality. Comm. ACM., 21(2):583-- 585, July 1978.

....In the tree automaton formalism the value of find(s) is the grammar nonterminal representing the equivalence class of s. The algorithms underlying the construction and maintenance of congruence grammars are quite similar to congruence closure algorithms [Kozen, 1977] Downey et al. 1980] [Shostak, 1978]. The details can be found in [McAllester, 1992] Although for efficiency reasons the actual equality data structures used in the implemen22 tation of the VAGCAD system are somewhat subtle, the VAG programmer need not understand these mechanisms. To understand the content of the equality reasoning ....

R. Shostak. An algorithm for reasoning about equality. Comm. ACM., 21(2):583--585, July 1978.

....combined procedure. 1 Introduction The construction of program verification tools in the late 1970 s and early 1980 s drove the development of decision procedures for quantifier free formulas over various theories and combinations of theories. Much of this work was in two streams, one by Shostak [13, 14, 15, 16, 17] and the other by Nelson and Oppen [10, 11, 12, 9] Shostak s work is the basis for the decision procedures in PVS [19] and the Eves verification environment [4] makes use of some of Nelson and Oppen s techniques. Decision procedures are an important tool in theorem provers and verification ....

R. E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583--585, July 1978.

....satisfied because of the non matchable leftmost constants. This is detected by chunk solve, thus a split up of x [n] can be avoided. 4.3.6 Phase 5: Propagation In this step the equality within each column is propagated. The applied method is to build up a union find structure (for example see [Sho78,GHR93] in a way that there is a procedure pair union and find, where find( maps each occurring chunk to a unique representative union( merges the representatives of both arguments In the beginning of the propagation, find maps each chunk to itself. When a union of two different ....

R.E. Shostak. An Algorithm for Reasoning about Equality. Communications of the ACM, 21(7):583--585, July 1978.

....the proof of the example given in Section 1 as well as all of the problems given in [21] IPR supplements this restricted substitution of equals with some Eunification. IPR s method of E unification is complete for ground equality. This is accomplished by the ground congruence closure technique [17, 18, 16]. Some simple equality reasoning is also applied when fetching theorems [23] 5 Closely Related Work Most of the closely related work in automating the use of a knowledge base relies on the use of rewriting or Horn clauses. IPR does not control rewriting very well and therefore does not perform ....

Robert E. Shostak. An algorithm for reasoning about equality. Comm. of the ACM, 21(2):583--585, 1978.

....the presence of uninterpreted This work was partially supported by the Air Force Office of Scientific research under contract F49620 95 C0044. 1 function symbols is crucial to most applications of mechanized formal methods, and efficient techniques for achieving this (such as congruence closure [14]) require that functions are total. However, it is draconian and rather unnatural to force all functions (including, for example, division) to be total, so an effectively mechanized formal method requires careful and integrated design choices to be made for both the specification language and its ....

Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583--585, July 1978.

....(2) re ordering of operations (3) replication of paths segments (4) loop transformations like loop unrolling, rotation and pipelining and (5) speculative execution 1. 1 Related Work A large body of related work exists on use of uninterpreted functions, with and without symbolic simulation [1, 2, 3], as well as on veri Thetacation techniques for high level synthesis and software [4, 5, 6, 7] We cannot do justice to it for lack of space. An extended version of the paper can be obtained from the authors. 2 Problem De Thetanition and Scope of this Paper In this paper, we address the problem ....

R. Shostak, #An algorithm for reasoning about equality,# Communications of the ACM, vol. 21, no. 7, pp. 583#585, 1978.

....decision procedures for semantic theories. For example, the equational inference rules of reflexivity, symmetry, transitivity, and substitutivity define a tractable inference relation that yields a decision procedure for the entailment relation between sets of ground equations [Kozen, 1977] [Shostak, 1978]. Another example is the set of equational Horn clauses valid in lattice theory. As a special case of the results in this paper one can show automatically that validity of a lattice theoretic Horn clause is decidable in cubic time. Deductive databases provide a second motivation for studying ....

R. Shostak. An algorithm for reasoning about equality. Comm. ACM., 21(2):583--585, July 1978.

....specifications. Significance of congruence closure algorithms on ground equations in compiler optimization and verification applications have long been recognized. Particularly, in the mid 70 s and early 80 s, a number of algorithms for computing congruence closure were reported in the literature [9, 3, 14]. Congruence closure has been used as a glue to tie different decision procedure for various theories arising in the application of verification and specification analyses. Two related but different approaches are discussed in [15, 11, 12] These approaches have been implemented in verification ....

....Shostak s framework for combining decision procedures, and argued that this framework is better from efficiency considerations than the one proposed by Nelson and Oppen based on propagation of ground equations. Shostak s congruence closure algorithm serves as the core of his combination framework [14, 15]. Decision procedures satisfying certain properties can be tightly integrated with the basic congruence closure algorithm to give a combination of decision procedures. In contrast, Nelson and Oppen s approach [11, 12] relies on decision procedures working independently and deriving ground ....

R.E. Shostak, "An algorithm for reasoning about equality," Communications of ACM, 21(7) (1978), 583-585.

....extracted from ground tableaux consist solely of ground terms. There are very efficient methods for solving these ground E unification problems, that are based on computing the equivalence classes of the terms to be unified (w.r.t. to the relation defined by the equalities on the branch) [14, 12]. Algorithms based on computing equivalence classes can be used as well to solve mixed E unification problems and, thus, to add equality to universal formula tableaux [4] However, for solving non ground problems, it is much better to use completion based methods. Unfortunately, the Unfailing ....

R. Shostak. An algorithm for reasoning about equality. Comm. of the ACM, 21(7):583--585, July 1978.

....paper are formulated in terms of unsatisfiability. For an equation s t and a multiset of equations E we write E s t to denote that the formula ( V e2E e) oe s t is provable in first order logic with equality. For such formulas provability can be tested by the congruence closure algorithm (Shostak, 1978). For the inclusion of multisets we shall use notation S 1 v S 2 . 3 Rigid basic superposition The term rigid paramodulation has already been used in (Becher and Petermann, 1994; Plaisted, 1995) for systems of inference rules in which all variables are treated as rigid . For example, rigid ....

R. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21:583--585, July 1978.

....problems extracted from ground tableaux consist solely of ground terms. There are very efficient methods for solving these ground E unification problems, that are based on computing the equivalence classes of the terms to be unified (w.r.t. to the relation defined by the equalities on the branch) [21, 16]. Algorithms based on computing equivalence classes can be used as well to solve rigid and mixed E unification problems, i.e. to add equality to free variable tableaux with universal formulae [7] However, for solving non ground problems, it is much better to use completion based methods. ....

R. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-- 585, July 1978.

....interpreted function symbols 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 . ....

R. Shostak. An algorithm for reasoning about equality. Comm. ACM, 21:583--585, July 1978.

....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 ....

Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-585, July 1978. 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 ....

Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-585, July 1978.

....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 f 3 (x) f(x) f 5 (x) f(x) the equivalence classes generated by the antecedent equality are fxg; ff(x) f 3 (x) f 5 (x)g; and ff 2 (x) ....

Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-585, July 1978.

.... rule: a 1 = b 1 : an = b n f(a 1 ; an ) f(b 1 ; b n ) 2 The congruence closure decision procedure for equality decides judgements of the form T a = b by partitioning the term universe (the set of subterms of T , a, and b) into equivalence classes generated from T [NO77,Sho78,DST80]. The partitioning is congruence closed if whenever for each i, 1 i n, a i and b i are in the same equivalence class, then the terms f(a 1 ; an ) and f(b 1 ; b n ) are also in the same equivalence class. Shostak s algorithm is used for deciding equality in the presence of ....

Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-585, July 1978.

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R. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21:583--585, July 1978.

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Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583--585, July 1978.

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Robert E. Shostak. An algorithm for reasoning about equality. Communications of the ACM, 21(7):583-585, July 1978.

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