W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954. 12  Home/Search   Document Not in Database   Summary   Related Articles   Check

....procedures for combinations of theories Before considering the problem of nding small connections in unsatis able conjunctive sets, it is useful to remind some facts about the somewhat simpler problem of deciding whether a conjunctive set of literals is satis able or not. In 1954 Ackermann [1] showed that quanti er free rst order 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 ....

W. Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....of the model. Finally, variables that range over a set of size n are encoded by log(n) propositional variables. The resulting formula can be checked for satisfiability with any existing SAT technique, for instance based on resolution or on BDDs. An early example is Ackermann s reduction [1], by which second order variables can be eliminated. More optimal versions can be found in [15, 20, 11] Recently, this method is applied in [24] to boolean combinations over successor, predecessor, equality and inequality over the integers, in [28] it is applied to separation predicates x y ....

....introduced is preferable in practice. Another line of future research would be the extension of our result to other algebras. An interesting extension would be the incorporation of uninterpreted functions directly (they can already be dealt with by first eliminating them by Ackermann s reduction [1, 24]) Other interesting extensions are the incorporation of addition ( or an investigation of other free algebras (such as LISP list structures based on null and cons) It should be straightforward to extend our method in case all constructors are unary. This would yield a decision procedure for ....

W. Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....of the model. Finally, variables that range over a set of size n are encoded by log(n) propositional variables. The resulting formula can be checked for satisfiability with any existing SAT technique, for instance based on resolution or on BDDs. An early example is Ackermann s reduction [1], by which second order variables can be eliminated. More optimal versions can be found in [15, 20, 11] Recently, this method is applied in [24] to boolean combinations over successor, predecessor, equality and inequality over the integers, in [28] it is applied to separation predicates x y c, ....

....introduced is preferable in practice. Another line of future research would be the extension of our result to other algebras. An interesting extension would be the incorporation of uninterpreted functions directly (they can already be dealt with by first eliminating them by Ackermann s reduction [1, 24]) Other interesting extensions are the incorporation of addition ( or an investigation of other free algebras (such as LISP list structures based on null and cons) It should be straightforward to extend our method in case all constructors are unary. This would yield a decision procedure for ....

W. Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....This terminates because there are only finitely many of them in fact, in non deterministic (because of splitting) exponential time. It is also easy to check that clauses (4) and (5) are exactly what is needed to write a definitional clausal form [2] of skolemized formulas from the monadic class [1]. This gives another proof that the monadic class is decidable, similar to [25] In particular, this decides positive set constraints, as noticed by [3] Using splitting is more efficient in practice than Joyner s condensing rule [25] The same technique can be used, at least in principle, to ....

W. Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....form v i v j c where 2 f ; g, c is a constant, and v i ; v j are variables of type real or integer. The other inequality signs as well as equalities can be expressed in this logic. Uninterpreted functions can be handled as well since they can be reduced to Boolean combinations of equalities[1]. Separation predicates are used in veri cation of timed systems, scheduling problems, and more. Hardware models with ordered data structures have inequalities as well. For example, if the model contains a queue of unbounded length, the test for head tail introduces inequalities. In fact, most ....

W. Ackermann. Solvable cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....ne Boolean formulas based on the analysis of a conjunction of constraints corresponding to a Boolean assignment. In this way, domain speci c knowledge is added lazily in terms of Boolean clauses. For the special case of equality theories over terms with uninterpreted function symbols, Ackermann [1] already de ned a reduction to Boolean logic by adding all possible applications of the congruence axiom. Variations of Ackermann s trick have been used, for example, by Pnueli et al. [10] for validating compilation runs and by Bryant, German, and Velev [3] for equivalence checking of pipelined ....

W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundation of Mathematics, 1954.

....terminates because there are only finitely many of them in fact, in non deterministic (because of splitting) exponential time. 4 It is also easy to check that clauses (4) and (5) are exactly what is needed to write a definitional clausal form [2] of skolemized formulas from the monadic class [1]. This gives another proof that the monadic class is decidable, similar to [23] In particular, this decides positive set constraints, as noticed by [3] Using splitting is more efficient in practice than Joyner s condensing rule [23] The same technique can be used, at least in principle, to ....

W. Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....across individual theories, we obtain a simpli cation procedure for the combined theory. A fragment of the theory considered here, namely propositional logic with uninterpreted functions over arbitrary domains and equality over these domains, can also be decided using Ackermann s trick [Ack54] for eliminating function symbols at the expense of introducing new variables and congruence constraints. This equivalence preserving transformation may result in exponential blow up, but recent re nements [PRSS99, BGV99, GSZA98, GvdP00] have made it possible to use Ackermann s trick for some ....

W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundation of Mathematics, 1954.

....tautology) of formulae of propositional logic with equality and uninterpreted function symbols (EUF) Such a method usually proceeds in three steps: 1. Elimination of function symbols 1 2. Reduction to propositional logic 3. Check with existing BDD package Ad 1. By a result due to Ackermann [1], the function symbols can be eliminated, at the cost of introducing new variables and congruence constraints. In essence, subterms like F (x) and F (y) are replaced by new variables f 1 and f 2 , and the the functionality constraint x = y f 1 = f 2 is added. This yields a formula of ....

....that extendibility does not necessarily come with a loss in efficiency. 2 EQ BDDs Our aim is to check satisfiability and tautology of propositional formulae with equality. In this paper, we assume that function symbols have been eliminated, for instance with Ackermann s function elimination [1]. We now define a syntax for formulae. First assume a set P of proposition (boolean) variables (typically p, q, and a set V of domain variables (typically x, y, z, Definition 1 Formulae are expressions satisfying the following syntax: Phi : 0 j 1 j P j V = V j : Phi j Phi ....

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Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....symbols , and the numerals along with the usual axioms of rational arithmetic. Consider also the theory E with one uninterpreted unary function symbol f . The satis ability problems for each of these theories considered separately were solved long ago by Fourier for Q and by Ackermann for E [Ack54]. Consider now the following goal from the combined theory Q E 1 : f(f(x) f(y) 6= f(z) y x x y z z 0 (1) Informally, to demonstrate that the above set of literals is not satis able, we would rst use the two literals in the middle to infer in Q that 0 z and then the last ....

....the theory of equality. The free functions of the theory E are = and 6= along with any uninterpreted function symbols. The axioms of the theory are those shown in Figure 3. There is one congruence rule for each function symbol in the system. The theory E was rst shown decidable by Ackermann [Ack54] by reducing the problem to that of constructing the congruence closure of a relation on a graph. If R is an equivalence relation over a set of terms, we say that two terms f(t 1 ; t n ) and f(t 0 1 ; t 0 n ) are congruent if t i is related to t 0 i by R for all i = 1; ....

Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

....verification problem we solve is design equivalence; this includes such applications as verifying equivalence of pipelined and nonpipelined processors. This can be posed as a problem in satisfiability checking for quantifierfree formulas involving both equality and UIFs. As shown by Ackermann [1], this problem can be reduced to satisfiability checking of quantifier free formulas involving only equality through a suitable generalization of the following: given a formula OE containing terms f(x 1 ) and f(x 2 ) replace f(x 1 ) and f(x 2 ) and by fresh variables y 1 and y 2 to obtain a ....

.... of the same UIF are equal, then the outputs of the two instances are equal; this constraint can be added to the IE circuit using simple circuitry (an equality checker and a gate) As is the case for Shostak s procedure [13] the soundness and completeness of this construction follows from [1]. 3 IE Netlist Satisfiability Checking IE Netlist Satisfiability Checking consists of taking an IE netlist and determining if an input assignment exists for which a specified Boolean valued output can take the value 1. Note that the usual product construction for checking the equivalence of ....

Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

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Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, 1954.

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W. Ackermann. Solvable Cases of the Decision Problem. North-Holland Publishing Co., Amsterdam, 1954. Studies in Logic and the Foundations of Mathematics.

....advantages. First, the decidability and complexity of logical implication for DLF and its variants are closely related to reasoning in Datalog # nS , a logic programming language of monadic predicates and functions [9, 10] and in turn to the classical Ackermann case of the Decision Problem [2, 4]. This connection allows us to draw on powerful techniques developed for the classical decision problems. Second, the use of Datalog # nS provides an uniform framework in which we can study various extensions of DLF , in particular how roles can be simulated by attributes or how dependencies can ....

Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, 1954.

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W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954. 12

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W. Ackermann, Solvable Cases of the Decision Problem. Studies in Logic and the Foundation of Mathematics. North-Holland, Amsterdam, 1954.

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W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundation of Mathematics, 1954.

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Ackermann, W.: Solvable cases of the decision problem. Studies in Logic and the Foundation of Mathematics (1954)

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Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

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Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

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W. Ackermann. Solvable cases of the decision problem. Studies in Logic and the Foundation of Mathematics, 1954.

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W. Ackermann. Solvable cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1954.

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W. Ackermann, Solvable Cases of the Decision Problem. Studies in Logic and the Foundation of Mathematics. North-Holland, Amsterdam, 1954.

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Wilhelm Ackermann. Solvable Cases of the Decision Problem. Studies in Logic and the Foundations of Mathematics. North-Holland, 1954.

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