U. Martin and T. Nipkow, "Unification in Boolean Rings", in Proc. 8th International Conference on Automated Deduction, Oxford, July 1986. Springer-Verlag LNCS vol. 230, pp. 506-513.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....E is said to be unitary if any two terms t 1 and t 2 that are E unifiable have a unique (modulo equivalence under E) most general unifier. Apart from the usual notion of unification of first order terms, equational theories that admit unique most general unifiers include that of Boolean rings [28], and theories that are either left distributive or right distributive (but not both) 35] We assume that the equational theory E under consideration is unitary. Given two E unifiable terms t 1 and t 2 , let mgu E (t 1 ; t 2 ) denote the (E unique) most general unifier of t 1 and t 2 . The ....

U. Martin and T. Nipkow, "Unification in Boolean Rings", in Proc. 8th International Conference on Automated Deduction, Oxford, July 1986. Springer-Verlag LNCS vol. 230, pp. 506-513.

....by the equational theory. Unification problems are translated into equations over certain algebraic structures, which (in some cases) can be solved using known results from mathematics. The two most prominent instances of the semantic approach are 1. unification in Boolean algebras and rings [6, 16, 15], and its generalization to finite and to primal algebras [4, 5, 17, 11] 2. unification modulo the theories ACU (associativity commutativity with unit [14, 20] ACUI (associativity commutativity idempotency with unit [14, 2] and AG (the theory of Abelian groups [12] It has turned out that ....

U. Martin and T. Nipkow. Unification in Boolean rings. J. Automated Reasoning, 4:381--396, 1989.

....Three topics of combinational logic synthesis: redesign, multi level logic minimization and minimization of Boolean relations are discussed. All these problems can be uniformly formalized as Boolean unification problems. Experimental results are also reported. 1 Introduction Boolean Unification[9, 8] is a procedure to obtain the general solution of a given Boolean equation or formula. In the field of CAD for integrated circuit design, it has been applied to logic verification and test pattern generation [11, 4] combined with logic programming. In this paper we present various applications of ....

....Three topics: redesign [5] multi level logic minimization and minimization of Boolean relations [12] are discussed. All these problems can be uniformly formalized as Boolean unification problems. The general solutions of these problems can be obtained by using Boolean unification algorithms [9, 8]. These solutions express complete don t cares, which enable us to explore larger design space. In section 2, we briefly review a Boolean unification problem and its algorithm and show that the algorithm can be easily implemented by Binary Decision Diagrams [3] In section 3, we present three ....

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U. Martin and T. Nipkow. Unification in boolean rings. Journal of Automated Reasoning, 4:381--396, 1988.

....light on the power of ACB matching. The easy combination of ACB matching problems is also an advantage of our approach. 1 Introduction Extended unification (see [3] has been investigated for many theories. Well known positive results for associativity, commutativity [21] for boolean rings [14], or for convergent rewrite systems [10, 4] demonstrate the importance of extended unification in many application fields, like theorem proving, logic programming, etc. The congruence relations induced by equational theories can be interpreted as syntactical similarity that makes them also very ....

Martin, U. and Nipkow, T.: Unification in Boolean Rings. Proc. 8th International Conference on Automated Deduction, LNCS 230, 506-513, (1986).

....a relational description (an equation) into a functional description by introducing new variables. The variables p 1 ; p 2 : p n are said to have been eliminated by Boolean unification. 3. 2 Algorithms There are two major Boolean Unification algorithms: Boole s method and Lowenheim s method [10, 11]. Boole s method is recursive and is based on the Shannon expansion of a given formula. Lowenheim s method constructs an mgu from a set of particular solutions instead of working from a given formula. In this report we shall only treat Boole s method and its variants. See [11] for a fuller ....

Ursula Martin and Tobias Nipkow. Unification in boolean rings. Journal of Automated Reasoning, 4:381--396, 1988.

....of the theorems pertaining to the justification of freeness. The unification of terms built up from scons is a very special case of ACI unification [KN93] Examples of ACI unification include the unification of set terms built up the [ union) constructor and unification in boolean rings [MN86]. The interested reader may consult the reference [Sie90] for a catalog of several different types of unification problems, including their complexity. Note that scons is not associative, commutative, or idempotent, but enjoys closely related properties, namely, scons(x; scons(y; s) scons(y; ....

Martin, U., and Nipkow, T.: Unification in Boolean Rings, Proc. 8th Conf. on Automated Deduction, pp. 506--513, Springer LNCS 230, 1986.

....algorithms (Crone Rawe 1989) We chose the one published by (Buttner Simonis 1987) the origin of the method goes back to (Boole 1947) The algorithm computes the most general boolean ring unifier of two terms t 1 and t 2 . It operates on a deterministic disjunctive minimal normal form (Martin Nipkov 1986) for terms in the boolean ring hV , Phi, 0,1i, i.e. all boolean functions are expressed in terms of Phi and . Example. X 1 Phi X X Y X Phi Y Phi XY The use of CLP(B) is limited to discrete devices which compute some boolean function. However, it allows for a compact representation of ....

Martin, U., Nipkov, T. (1986). Unification in boolean rings. Proc. 8th Intl. Conference on Automated Deduction, pp. 506-513.

....applied to a function f with respect to a variable x i is denoted by S x i f and defined as: S x i f = f x i f x i . Both C x i f and S x i f are functions independent of x i . 3 Boolean Unification Algorithms Boolean unification (i.e. solution of Boolean equations) as discussed in [6, 7], is a basic inference mechanism in algebraic manipulation of formulas. Given a formula, Boolean unification derives a general solution (called the most general unifier, abbreviated as mgu henceforth) which contains some newly introduced variables to represent all the possible solutions (if there ....

....are the functions of q 1 and q 2 . By substituting any functions (or constants) into r 1 and r 2 , particular solutions can be obtained. For example, let r 1 = q 1 and r 2 = q 2 , we have: p 1 = q 1 q 2 p 2 = q 1 q 2 . Now we discuss a Boolean unification algorithm developed by Boole [6, 7] which is the basis for our Boolean matching algorithm. Boole s algorithm is based on recursive Shannon expansion of a given formula, and it derives the most general unifiers in a bottom up fashion. Shown in Figure 1 is Boole s unification algorithm (it is modified from the original algorithm for ....

U. Martin and T. Nipkow, "Unification in Boolean Ring", Journal of Automated Reasoning, Vol. 4, pp. 381-396, 1988.

....decision problem for elementary unification (where the terms to be unified contain only symbols of the signature of Boolean algebras) is only NP complete. 1 Introduction Boolean unification, i.e. unification modulo the theory of Boolean algebras or rings, has been considered by several authors [5, 14, 13]. On the one hand, this problem is of interest for research in unification theory since, unlike theories such as associativity commutativity, the theory of Boolean algebras is unitary even for unification with constants (where the terms to be unified may contain additional free constant symbols) ....

....can, e.g. be used to support hardware verification and design tasks [5, 18] Partially supported by the EC Working Group CCL II. The emphasis in the work on Boolean unification was on developing algorithms that compute a most general unifier for unification problems with constants [5, 14, 13], or finite complete sets of unifiers for general unification problems [17, 3] Of course, such algorithms can also be used to decide solvability of a given unification problem. However, the complexity of a decision procedure obtained this way need not be optimal. In fact, to the best of our ....

U. Martin and T. Nipkow. Unification in Boolean rings. J. Automated Reasoning, 4:381--396, 1989.

....[Crone Rawe, 1989] We chose one that was published by [B uttner and Simonis, 1987] the origin of the method goes back to [Boole, 1947] The algorithm computes the most general boolean ring unifier of two terms t 1 and t 2 . It operates on a deterministic disjunctive minimal normal form [Martin and Nipkov, 1986] for terms in the boolean ring hV , Phi, 0,1i, where V is the set of variables and Phi is the xor operator. 5.1 Reformulation of the models Instead of using extensional descriptions of the structural, weak and strong fault models we can formulate them in terms of boolean algebra expressions. For ....

Martin, U., Nipkov, T. Unification in boolean rings. Proc. 8th Intl. Conference on Automated Deduction, pp. 506-513, 1986.

....determines equations 0 = 00 to be unified. The equational problem 0 = 00 is decomposed in the resolution of equations on data types and on clocks. Unification of boolean clock equations is decidable and unitary (i.e. boolean equations have a unique most general unifier [12]) In inf, we refer to mgu Delta ( 0 ) as the boolean unification algorithm of [12] determining the most general unifier of the equation = 0 with constants Delta. Similarly, causality information ; OE is represented by a simple form of extensible record [16] of prefix and ....

.... 00 is decomposed in the resolution of equations on data types and on clocks. Unification of boolean clock equations is decidable and unitary (i.e. boolean equations have a unique most general unifier [12] In inf, we refer to mgu Delta ( 0 ) as the boolean unification algorithm of [12] determining the most general unifier of the equation = 0 with constants Delta. Similarly, causality information ; OE is represented by a simple form of extensible record [16] of prefix and members OE, for which we know the resolution of equations OE = OE 0 to be a decidable and ....

U. Martin, T. Nipkow. Unification in boolean rings. In Journal of Automated Reasoning, v. 4. Kluwer Academic Press, 1988.

....be evaluated bottom up in closed form and LOGSPACE (PTIME) data complexity. This extends the approach to safe queries of [3, 25, 31, 44] Section 4) 6. Finally, Datalog with boolean equality constraints can be evaluated bottom up and in closed form. For the definitions we refer to Section 5 and [10, 34, 40]. The data complexity here is higher than in the previous cases and it depends on the use of free boolean algebras with m generators. We partly analyze this data complexity and show it to be Pi p 2 hard (Section 5) 2 Real Polynomial Inequality Constraints Throughout Section 2, we assume ....

.... 0; 1i, is called a boolean ring with unity if we define for any elements x and y the binary function x Phi y (exclusiveor) as (x y 0 ) x 0 y) Because of one to one correspondence between boolean algebras and boolean rings with unity the theory here can be developed in either setting [40]. In what follows we use algebras and, sometimes, the exclusive or as an abbreviation. Boolean Terms: We use T (F; V [ C) for the set of terms built in the usual way, from F the set of function symbols f; 0 ; 0; 1g, V a set of variable symbols, and C a of constant symbols distinct from 0; 1. ....

[Article contains additional citation context not shown here]

U. Martin, T. Nipkow. Unification in Boolean Rings. Journal of Automated Reasoning, 4:381-396, 1988.

....be evaluated bottom up in closed form and LOGSPACE (PTIME) data complexity. This extends the approach to safe queries of [3, 23, 29, 42] Section 4) 6. Finally, Datalog with boolean equality constraints can be evaluated bottom up and in closed form. For the definitions we refer to Section 5 and [8, 32, 38]. The data complexity here is higher than in the previous cases and it depends on the use of free boolean algebras with m generators. We partly analyze this data complexity and show it to be Pi p 2 hard (Section 5) 2 Real Polynomial Inequality Constraints Throughout Section 2, we assume ....

.... 0; 1i, is called a boolean ring with unity if we define for any elements x and y the binary function x Phi y (exclusiveor) as (x y 0 ) x 0 y) Because of one to one correspondence between boolean algebras and boolean rings with unity the theory here can be developed in either setting [38]. In what follows we use algebras and, sometimes, the exclusive or as an abbreviation. Boolean Terms: We use T (F; V [ C) for the set of terms built in the usual way, from F the set of function symbols f; 0 ; 0; 1g, V a set of variable symbols, and C a of constant symbols distinct from 0; 1. ....

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U. Martin, T. Nipkow. Unification in Boolean Rings. Journal of Automated Reasoning, 4:381-396, 1988.

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Martin U. and Nipkow T. [1989b], `Unification in Boolean rings', J. Automated Reasoning 4, 381--396.

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