W. V. Quine. A way to simplify truth functions. American Mathematical Monthly, 62(9):627--631, November. 1955.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....large , using the clauses fC i (x) j i 2 I( y)g as premises. In other words, it must show that fC i (x) j i 2 I( y)g is unsatis able. This can be done by applying a complete inference method for logical clauses, such as the resolution method discovered by W. V. Quine over 40 years ago [37, 38] (known as consensus when applied to formulas in disjunctive normal form) The name resolution derives from Robinson [39] who developed the method for rst order predicate logic. However, this tends to be a very inecient approach [18, 19] It is more practical to observe that the implicit ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

....[8] discussed the complexity of identification of cellular automata and presented sequential and parallel algorithms for computing the local transition table. However, neither of these authors obtained a clear neighborhood structure or parsimonious rule expression. In digital circuit design [9] [14] small Boolean expressions are searched to reproduce given data tables. But these solutions are Manuscript received July 10, 1998; revised February 14, 1999 and April 23, 2000. This work was supported by the University of Sheffield and U.K. EPSRC. This paper was recommended by Associate ....

....This difference increases the difficulty of the CA term selection problem. Note that CA term selection is different from and more difficult than Boolean function minimization for which many useful techniques are available, especially in the digital circuit design literature [9] [14] CA term selection involves both identifying and minimizing Boolean functions while methods related to Boolean function minimization usually only consider deriving the equivalent minimum expression from already known Boolean functions. The identification is difficult because of the logical ....

W. V. Quine, "A way to simplify truth functions," Amer. Math. Monthly, vol. 62, pp. 627--631, 1955.

....large number of spanned patterns, this task can only be accomplished with the help of efficient algorithms running in total polynomial time. The description of such an algorithm, running in fact in incremental polynomial time, and resembling the consensus method of Blake [5] and Quine [17] for finding the prime implicants of a Boolean function, is the aim of this paper. This paper is organized as follows. After introducing in Section 2 several definitions and notations, we present in Sections 3 and 4 a consensus type algorithm and an accelerated version of it for the generation of ....

....All Positive In this section we shall describe a consensus type method for solving the SPPG problem, along with an implementation of it, which runs in incremental polynomial time. Since the introduction of the consensus method for finding the prime implicants of a Boolean function ( 5] [17]) several other consensus methods appeared in the literature. Malgrange [16] uses a consensus type approach to find all the maximal submatrices consisting of ones of a 0 1 matrix. Another consensus type algorithm is developed in [1] for finding all maximal bicliques of a graph. A generalization ....

Quine, W. A way to simplify truth functions. American Mathematical Monthly, 62 (1955), 627-631.

....larger numbers of observations unclassified. We present below a consensus type algorithm for the generation of all spanned patterns, along with an implementation of it, which runs in incremental polynomial time. The method is similar to the well known consensus method of Blake ( 5] and Quine ([12]) for finding prime implicants of a Boolean function. Malgrange ( 13] used a consensus type approach to find all the maximal submatrices consisting of ones of a 0 1 matrix. Also, a consensus type algorithm for finding all maximal bicliques of a graph is presented in [1] An important feature of ....

....we are not able to guarantee yet a total polynomial time for producing all strong spanned patterns. In fact, the dualization problem of a monotone non decreasing Boolean function can be reduced in quadratic time to the problem of generating all strong spanned patterns of a certain dataset (see [12]) Thus, the existence of a total polynomial time algorithm for generating all strong spanned patterns would imply the existence of a total polynomial time algorithm for the dualization problem mentioned above; until now, the best known algorithms are pseudopolynomial. 5 Evaluation ....

W. Quine. A way to simplify truth functions. American Mathematical Monthly, 62 (1955), 627-631.

....4 x 6 . Note that, in this definition, the roles of t 1 and t 2 are asymmetric; t 1 (resp. t 2 ) is called the left parent (resp. right parent) of t 3 , and t 3 is the child of t 1 as well as being the child of t 2 . Let = i t i be an arbitrary DNF expression of a function f . It is known [30] that every prime implicant t of f can be derived from the terms in by applying a consensus procedure. In other words, there is a sequence t = t of terms such that each t is either in (i.e. t = t i for some i) or the consensus of two terms t such that k 1 ; k 2 k. Since ....

W. Quine, A way to simplify truth functions, American Mathematical Monthly, 62 (1955) 627-631.

....reduce it to its simplest equivalent DNF, i.e. to the simplest DNF realizing the function that g realizes. Simplest meant the DNF with fewest terms. His procedure consists of two parts; nding the so called prime implicants of the DNF and covering the points of the DNF by prime implicants. In [19] he re ned his procedure. The re nement meant that it was no longer necessary to expand the DNF into its points when nding the prime implicants. This re nement made the covering part more complicated. We do not present all of the re ned procedure, but take our situation into account and ....

....ned procedure, but take our situation into account and present what we need. Essentially, the procedure we present for computing a minimum term DNF consists of the re ned rst part and the unre ned second part. 8 We need some de nitions. These are de nitions introduced by Quine in [18] and [19]. A term t subsumes term t if t contains all literals that t contains. Furthermore, if t is a point and t subsumes t we will say that t covers t. Abusing terminology slightly, we will sometimes say that t covers a 2 f0; 1g by this we mean that t covers the point X . A term ....

Quine, W.V., \A Way to Simplify Truth Functions", in: American Mathematical Monthly 62 (1955) 627-631.

....time. 11 A prime implicant (resp. a prime implicate) # of # is essential iff the disjunction (resp. conjunction) of all prime implicants (resp. prime implicates) of # except # is not equivalent to #. 12 The correctness of (the dual of) Quine s consensus algorithm for computing prime implicants [28] ensures it, since no clause of #n is subsumed by another clause and no consensi can be performed since there are no negated variables. 15 L NNF DNNF d DNNF FBDD OBDD OBDD DNF CNF PI IP MODS sd DNNF NNF # # # # # # # # # # # # DNNF # # # # # # # # # d DNNF # # # # # # FBDD # # # OBDD ....

W.V.O. Quine. A way to simplify truth functions. American Mathematical Monthly, 52:627-631, 1955.

....terminal nodes visit exactly n non terminal nodes. In an AND TDD, each 1 path corresponds to an implicant of f . In general, RO AND TDDs do not represent all the implicants, while the QR AND TDDs represent all the implicants. 2. 6 Prime TDDs A prime TDD represents all the prime implicants (PIs) [4, 13, 39, 42] of a two valued logic function f.A prime TDD represents F : T n B, where F (ff) 1iff the product x ff 1 1 x ff 2 2 111x ff n n isaPIoff.AprimeTDD is a special case of SOP TDDs and is unique for each f . By using prime TDDs, we can efficiently generate all the PIs. The prime TDD can be ....

W. V. Quine, "A way to simplify truth functions," Amer. Math. Mon.,Vol. 62, Nov. 1955, pp. 627-631.

....one and negatively in the other, the resolvent of the clauses is the clause consisting of all the literals in either clause except x j and x j . For example, the resolvent of the clauses x 1 # x 2 # x 3 x 1 # x 2 # x 4 (5) is x 2 #x 3 #x 4 . The logician Quine showed in the 1950s [123, 124] that repeated application of this resolution step to a clause set, and to the resolvents generated from the clause set, derives all clauses that are implied by the set. To see the connection with Fourier Motzkin elimination, write the clauses (5) as 0 1 inequalities: x 1 x 2 (1 x 3 ) # ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955): 627-631.

....are no other distinct implicants g 0 of f with g g 0 . In the lattice F n , every Boolean function f is the join of its prime implicants. A DNF of f that is expressed as the join of elementary conjunction DNFs of its prime implicants can be obtained using the consensus algorithm (see Quine [11]) We assume that the reader is familiar with this algorithm, and with the operation of adjunction of consensus applied to elementary conjunctions or to the Boolean functions they represent (see [11] A set S of Boolean functions in F n represented by elementary conjunctions is the set of prime ....

.... conjunction DNFs of its prime implicants can be obtained using the consensus algorithm (see Quine [11] We assume that the reader is familiar with this algorithm, and with the operation of adjunction of consensus applied to elementary conjunctions or to the Boolean functions they represent (see [11]) A set S of Boolean functions in F n represented by elementary conjunctions is the set of prime implicants of some f 2 F n if and only if (i) distinct members of S are incomparable in F n , ii) S is closed under adjunction of consensus. In one phrase, S is to be an antichain closed under ....

W.V. Quine, "A way to simplify truth functions", American Mathematical Monthly, 62 (1955), 627--631. -- 20 --

....for C = C ) for C = CHorn (and for all related classes, such as k quasi Horn and k quasi reverse Horn for any fixed k) and for C = C k DNF with k fixed. To discuss properties of very robust extensions, we shall recall the consensus method and some of its properties (see e.g. [17, 18]) Given two terms t = V j#P x j V j#N x j and t # = V j#P # x j V j#N # x j , we say that they are in conflict at variable x j if j # (P #N # )#(N #P # ) i.e. if x j appears in one and x j appears in the other) If t and t # are in conflict at exactly one of the variables, then ....

....# are implicants of the Boolean function f , then their consensus t ## = t, t # ] when exists) is also an implicant of f . The consensus method is the algorithm, in which consensuses of implicants of a given DNF of f are formed as long as new implicants are generated. It is well known (see e.g. [17]) that this method is complete in the sense that all prime implicants of f will be obtained in this way, starting from any DNF representation of f . Of course, all these notions and results can straightforwardly be translated for CNF representations using De Morgan s laws. The corresponding ....

W. Quine. A way to simplify truth functions. American Mathematical Monthly, 62:627-- 631, 1955.

....k quasi reverse Horn for any fixed k) and for C = C k DNF with k fixed. Let us recall that a class C is minor closed, if f # C and g # f imply g # C. To discuss properties of very robust extensions, we shall further recall the consensus method and some of its properties (see e.g. [14, 15]) Given two terms t = # j#P x j # j#N x j and t # = # j#P # x j # j#N # x j , we say that they are in conflict at variable x j if j # (P #N # ) # (N #P # ) i.e. if x j appears in one and x j appears in the other) If t and t # are in conflict at exactly one of the variables, ....

....# are implicants of the Boolean function f , then their consensus t ## = t, t # ] when exists) is also an implicant of f . The consensus method is the algorithm, in which consensuses of implicants of a given DNF of f are formed as long as new implicants are generated. It is well known (see e.g. [14]) that this method is complete in the sense that all prime implicants of f will be obtained in this way, starting from any DNF representation of f . Of course, all these notions and results can straightforwardly be translated for CNF representations using De Morgan s laws. The corresponding ....

W. Quine. A way to simplify truth functions. American Mathematical Monthly, 62:627--631, 1955.

....for C = C ) for C = CHorn (and for all related classes, such as k quasi Horn and k quasi reverse Horn for any fixed k) and for C = C k DNF with k fixed. To discuss properties of very robust extensions, we shall recall the consensus method and some of its properties (see e.g. [18, 19]) Given two terms t = # j#P x j # j#N x j and t # = # j#P # x j # j#N # x j , we say that they are in conflict at variable x j if j # (P #N # )#(N #P # ) i.e. if x j appears in one and x j appears in the other) If t and t # are in conflict at exactly one of the variables, ....

....are implicants of the Boolean function f , then their consensus t ## = t, t # ] when exists) is also an implicant of f . The consensus method is the algorithm, in which consensuses of implicants of a given DNF of f are formed as long as new implicants are generated. It is well known (see e.g. [18]) that this method is 15 complete in the sense that all prime implicants of f will be obtained in this way, starting from any DNF representation of f . Of course, all these notions and results can straightforwardly be translated for CNF representations using De Morgan s laws. The corresponding ....

W. Quine. A way to simplify truth functions. American Mathematical Monthly, 62:627--631, 1955.

.... logic circuits has evolved from its first notions [36] to a standard element of undergraduate computing curricula [34] Standard graphical design aids such as Karnaugh Maps [18, 41] are widely used and tools suitable for computer implementation have evolved from the Quine McCluskey Method [32, 26] to freely available tools such as Espresso [2] and MisII [3] and many commercial products. Probably the earliest attempt to evolve circuits is Friedman s thesis, that dates back to the mid 1950s [8] In his thesis, Friedman proposed that a series of control circuits, similar to what we now call ....

....the minimization process. Such recognition may not be obvious. Furthermore, there is no general set of rules to aid that recognition. Two popular minimization techniques are the Karnaugh Map [18] which is based on a graphical representation of Boolean functions, and the QuineMcCluskey Procedure [32, 26], which is a tabular method. Both of these methods are mechanical in nature. Karnaugh Maps are useful in minimizing functions with up to five or six variables. The Quine McCluskey Procedure is useful for functions of any number of variables and can easily be programmed to run on a digital ....

W. V. Quine. A way to simplify truth functions. American Mathematical Monthly, 62 (9):627--631, 1955. 34

....resolvent, x 1 :x 2 x 3 : 4) 3 (3) however, can be resolved with neither (1) nor (2) The resolution algorithm generates all logical implications of a clause set S, in the following sense. A prime implication of S is a clause implied by S that is absorbed by no other such clause. Quine [17, 18] showed that if the resolution algorithm begins with a satisfiable set S, it produces the set S 0 of all prime implications of S. Any clause implied by S is absorbed by some clause in S 0 . If S is unsatisfiable, the algorithm produces the empty clause, which in this case is the only prime ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

....of its performance. Results are compared against those produced by our previous approach (a GA with an n cardinality alphabet and a two stage fitness function that we will simply denote as NGA) and against designs produced by humans (using Karnaugh Maps [14] and the QuineMcCluskey Procedure [20, 15]) Then, we present our conclusions and some of the possible paths of future research. 2. Related Work The idea of using multiobjective optimization techniques to handle constraints is not new. Some researchers have proposed to redefine the single objective optimization of f as a multiobjective ....

W. V. Quine. A Way to Simplify Truth Functions. American Mathematical Monthly, 62 (9):627--631, 1955.

....bx fi defines a facet if and only if it is a valid cut and there are affinely independent points y 1 ; y d such that Ay j a and by j = fi for j = 1; d. To define the strongest possible logic cut, I will first review the notion of a prime implication defined by Quine 50 years ago [47, 48]. Given a set S of logical clauses, clause C is a prime implication of S if S implies C but implies no other clause that implies C. For instance, the prime implications of the set x 1 x 2 x 1 x 2 x 3 x 4 x 3 x 5 (20) are: x 1 x 3 x 4 x 3 x 5 x 4 x 5 (21) Every clause implied by ....

....is exactly one variable x j that appears positively in one and negatively in the other. The resolvent contains all the literals in either clause except x j and x j . For instance, the third clause below is the resolvent of the first two. x 1 x 2 x 3 x 1 x 2 x 4 x 2 x 3 x 4 26 Quine [47, 48] showed that repeated resolutions generate all prime impolications of a set of clauses. To see that resolvents are rank 1 cuts, write the above clauses as the first, second and fifth inequalities below. x 1 x 2 x 3 1 Gammax 1 x 2 Gamma x 4 Gamma1 x 3 0 Gamma x 4 Gamma1 x 2 x ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

....but only linear time otherwise [6] Formula A implies formula B if and only if A:B is unsatisfiable. Thus the satisfiability problem and the inference problem are essentially the same in propositional logic. A and B are equivalent if they imply each other. 2. 2 The Resolution Procedure Quine [43, 44] proved that the satisfiability problem can be solved by the resolution procedure, which Robinson generalized to first order logic [45] When two clauses have the property that exactly one atomic proposition x j occurs posited in one and negated in the other, the resolvent of the clauses is a ....

....in the worst case [20] and can be very slow even for random problems [24] A clause C is a prime implication of a set S of clauses if S implies C and implies no other clause that absorbs C. Thus every implication of S is absorbed by some prime implication of S. Quine proved, Theorem 1 (Quine [43, 44]) If a set S of clauses is satisfiable, the resolution procedure generates precisely the set of prime implications of S. It generates the empty clause if and only if S is unsatisfiable. Horn clauses have received much attention because unit resolution alone is enough to check whether a set of ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

....problems in first order predicate logic [4, 7] Partially supported by U. S. Air Force Office of Scientific Research grant AFOSR 910287. GSIA working paper 1991 9. 1 Our approach is first to solve the original satisfiability problem with the classical Davis Putnam Loveland (DPL) algorithm [1, 3, 9, 11, 12]. We then solve the incremented problem with a modified DPL algorithm that takes advantage of the information in the data structure generated during the solution of the original problem. We choose DPL because, as has been reported elsewhere [2, 5, 10] it is quite competitive with more recent ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

.... logic circuits has evolved from its first notions [3] to a standard element of undergraduate computing curricula [4] Standard graphical design aids such as Karnaugh Maps [5, 6] are widely used and tools suitable for computer implementation have evolved from the Quine McCluskey Method [7, 8] to freely available tools such as Espresso [9] and MisII [10] and many commercial products. Louis [11] is one of few sources found in the literature to address the use of GAs for the combinational logic design problem. In his dissertation [12] Louis combines knowledgebased systems with the ....

....the minimization process. Such recognition may not be obvious. Furthermore, there is no general set of rules to aid that recognition. Two popular minimization techniques are the Karnaugh Map [5] which is based on a graphical representation of Boolean functions, and the Quine McCluskey Procedure [7, 8], which is a tabular method. Both of these methods are mechanical in nature. Karnaugh Maps are useful in minimizing functions with up to five or six variables. The QuineMcCluskey Procedure is useful for functions of any number of variables and can easily be programmed to run on a digital computer. ....

Quine, W. V. (1955) A way to simplify truth functions, American Mathematical Monthly, 62, No. 9, pp. 627--631.

....If a term T of Phi absorbs a term T 0 of Phi, delete T 0 . It is easy to notice that all the DNFs produced at every step of the consensus method represent the same function as the original DNF. The following result plays a central role in the theory and applications of Boolean functions [1, 11]: Proposition 2.1 (Blake, Quine) The consensus method applied to an arbitrary DNF of a Boolean function f results in the DNF which is the disjunction of all the prime implicants of f . A classical hard problem concerning Boolean formulae is the satisfiability problem (SAT) When working with ....

W. Quine. A way to simplify truth functions, American Mathematical Monthly, 62 (1955), 627--631.

....clause C 3 : C 3 = j2(P 1 nflg) P 2 x j j2N 1 [ N 2 nflg) x j ; 9) is called the resolvent of C 1 and C 2 . For example, x 2 x 3 x 4 x 5 x 6 is the resolvent of x 1 x 3 x 4 x 5 and x 1 x 2 x 3 x 4 x 6 . Let F = V i C i be an arbitrary CNF of a theory Sigma. It is known (e.g. [41]) that every prime implicate C of Sigma can be derived from the clauses in F by applying a resolution procedure which iteratively generates resolvents. In other words, there is a sequence C (1) C (2) C (m) C of clauses such that each C (k) is either in F or the resolvent of ....

W. Quine. A way to simplify truth functions, American Mathematical Monthly , 62 (1955), 627 - 631.

....If a term T of Phi absorbs a term T 0 of Phi, delete T 0 . It is easy to notice that all the DNFs produced at every step of the consensus method represent the same function as the original DNF. The following result plays a central role in the theory and applications of Boolean functions [4, 15]: Proposition 2.1 (Blake, Quine) The consensus method applied to an arbitrary DNF of a Boolean function f results in the DNF which is the disjunction of all the prime implicants of f . Throughout the text, the following notation will be used to represent terms: Definition 2.2 If S = fi 1 ; ....

W. Quine. A way to simplify truth functions, American Mathematical Monthly, 62 (1955), 627--631.

.... logic circuits has evolved from its first notions [29] to a standard element of undergraduate computing curricula [27] Standard graphical design aids such as Karnaugh Maps [14, 32] are widely used and tools suitable for computer implementation have evolved from the Quine McCluskey Method [26, 22] to freely available tools such as Espresso [2] and MisII [3] and many commercial products. Probably the earliest attempt to evolve circuits is Friedman s thesis, that dates back to the mid 1950s [8] In his thesis, Friedman proposed that a series of control circuits, similar to what we now call ....

....the minimization process. Such recognition may not be obvious. Furthermore, there is no general set of rules to aid that recognition. Two popular minimization techniques are the Karnaugh Map [14] which is based on a graphical representation of Boolean functions, and the Quine McCluskey Procedure [26, 22], which is a tabular method. Both of these methods are mechanical in nature. Karnaugh Maps are useful in minimizing functions with up to five or six variables. The QuineMcCluskey Procedure is useful for functions of any number of variables and can easily be programmed to run on a digital computer. ....

W. V. Quine. A way to simplify truth functions. American Mathematical Monthly, 62 (9):627--631, 1955.

....absorbs C. Because any constraint set in binary variables is equivalent to a clause set, resolution can generate all valid clausal cuts for such a constraint set. We will also see that resolution can modified so that it achieves various kinds of consistency. Classical resolution was developed by Quine (1952,1955) for propositional logic and extended by Robinson (1965) to predicate logic. Various forms of it are widely used in logic programming and theorem proving systems. Its worstcase complexity was first investigated by Tseitin (1968) and shown by Haken (1985) to be exponential. In practice, ....

Quine, W. V. (1952). A way to simplify truth functions, American Mathematical Monthly 62, 627-631.

....C 3 : C 3 = j2(P 1 nflg) P 2 x j j2N 1 [ N 2 nflg) x j ; 9) is called the resolvent of C 1 and C 2 . For example, x 2 x 3 x 4 x 5 x 6 is the resolvent of x 1 x 3 x 4 x 5 and x 1 x 2 x 3 x 4 x 6 . Let F = V i C i be an arbitrary CNF of a theory Sigma. It is known (e.g. [28]) that every prime implicate C of Sigma can be derived from the clauses in F by applying a resolution procedure which iteratively generates resolvents. In other words, there is a sequence C (1) C (2) C (m) C of clauses such that each C (k) is either in F (i.e. C (k) C ....

W. Quine. A way to simplify truth functions, American Mathematical Monthly, 62 (1955), 627 - 631.

....appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. DAC 97, Anaheim, California c fl 1997 ACM 0 89791 920 3 97 06 . 3. 50 logic minimization [8] [10] [11] The basic idea is to reduce the size of the problem via essentiality, column dominance, and row dominance. Gimpel devised another form of reduction which is applicable when two columns have the same cost and the rows they cover show certain properties [5] Reductions are applied ....

W. V. O. Quine. A way to simplify truth functions. Am. Math. Monthly, 62:627--631, 1955.

....If a term T of Phi absorbs a term T 0 of Phi, delete T 0 . It is easy to notice that all the DNFs produced at every step of the consensus method represent the same function as the original DNF. The following result plays a central role in the theory and applications of Boolean functions [2] [10]: Proposition 2.1 (Blake, Quine) The consensus method applied to an arbitrary DNF of a Boolean function f results in the DNF which is the disjunction of all the prime implicants of f . Throughout the text, the following notation will be used to represent terms: Definition 2.3 If S = fi 1 ; ....

W. Quine, "A way to simplify truth functions", American Mathematical Monthly, 62 (1955), 627--631.

....The application of the techniques presented in this paper to circuit design is the subject of future study. The classical approach to tackle the Boolean Function Minimization Problem (where one wishes to minimize the number of disjuncts in the sum of products form) was developed by Quine [19] [20] and McCluskey [14] To describe the idea behind this approach, a few definitions are first needed. If a product term evaluates to 1 for a given minterm, it is said to contain that minterm. An implicant of a function F is a product term that does not contain any minterm of the off set of F . A ....

W.V. Quine. A way to simplify truth functions. Am. Math. Monthly, 62, 1955.

....by W. V. Quine over 40 years ago Let S be a set of logical clauses. Perform resolve(S) Procedure resolve(S 0 ) If S 0 contains clauses C; D that have a resolvent R that no clause in S 0 absorbs then Perform resolve(S 0 [ fRg) Else stop. Figure 2: The resolution algorithm. [18, 19]. It is the resolution method, known as consensus when applied to formulas in disjunctive normal form. The name resolution actually derives from Robinson [20] who developed the method for first order predicate logic. Given two clauses for which exactly one variable x j occurs positively on one ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

.... [1] 2] The classical approach to tackle the Boolean Function Minimization Problem (where one wishes to minimize the number of disjuncts in the sum of products form) was developed by 1 Stanford University, Stanford CA 94305 USA 2 AT T Bell Laboratories, Murray Hill NJ 07974 USA Quine [3] [4] and McCluskey [5] Because exact versions of the Quine McCLuskey method fail to handle large instances, many heuristic approaches have been developed. They include mini [6] presto [7] and espresso mv [1] espresso mv is widely used in the circuit design industry. In [8] Kamath, Karmarkar, ....

W. Quine, "A way to simplify truth functions," Am. Math. Monthly, vol. 62, 1955.

....F above. Actually the first clause can be dropped without effect, because it is implied by the second. Clause C 1 implies clause C 2 if and only if C 1 absorbs C 2 , which is to say that all the literals of C 1 occur in C 2 . The empty clause contains no literals and is necessarily false. Quine [18, 19] showed that a simple inference method, now called resolution, derives all implications of a given set of clauses. If two clauses have the property that exactly one atomic proposition x j occurs positively in one and negatively in the other, their resolvent (on x j ) consists of the disjunction ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

....one clause of F(C) is used in the procedure, we have #(F(C) n 0 jSj n 0 1 (the set S is not empty, since the function f has no unit prime implicates) Let C be a negative clause in F 3 , and C = W x2S 1 :x. The structure and completeness of the resolution (consensus) procedure (see e.g. [12]) imply (similarly to [9] that F(C) contains just one negative clause, i.e. F(C) C 0 V F h (C) where C 0 = W x2S 2 :x 2 F n and F h (C) F h . By Lemma 3.2 the forward chaining procedure for F h (C) starting with the set S 1 includes eventually all the variables in S 2 into the set R. ....

W. Quine. A way to simplify truth functions. American Mathematical Monthly, 62(1955), 627--631.

....implication. The problem with the normalization to primitive parts is that we have to renormalize the left hand sides for smart simplification destroying the ordering of the conjunction. Boolean normal form computation in propositional logic has been tackled in several papers by Quine (1952, 1955, 1959) He has shown how minimal Boolean normal forms can be obtained. All methods described by Quine have to combine a Boolean variable fi with its negation :fi in some way, where the point is that fi :fi ( true. Subsumption is used as test for implication between clauses. In the case of ....

Quine, W. V. (1955). A way to simplify truth functions. American Mathematical Monthly, 62, 627--631.

....the primal solution is degenerate. Each solution gives rise to a different sensitivity analysis. A more complete exposition of linear programming sensitivity analysis may be found in [2] 4 A Multivalent Resolution Method It is well known that the resolution method originally developed by Quine [9, 10] (and later extended to first order logic by Robinson [11] provides a complete refutation method for propositional logic in conjunctive normal form. 2 The resolution method is readily generalized to problems in which the variable domains contain more than two discrete values. The method that ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

....that seem most useful for the problem at hand. Valid (and nonvalid) logic constraints can also be derived from the special structure of a problem, much as is done for polyhedral cuts. These constraints may be valid or nonvalid and are discussed briefly below. 4. 1 Resolution Resolution [51, 52, 58] was originally defined for logical clauses, which are disjunctions of literals (atomic propositions or their negations) Resolution can derive valid logic constraints for any set of formulas q(y) in which the variables y are atomic propositions, because any such formula is equivalent to a finite ....

....of y 1 y 2 and :y 1 :y 3 . Given a set S of clauses, the resolution algorithm picks a pair of clauses in S that have a resolvent that is implied by no clause in S, and adds the resolvent to S. It repeats until there is no such pair, which occurs after finitely many iterations. Theorem 6 (Quine [51, 52]) A clause set S implies clause C if and only if the resolution algorithm applied to S generates a clause that implies C. In particular, S is unsatisfiable if and only if resolution generates the empty clause. Thus resolution is somewhat analogous to Chv atal s cutting plane procedure, because it ....

Quine, W. V., A way to simplify truth functions, American Mathematical Monthly 62 (1955) 627-631.

.... MSCAS (1993) 2 Stanford University, Stanford CA 94305 USA 3 AT T Bell Laboratories, Murray Hill NJ 07974 USA The classical approach to tackle the Boolean Function Minimization Problem (where one wishes to minimize the number of disjuncts in the sum of products form) was developed by Quine [3] [4] and McCluskey [5] Because exact versions of the Quine McCLuskey method fail to handle large instances, many heuristic approaches have been developed. They include mini [6] presto [7] and espresso mv [1] espresso mv is widely used in the circuit design industry. In [8] Kamath, Karmarkar, ....

W. Quine, "A way to simplify truth functions," Am. Math. Monthly, vol. 62, 1955.

No context found.

W. V. Quine. A way to simplify truth functions. American Mathematical Monthly, 62(9):627--631, November. 1955.

No context found.

W. Quine, "A way to simplify truth functions," Amer. Math. Monthly, vol. 62, pp. 627--631, 1955.

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