A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....view of the exponentially 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 ....

....for Generating 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 ....

Blake, A. Canonical expressions in Boolean algebra. Ph.D. Thesis. University of Chicago, 1937.

....leaving somewhat 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 ....

A. Blake. Canonical Expressions in Boolean Algebra. Ph.D. Thesis, University of Chicago, 1937.

....and somewhat surprising connection of the variable space of a proof, to the black white pebbling measure of an underlying graph. This research was supported by the Clore Foundation Doctoral Scholarship fund. 1 Introduction The resolution proof system was initially introduced by Blake in 1937 [8], and further developed in the influential work of Robinson [32] and Davis and Putnam [22] This system is one of the weakest in the world of propositional proof systems, and therein lies its strength. Because of its simplicity, and because all lines in a proof are clauses, this system yields ....

A. Blake. Canonical Expressions in Boolean Algebra. PhD Thesis, University of Chicago, 1937.

....exists a resolution proof of C. Resolution is implicationally complete for CNF formulas, meaning for any CNF C, and any clause A, there is a derivation of A from C i C j= A. 2.2 Survey of Results 2.2. 1 History and Previous Lower Bounds The resolution proof system was initially introduced by [Blake ( 37)] and further developed in the in uential work of [Robinson ( 65) and [Davis and Putnam ( 60) This system is one of the weakest in the world of propositional proof systems, and therein lies its strength. Because of its simplicity, and because all lines in a proof are clauses, this system ....

A. Blake. Canonical Expressions in Boolean Algebra. PhD Thesis, University of Chicago, 1937.

....of products is a sum ofproducts expression (SOP) Definition 2.3. A prime implicant (PI) of a function f is a product that implies f such that the deletion of any literal from the product results in a new product that does not imply fi Definition 2.4. A complete sum of products expression (CSOP) [2], 221 of a function f is the SOP of all PIs of f. Definition 2.5. An irredundant sum of products expression (ISOP) is an SOP where each product is a PI and no PI can be deleted without changing the function represented by the expression. Definition 2.6. Among the ISOPs for f, the one with the ....

A. Blake, "Canonical Expressions in Boolean Algebra," dissertation, Dept. of Math., Univ. of Chicago, 1937.

....of f . Proof: It follows from Lemma 4.1, Lemma 4.6 and Lemma 4.7 that u atom x2t dxe is the best upper bounding box approximation to a term t. The result therefore follows from Lemma 4.8. We now present an algorithm for computing L f and U f . This algorithm makes use of the Blake canonical form [1], BCF (f) of a Boolean function f , which consists of the sum of all prime implicants of f . Definition. A prime implicant of a function f is a term p such that p f and q 6 f for all true subterms q of p. A sum of products formula f is syllogistically less than a sum of products formula g (in ....

A. Blake. Canonical Expressions in Boolean Algebra. PhD thesis, Uni. of Chicago, 1937.

....deleted obtaining a new formula which is equivalent, in the sense specifled above, to the original one. This operation is particularly useful in reducing the number of monomials in the formula. 5. 3 Synthesis Resolution This operation is a special case of the general operation called resolution [9, 1] in the case of CNF, and consensus [8] in the case of DNF. Given a formula containing two monomials which are identical except for one literal x i appearing positive in one monomial and negative in the other, hence with the following structure: m 1 = x i x h : x k ) m 2 = x i x h : ....

A. Blake. Canonical Expressions in Boolean Algebra. Ph.D. thesis, University of Chicago, 1937.

.... 1 in a sense complementary to the (non uniform) computational complexity; moreover, there exist extremely rich and productive relations between the two areas (see e.g. Raz96, BP98] Much of the research in proof complexity is centered around the resolution proof system that was introduced in [Bla37] and further developed in [DP60, Rob65] In fact, it was for a subsystem of this system (nowadays called regular Resolution) that Tseitin proved the first non trivial lower bounds in his seminal paper of more than 30 years ago [Tse68] Despite its apparent (and deluding) simplicity, the first ....

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

....exponential complexity. In general it may be observed that if each variable occurs only once in an expression then evaluating it will yield an optimal result. Also if we start with thin intervals then we will also get an optimal result. Finally, for a propositional formula in Blake canonical form [3] evaluation with intervals always yields an optimal result [7] Moreover, all propositional formulas can be converted to this form. Thus we can evaluate propositional formulas in linear time and get optimal results. The catch is that converting propositional formulas to Blake canonical form is ....

A. Blake. Canonical Expressions in Boolean Algebra. PhD thesis, University of Chicago, 1938. Published by University of Chicago Librarries, 1938.

....from F , the root is labeled by C, and the clause labeling an inner node is the resolvent of the clauses labeling its two direct successors. We use T : F C 0 C to express that T : F C 0 for some sub clause C 0 C of C. As usual, F j= C denotes 8 : F = fCg = Since [7] it is known that F j= C holds if and only if F C 0 C. 14 A resolution tree T is called regular ( 87] if there is no subtree T 0 : F C 0 of T containing a resolution variable v with v 2 var(C 0 ) And T is called an input resolution tree if for each node the distance to a leaf is ....

Archie Blake. Canonical expressions in Boolean algebra. PhD thesis, Chicago, 1937. See [68].

....are given in Appendix A. 2 Known results Resolution In 1934 Gentzen [19] discovered the fundamental importance of (symmetric) cuts in proofs. Resolution is cut applied to simplest logical components (i.e. clauses) closed under this operation. Apparently it was first used in 1937 by A. Blake [2, 3], who proved its (strong) completeness. Quine 1955 59 [47, 4] called this concept (iterated) consensus, while the notion resolution is due to Robinson 1965 [49] who combined it with unification (introduced by Herbrand) improving the search through all possible Herbrand terms used before. ....

Archie Blake. Canonical expressions in Boolean algebra. PhD thesis, Chicago, 1937. See [40].

....allow straight line proofs (that is, every deduced formula can be used more than once in forthcoming inferences) Missing details of definitions as well as more information about these p.p.s. and their modifications can be found e.g. in [22, 44] Resolution. This proof system was introduced in [8] and further developed in [16, 38] Let C be the class of all disjunctive normal forms. A resolution proof of a tautology OE = K 1 : Km ) from TAUT C is actually a resolution refutation of the set fD 1 ; Dmg, where D i is the clause (elementary disjunction) that is the negation of ....

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

....the same number from [n] 1. Introduction Complexity of propositional proofs is rapidly taking on as important a role in the theory of feasible proofs as the role played by the complexity of Boolean circuits in the theory of efficient computations. The resolution proof system introduced in [Bla37] and further developed in [DP60, Rob65] is one of the first and simplest in the hierarchy of propositional proof systems; it is also of importance for various automatic theorem proving procedures. Tseitin [Tse68] proved, almost 30 years ago, the first exponential lower bound for regular ....

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

....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 ....

A. Blake, canonical expressions in Boolean algebra, Ph.D. Thesis, University of Chicago (Aug. 1937).

....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 ; ....

A. Blake. Canonical expressions in Boolean algebra, Ph.D. Thesis, University of Chicago, August 1937.

....of f . Proof: It follows from Lemma 4.1, Lemma 4.6 and Lemma 4.7 that u atom x2t dxe is the best upper bounding box approximation to a term t. The result therefore follows from Lemma 4.8. We now present an algorithm for computing L f and U f . This algorithm makes use of the Blake canonical form [1], BCF (f) of a Boolean function f , which consists of the sum of all prime implicants of f . Definition. A prime implicant of a function f is a term p such that p f and q 6 f for all true subterms q of p. A sum of products formula f is syllogistically less than a sum of products formula g (in ....

A. Blake. Canonical Expressions in Boolean Algebra. PhD thesis, Uni. of Chicago, 1937.

....exponential complexity. In general it may be observed that if each variable occurs only once in an expression then evaluating it will yield an optimal result. Also if we start with thin intervals then we will also get an optimal result. Finally, for a propositional formula in Blake canonical form (Blake 1938) evaluation with intervals always yields an optimal result (H. 1997) Moreover, all propositional formulas can be converted to this form. Thus we can evaluate propositional formulas in linear time and get optimal results. The catch is that converting propositional formulas to Blake canonical form ....

Blake, A. 1938. Canonical Expressions in Boolean Algebra.

....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 ; ....

.... structure terms string term string mode pointer to terms next list of terms List1,List2 terms headlist1,current,convhull List1 = T 1 ; T 2 ; Tm ] for i = 1 to m List1[i] mode=oldterm List2 = Algorithm: begin fmaing while (List1 [ do headlist1=List1[1] current=List1[2] while (current ; do if d(headlist1.term,current.term) k let convhull= headlist1.term,current.term] delete headlist1 and current from List1 push convhull to List1 List1[1] mode=newterm headlist1=List1[1] current=List1[2] else current=current.next if headlist1.mode=newterm ....

[Article contains additional citation context not shown here]

A.Blake, Canonical expressions in Boolean algebra, Ph.D. Thesis, University of Chicago (Aug. 1937).

....same number from [n] 2 Introduction Complexity of propositional proofs is rapidly becoming to play as important a role in the theory of feasible proofs as the role played by the complexity of Boolean circuits in the theory of efficient computations. And the resolution proof system introduced in [Bla37] and further developed in [DP60, Rob65] is one of the first and simplest in the hierarchy of propositional proof systems; it is also of invaluable importance for various automatic theorem proving procedures. Tseitin [Tse68] proved, almost 30 years ago, the first exponential lower bound for ....

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

No context found.

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

No context found.

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

No context found.

A. Blake. Canonical expressions in Boolean algebra. PhD thesis, University of Chicago, 1937.

No context found.

A. Blake. Canonical Expressions in Boolean Algebra. PhD thesis, University of Chicago, 1937.

No context found.

A. Blake, Canonical expressions in Boolean algebra, Ph.D. thesis, Department of Mathematics, Chicago Univesity, Chicago, 1937.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC