1 Fundamental Concepts.- 1.1 Formulas.- 1.2 Propositions and Predicates.- 1.3 Sets.- 1.4 Operations on Sets.- 1.5 Partitions.- 1.6 Relations.- 1.7 Functions.- 1.8 Operations and Algebraic Systems.- 2 Boolean Algebras.- 2.1 Postulates for a Boolean Algebra.- 2.2 Examples of Boolean Algebras.- 2.2.1 The Algebra of Classes (Subsets of a Set).- 2.2.2 The Algebra of Propositional Functions.- 2.2.3 Arithmetic Boolean Algebras.- 2.2.4 The Two-Element Boolean Algebra.- 2.2.5 Summary of Examples.- 2.3 The Stone Representation Theorem.- 2.4 The Inclusion-Relation.- 2.4.1 Intervals.- 2.5 Some Useful Properties.- 2.6 n-Variable Boolean Formulas.- 2.7 n-Variable Boolean Functions.- 2.8 Boole's Expansion Theorem.- 2.9 The Minterm Canonical Form.- 2.9.1 Truth-tables.- 2.9.2 Maps.- 2.10 The Lowenheim-Muller Verification Theorem.- 2.11 Switching Functions.- 2.12 Incompletely-Specified Boolean Functions.- 2.13 Boolean Algebras of Boolean Functions.- 2.13.1 Free Boolean Algebras.- 2.14 Orthonormal Expansions.- 2.14.1 Lowenheim's Expansions.- 2.15 Boolean Quotient.- 2.16 The Boolean Derivative.- 2.17 Recursive Definition of Boolean Functions.- 2.18 What Good are "Big" Boolean Algebras?.- 3 The Blake Canonical Form.- 3.1 Definitions and Terminology.- 3.2 Syllogistic & Blake Canonical Formulas.- 3.3 Generation of BCF(f).- 3.4 Exhaustion of Implicants.- 3.5 Iterated Consensus.- 3.5.1 Quine's method.- 3.5.2 Successive extraction.- 3.6 Multiplication.- 3.6.1 Recursive multiplication.- 3.6.2 Combining multiplication and iterated consensus.- 3.6.3 Unwanted syllogistic formulas.- 4 Boolean Analysis.- 4.1 Review of Elementary Properties.- 4.2 Boolean Systems.- 4.2.1 Antecedent, Consequent, and Equivalent Systems.- 4.2.2 Solutions.- 4.3 Reduction.- 4.4 The Extended Verification Theorem.- 4.5 Poretsky's Law of Forms.- 4.6 Boolean Constraints.- 4.7 Elimination.- 4.8 Eliminants.- 4.9 Rudundant Variables.- 4.10 Substitution.- 4.11 The Tautology Problem.- 4.11.1 Testing for Tautology.- 4.11.2 The Sum-to-One Theorem.- 4.11.3 Nearly-Minimal SOP Formulas.- 5 Syllogistic Reasoning.- 5.1 The Principle of Assertion.- 5.2 Deduction by Consensus.- 5.3 Syllogistic Formulas.- 5.4 Clausal Form.- 5.5 Producing and Verifying Consequents.- 5.5.1 Producing Consequents.- 5.5.2 Verifying Consequents.- 5.5.3 Comparison of Clauses.- 5.6 Class-Logic.- 5.7 Selective Deduction.- 5.8 Functional Relations.- 5.9 Dependent Sets of Functions.- 5.10 Sum-to-One Subsets.- 5.11 Irredundant Formulas.- 6 Solution of Boolean Equations.- 6.1 Particular Solutions and Consistency.- 6.2 General Solutions.- 6.3 Subsumptive General Solutions.- 6.3.1 Successive Elimination.- 6.3.2 Deriving Eliminants from Maps.- 6.3.3 Recurrent Covers and Subsumptive Solutions.- 6.3.4 Simplified Subsumptive Solutions.- 6.3.5 Simplification via Marquand Diagrams.- 6.4 Parametric General Solutions.- 6.4.1 Successive Elimination.- 6.4.2 Parametric Solutions based on Recurrent Covers.- 6.4.3 Lowenheim's Formula.- 7 Functional Deduction.- 7.1 Functionally Deducible Arguments.- 7.2 Eliminable and Determining Subsets.- 7.2.1 u-Eliminable Subsets.- 7.2.2 u-Determining Subsets.- 7.2.3 Calculation of Minimal u-Determining Subsets.- 8 Boolean Identification.- 8.1 Parametric and Diagnostic Models.- 8.1.1 Parametric Models.- 8.1.2 The Diagnostic Axiom.- 8.1.3 Diagnostic Equations and Functions.- 8.1.4 Augmentation.- 8.2 Adaptive Identification.- 8.2.1 Initial and Terminal Specifications.- 8.2.2 Updating the Model.- 8.2.3 Effective Inputs.- 8.2.4 Test-Procedure.- 9 Recursive Realizations of Combinational Circuits.- 9.1 The Design-Process.- 9.2 Specifications.- 9.2.1 Specification-Formats.- 9.2.2 Consistent Specifications.- 9.3 Tabular Specifications.- 9.4 Strongly Combinational Solutions.- 9.5 Least-Cost Recursive Solutions.- 9.6 Constructing Recursive Solutions.- 9.6.1 The Procedure.- 9.6.2 An Implementation using BORIS.- A Syllogistic Formulas.- A.1 Absorptive Formulas.- A.2 Syllogistic Formulas.- A.3 Prime Implicants.- A.4 The Blake Canonical Form.

Boolean Reasoning: The Logic of Boolean Equations

Polynomial-time algorithms for generation of prime implicants

A new method is presented for determining all minimal dependence sets, irredundant normal forms, and irredundant normal forms of minimal dependence sets of a Boolean function f. The method reduces the above problems into those of determining all minimal positive dependence sets, irredundant positive normal forms, and irredundant positive normal forms of minimal positive dependence sets, respectively, of a Boolean function f* corresponding to f. For each problem a corresponding cover formula is developed such that the prime implicants of it are in one-to-one correspondence to all possible solutions.

Irredundant Normal Forms and Minimal Dependence Sets of a Boolean Function

This note describes an iterative procedure for generating the prime implicants of switching functions by utilizing a new tabular mode of functional representation called clause-column table. The procedure generates all the prime implicants and can be applied equally well to functions given in the sum-of-products or in the product-of-sums froms, both canonical and noncanonical. The procedure can also be readily adapted to determine the prime implicants of functions having a large number of unspecified or DON'T CARE terms.

Clause-Column Table Approach for Generating All the Prime Implicants of Switching Functions

Fundamentals of threshold logic

One of the basic problems of combinational switching circuit theory is that of designing circuits with a minimum number of AND-gates or prime implicants. Algorithms have been formulated for this purpose which first generate all possible prime implicants corresponding to a specified switching function and then select minimal subsets of these prime implicants for use in the formation of the minimal networks [1]-[6]. In practically all the currently available methods of simplification of switching functions, use is made of the minterm type expression specified in the algebraic or its equivalent binary or decimal form. Operations with binary or decimal numbers have become very popular because of their inherent advantages.

Maxterm Type Expressions of Switching Functions and Their Prime Implicants

[5]-, Decision-Making Processes in Pattern Recogplying Quine's technique4 of iterated consential prime implicants occur and other nition. New York: Macmillan, 1962, Ch. 4. [6] 0. Firschein and M. Fischler, \"Automatic subsensus-taking to the set of prime implicants, dominated columns, if any. The necessary class determination for pattern-recognition appliMott derived a set of implication relations and sufficient conditions of eliminability of cations,\" IEEE Trans. on Electronic Computers (Correspondence), vol. EC-12, pp. 137-141, April that determine for every originally elimina nonessential prime implicant have next 1963. F a J L able prime implicant the necessary and sufbeen found by considering from the minimiz[7J E. Fix and J. L. Hodges, \"Discriminatory analysis: 6.... Non-parametric discrimination,\" USAF School of ficent conditions of its eliminability. The ing function6 corresponding to the reduced Aviation Medicine, Randolph AFB, Tex., Rept. different irredundant forms were than found, prime implicant table only the disjunction 4, 1951, and Rept. 11, 1952, as summarized below [81. on the basis of these implication relations, terms which contain the prime implicant [8] L. N. Kanal and K. K. Nambiar, \"On the applicaby writing a conjunctive form, one conjunct whose eliminability conditions have to be tion of discriminant analysis to identification in aerial photography,\" Philco Sci. Lab., Blue Bell, for each prime implicant, and simplifying it found, deleting the appearance of that parPa., presented at the 7th Nat'l Convention on by the use of distributive laws of Boolean ticular prime implicant from these disjuncMilitary Electronics, Washington', D. C., Septem algebra. In the aforementioned paper by tion terms and then taking the conjunction her 1963. Chang and Mott, it has been shown that the of these reduced disjunction terms. This complete set of irredundant normal equiproduct of sums expression has been simvalents of a formula can be obtained by conplified by the applications of the laws sidering only an \"abbreviated presence funcof Boolean algebra to yield the required sum tion\"-an expression in conjunctive form in of products form which corresponds to the which the different conjuncts give the presset of irredundant consequents implied by ence relations (i.e., those involving eliminthe prime implicant in question. Due to the

A. K. Choudhury

Papers

Maxterm Type Expressions of Switching Functions and Their Prime Implicants

Computing irredundant normal forms from abbreviated presence functions

A Note on Testing and Realization of Threshold Functions

Comment on "Detection of Totally Symmetric Boolean Functions''