[...]... of dense linear order) The purpose of this book is to use logic for H.P Williams, Logic and Integer Programming, International Series in Operations Research & Management Science 130, DOI 10.1007/97 8-0 -3 8 7-9 228 0-5 1, C Springer Science+Business Media, LLC 2009 1 2 1 An Introduction to Logic modelling and computational purposes The methods of logic can be used to solve less ambitious problems than the... inference We formalise this in Sects 1.3 and 1.4 where we represent statements symbolically and show how to manipulate them Sometimes this is referred to as symbolic logic For example I s it valid to make the f ollowing in f er ence ? (1.1) Glasgow is in Scotland and England and Scotland are part of Britain Manchester is in England or Scotland (1.2) Therefore Manchester and Glasgow ar e both in Britain (1.3)... one problem, to what extent one uses logic or to what extent one uses more traditional methods There is great advantage, however, in being able to move between the two and recognise the relationships between them In this sense discrete optimisation (usually known as integer programming) and logic are symbiotic 1.2 Logical Inference and Consistency As already stated logic is concerned with deducing statements... atomic statements) They are represented by the symbols ‘↓’ and ‘|’ and are referred to as the connective arrow and the Sheffer stroke, respectively They are defined in the following truth table (Table 1.4) In intuitive terms ‘↓’ and ‘|’ represent the logical connectives ‘nor’ and ‘nand’ (‘not and ), respectively They are sometimes manufactured as logical ‘gates’ for electrical circuits, as discussed in... Example 2.1 A linear programme Maximise 2x1 + 3x2 − x3 (2.1) subject to x1 + x2 3 (2.2) −2 (2.3) −2x1 + x2 − x3 = 0 (2.4) −x1 + 2x3 H.P Williams, Logic and Integer Programming, International Series in Operations Research & Management Science 130, DOI 10.1007/97 8-0 -3 8 7-9 228 0-5 2, C Springer Science+Business Media, LLC 2009 25 ... equivalent expressions and standardise them into normal forms as described below Repeated application of De Morgan’s laws to a statement, in order to negate it, changes all the ‘∨’ connectives to ‘·’ and vice versa and negates the unnegated literals and unnegates the negated ones The resultant statement is said to be the logical dual of the original and vice versa 6 1 An Introduction to Logic Equation (1.12)... statements Equations (1.4) and (1.5) are known as the commutative laws and (1.6) and (1.7) as the distributive laws (They are analogous to similar laws for addition and multiplication in arithmetic, although if ‘∨’ is analogous to addition and ‘·’ to multiplication (1.7) does not apply.) Equations (1.8) and (1.9) are known as De Morgan’s laws They demonstrate a symmetry between the ‘∨’ and ‘·’ connectives... rationals) that lies strictly between −x + 1 and both of 2 x − 2 and − 1 x + 1 The condition for this is that 3 3 2 −x +1< 2 2 x− 3 3 (1.73) and 1 −x +1 . St. London United Kingdom WC2A 2AE h.p .williams@ lse.ac.uk ISBN 97 8-0 -3 8 7-9 227 9-9 e-ISBN 97 8-0 -3 8 7-9 228 0-5 DOI 10.1007/97 8-0 -3 8 7-9 228 0-5 Springer Dordrecht Heidelberg London New York Library of Congress. propositional and predicate calculus. Chapter 2 explains linear and integer programming (LP and IP) using the machinery of logic. It also explains the funda- mental structural and mathematical. (usually known as integer programming) and logic are symbiotic. 1.2 Logical Inference and Consistency As already stated logic is concerned with deducing statements from other state- ments. This is