Predicate Logic for Software Engineering

1. Predicate Logic for Software Engineering David Lorge Parnas McMaster University, Ontario, Canada

2. Presentation for SwE Readings Class By: Sridhar Pentapati

3. About Dave Parnas • His insights have changed the way that we specify, design, document, build, and maintain software • His techniques have changed modern programming languages • His wisdom has steered software engineering field Progress in a scientific discipline can be measured by how quickly its founders are forgotten

4. Abstract Conventional interpretations of expressions that describe predicates are not suitable for use in software engineering because they do not deal with partial functions. Parnas’ team defines an interpretation for predicate expressions that is suitable for use in software documentation Logic is in the eye of the logician. *Gloria Steinem

5. Introduction • Predicate logic is a development of propositional logic, which we’re all well acquainted with • Sentences in predicate calculus are built up from atomic sentences 'Contrariwise', continued Tweedledee, 'if it was so, it might be, and if it were so, it would be; but as it isn't, it ain't. That's logic!' . *Lewis Carroll

6. Introduction contd... • Why logic in SwE? • Engineers need mathematical tools • for the description and analysis of their products • Mathematical logic is the basis of all proposed tools • A solid foundation of logic notation will be essential for anyone who hopes to be recognized as a software engineer Predicate Logic For Software Engineering

7. Introduction Contd... • Focus • It is essential to have a precise meanings for logical expressions, one that unambiguously yields a value of true or false for every statement of values to the variables that appear in an expression Logic is one thing, the human animal another. You can quite easily propose a logical solution to something and at the same time hope in your heart of hearts it won't work out. * Luigi Pirandello

8. Problems with Existing Logic • Logicians are conservative? Believes Martin Van Emden • Lacks precise meaning for logical expression (* values) • Complexity of expressions • Assumption of total functions Logic: The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. *Ambrose Bierce

9. Parnas Says: • Change predicate logic to allow functions to be partial • To do away with the resulting truth value “undefined” The want of logic annoys. Too much logic bores. Life eludes logic, and everything that logic alone constructs remains artificial and forced. * André Gide

10. Structure of The Paper • Introduction • Basic definitions • Syntax of Logic Expressions • Meanings of Logical Expressions • Examples of the Use of This Logic in Software Documentation • Conclusions • References Predicate Logic For Software Engineering

11. Reasons and Goal • Since practitioners do not want to use methods that require them to use many symbols to say simple things • They will not read expressions that are lengthy or deeply nested • A full, formal definition of a logic that permits concise expressions is a prerequisite for practical use The fact that logic cannot satisfy us awakens an almost insatiable hunger for the irrational. * A. N. Wilson

12. Introduction contd… • In one of his earlier papers’ Parnas reminded us that • functions and relations can be viewed as sets of ordered pairs, • sets can be characterized by predicates and described by logical expressions, • predicates can be represented in more readable way using multidimensional expressions, and • the meaning of these tables can be defined by rules for translating those tables into more conventional expressions Predicate Logic For Software Engineering

13. Introduction Contd… • Many researchers are developing mathematical methods for use by software developers. It is hoped that these methods would do for SwE, what differential and integral calculus did for other areas of engineering From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. * Sir Arthur Conan Doyle

14. Introduction contd…Crux of the problem • Conventional formal interpretations of logical expressions (e.g. [Mendelson]) assume that all functions are total, i.e. defined on a domain that includes all possible values of their arguments • Those interpretations are not intended to deal with partial functions, functions whose value has not been defined for certain values of the argument Predicate Logic For Software Engineering

15. Basics • A partial function ƒ from a set A to a set B is an assignment to each element a in a subset of A, called the domain of definition of ƒ, of a unique element b in B • The sets A and B are called the domain and codomain of ƒ, respectively Predicate Logic For Software Engineering

16. Basics Contd… • We say that ƒ is undefined for elements in A that are not in the domain of definition of ƒ • We write ƒ : A  B to denote that ƒ is a partial function from A to B • When the domain of definitions equals A, we say that ƒ is a total function Logic is like the sword--those who appeal to it, shall perish by it. * Samuel Butler

17. Another interpretation of Predicate Expressions!!! • But why? • Since under conventional interpretations a logical expression that includes partial functions will have a defined value only when the values assigned to all function arguments are within functions’ domains. Such interpretations are of limited usefulness when describing software because we frequently use partial functions to describe the behavior of programs Predicate Logic For Software Engineering

18. Motivating Example • Assume that  represents a function that is defined on a domain containing only non-negative real numbers. A s/w designer may write a boolean expression such as: ((x > 0)  (y = x))  ((x  0)  (y = -x)) (1) to describe the behavior of a computer program Predicate Logic For Software Engineering

19. The writer’s intent seems to be to write an expression that is equivalent to: y = |x| (2)the usual rules for evaluating such expressions require evaluation of all of the functions and relational operators before application of the logical operators to the truth-values that result. For every value of x other than 0, some component of expression (1) is undefined. With the standard interpretation of logical operators, which are defined only for two-value logics, the value of (1) is not defined except when x is assigned the value 0 Predicate Logic For Software Engineering

20. Comparison with Other Work • There is a huge and complex literature on the subject of logic with partial functions • This paper presents a specific proposal and its illustrated use • Confined to software engineering • Dijsktra’s cand • The interpretation for predicate expressions proposed here neither specifies the order of evaluation nor introduces new symbols into logic Predicate Logic For Software Engineering

21. Basic Definitions • Predicate – A predicate is a characteristic or attribute or property that the subject of the statement can have • “No lawyers are shysters ” A function that returns true or false. Conceptually it tests for a condition The property that a bird is an eagle can be expressed by the predicate: • eagle(x) • This predicate holds for all birds x which are eagles. For other birds, the predicate is false Predicate Logic For Software Engineering

22. Propositional Logic • Proposition: A proposition  is a statement that is either true or false, but not both • Washington, D.C., is the capital of the United States of America • 2 + 2 = 3 Proposition 1 is true, whereas 2 is false • More complex propositions can be formed by applying the logical operators (¬, , , etc..) • Propositional logic formula • Eg. ((p  q) ¬c) Predicate Logic For Software Engineering

23. Predicate Logic • Predicate logic allows us to represent fairly complex facts about the world, and to derive new facts in a way that guarantees that, if the initial facts were true then so are the conclusions. It is a well understood formal language, with well-defined syntax, semantics and rules of inference Predicate Logic For Software Engineering

24. Predicate Logic contd… • Predicate logic is obtained by extending propositional logic by using: • Predicates • Variables • Quantifiers Predicate Logic For Software Engineering

25. Quantifiers • Two types of Quantification: • Universal quantification •  • x P(x) - P(x) is true for every x • Existential quantification •  • x P(x) – There is an x for which P(x) is true Predicate Logic For Software Engineering

26. Tuples • A tuple is an ordered list of one or more simple tuples •      U = {true, false} •      Simple n-tuple • –   Ordered list of n members of U • Eg. <true, false, true, false> •    n-tuple • –   Ordered list of n simple tuples Eg. <<true, false>, true, false> You can only find truth with logic if you have already found truth without it

27. Tuples Contd… • When representing specific tuples, we separate the elements with commas and enclose tuples in <brackets> to make their structure clear • E.G. <True, false, true, false> represents a simple 4-tuple and <<true, false>, true, false> represents a 3-tuple that is not a simple 3-tuple • Sk is the set of all simple k-tuples: S1 is U. S is the union of S1, S2,…,Su ; u is the length of the longest tuple needed to apply the semantic model developed below • Tk is the set of all k-tuples. T1 is S. Tk includes Sk . T is the union of T1, T2, …, Tu Predicate Logic For Software Engineering

28. Relations, Functions • Binary relations represent relationships between the elements of two sets • Relations could be from one-to-many, whereas functions can’t be from one-to-many Predicate Logic For Software Engineering

29. Functions and Characteristic Predicate • We refer to two types: • Partial functions: functions whose domain is a proper subset of S • Total functions: functions whose domain includes all the members of S • Characteristic predicate: • For any set of simple tuples, X, the characteristic predicate of that set is a predicate whose domain is S, and whose value, for a simple tuple b, is true if and only if b is a member of X Predicate Logic For Software Engineering

30. The Syntax of Logical Expressions • Built-in functions and Predicates: • the strings f1, ………., fk are the names of functions • View functions as set of pairs • R1, …….., Rm are the names of characterizing sets of simple tuples • View predicates as characterizing sets of simple tuples Predicate Logic For Software Engineering

31. Syntax contd… • Terms: • A function application is a string of the form fj(V). Nothing else is a function • A term is either a member of U, a variable, or a function application. Nothing else is a term Predicate Logic For Software Engineering

32. Syntax contd… • Primitive Expressions: • Primitive Expression is a string of the form Rj(V) • Rj is a characteristic predicate • V is comma separated list of terms (called arguments) Predicate Logic For Software Engineering

33. Syntax contd… • Predicate Expressions • All primitive expressions are predicate expressions. • If P, Q: predicate expressions, x: variable, then following are also predicate expressions: (x , P), (P), (P)  (Q), (P)  (Q), ¬(P) There can never be surprises in logic

34. The Meaning of Logical Expressions • Denotation • each predicate expression is denoted as a set • subset of Su, where u is the number of variables that may appear in the expressions whose meaning is defined. • Assignment • simple u-tuple The mind has its own logic but does not often let others in on it. *Bernard Devoto

35. Meaning contd… • Evaluating terms for a given assignment: Mapping val for a term, t, and assignment, A: • If t is a constant representing t’, val(t, A) is t’ • If t is a variable, xk, val(t, A) is the kth element of the assignment A • If t is a function application, ƒk(V), let • n be the length of V, • if V’ is in the domain of ƒk, val(t, A) is ƒk(V’) • if V’ is not in the domain of ƒk, val(t, A) is * Predicate Logic For Software Engineering

36. Meaning contd… • Evaluating primitive expressions for a given assignment: • For a primitive expression Rj(V), and assignment A • Let Xj be the set of simple tuples characterized by Rj • n be the length of V, • Vi be the ith element of V, and V’ be simple tuple • If V’ is in Xj, tval(Rj(V), A) is true • If V’ is not in Xj, tval(Rj(V), A) is false Predicate Logic For Software Engineering

37. Meaning contd… • The denotation of primitive expressions: • tval(p, A) is true • The denotation of predicate expressions: • If P and Q are predicate expressions: • (xk , P), is the set of all assignments, A,  if c is any value in U, A[k  c] is in the denotation of P • (P)  (Q) is the union of P and Q • (P)  (Q) is the intersection of P and Q, and • ¬(P) is the set of all members of Su that are not in P Predicate Logic For Software Engineering

38. Meaning contd… • Satisfaction of an expression • Expressions that denote empty set are said to be false • Expressions that denote all of Su are said to be true • An expression, e, is said to be satisfied by an assignment, A, if A is a member of the denotation of e Logic is a poor guide compared with custom. *Winston Churchill

39. Meaning contd… • Notational conveniences: • Specify values for some and not for others • e.g. A list such as “x2 : x24” represents all assignments in which the second element is 4 and the 24th element is 96 Man has such a predilection for systems and abstract deductions that he is ready to distort the truth intentionally, he is ready to deny the evidence of his senses only to justify his logic. * Fyodor Mikhailovich Dostoyevsky

40. Use in Documentation • Palindrome • (i, 0 ≤ I < n  (B[l+i]=B[l+n-1-i])) • This expression gives the desired results even though the implication is evaluated outside the domain of B; the domain is characterized by the left-hand implication Logic, like whiskey, loses its beneficial effect when taken in very large quantities. * Lord Dunsany

41. Use in Documentation Relational description of a program that searches B for the value of x

42. Use in Documentation contd… • (i, B[i] = C[i]) • Here we are looking for matching elements in two arrays • When the value of i is outside the index set of either B or C, the value of B[i] = C[i] is false • This is the logic which has been basically presented in this paper Predicate Logic For Software Engineering

43. Conclusions • Not necessary to introduce either a third variable or conditional operators in order to deal with partial functions • Not only is the “motivating example”, eq (1), fully defined using the set-theoretic operations but also greatly simplified: (y = x)  (y = -x) Predicate Logic For Software Engineering

44. Conclusions contd… • Compact readable formulation is crucial • Easier to comprehend • Drawbacks with some complementary predicates – price for allowing partial functions • x > y would not denote the complement x ≤ y. both evaluate to false if either x or y is assigned negative value • a = a cannot be assigned a value of true if U includes negative values A mind all logic is like a knife all blade. It makes the hand bleed that uses it. *Rabindranath Tagore

45. Conclusions contd… • The properties of the functions used should be stated precisely • Axiom of reflection does not hold in this interpretation • Simplification is obtained by making primitive predicates evaluate to false whenever one or more of their arguments are undefined Logic is the art of going wrong with confidence

46. Conclusions contd… Parnas believes that these are proper decisions because: • Keeping logic simple is essential to practical application • The assigned meanings are consistent with intuitive interpretations, and • The formulae that results are relatively simple for cases arising frequently Predicate Logic For Software Engineering

47. Questions And Suggestions!!!!!!! Predicate Logic For Software Engineering

48. References • http://www.cee.hw.ac.uk/~alison/ai3notes • Discrete Mathematics and Its Applications - Kenneth Rosen, Fourth Edition • Software Fundamentals - Collected Papers by David L. Parnas Predicate Logic For Software Engineering