# Reasoning and programming. Lecture 5: Invariants and Logic. Boolean expressions. Reasoning. Examples

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1 Chir of Softwre Engineering Resoning nd progrmming Einführung in die Progrmmierung Introduction to Progrmming Prof. Dr. Bertrnd Meyer Octoer 2006 Ferury 2007 Lecture 5: Invrints nd Logic Logic is the sis of Mthemtics: proofs re only vlid if they follow the rules of logic. Softwre development: Conditions in contrcts: x must not e zero, so tht we cn clculte x+ 7 x. Conditions in progrm ctions: If i is positive, then execute this instruction. (to e introduced in lter lecture) Intro. to Progrmming, lecture 1: Overview 4 Boolen expressions A condition is expressed s oolen expression. It consists of oolen vriles (identifiers denoting oolen vlues) oolen opertors (not, or, nd, =, implies) nd represents possile oolen vlues (truth vlues, either or ). 2 Intro. to Progrmming, lecture 1: Overview 5 Resoning Exmples Progrmming is resoning. Logic is the science of resoning. We use logic in our every dys life: Socrtes is humn. All humns re mortl. Therefore Socrtes must e mortl. Exmples of oolen expressions (with rin_tody nd cuckoo_sng_lst_night s oolen vriles): rin_tody ( oolen vrile is oolen expression) not rin_tody (not cuckoo_sng_lst_night) implies rin_tody (Prentheses group su-expressions.) Intro. to Progrmming, lecture 1: Overview 3 Intro. to Progrmming, lecture 1: Overview 6 1

2 Negtion (not) Complex expressions not For ny oolen expression e nd ny vlues of vriles: Exctly one of e nd not e hs vlue. Exctly one of e nd not e hs vlue. One of e nd not e hs vlue. (Principle of the Excluded Middle.) Not oth of e nd not e hve vlue. (Principle of Non-Contrdiction.) Build more complex oolen expressions y using the oolen opertors. Exmple: nd ( nd (not c)) Intro. to Progrmming, lecture 1: Overview 7 Intro. to Progrmming, lecture 1: Overview 10 Disjunction (or) Truth ssignment nd truth tle or Truth ssignment for set of vriles: prticulr choice of vlues ( or ), for every vrile. or opertor is non-exclusive. or opertor is commuttive. Disjunction principle: An or disjunction hs vlue except if oth opernds hve vlue. A truth ssignment stisfies n expression if the vlue for the expression is. A truth tle for n expression with n vriles hs n+1 columns 2 n rows Intro. to Progrmming, lecture 1: Overview 8 Intro. to Progrmming, lecture 1: Overview 11 Conjunction (nd) Comined truth tle for sic opertors nd not or nd nd opertor is commuttive. Dulity of nd nd or: properties of either opertor yield properties of other (negting + swpping nd ) Conjunction principle: An nd conjunction hs vlue except if oth opernds hve vlue. Intro. to Progrmming, lecture 1: Overview 9 Intro. to Progrmming, lecture 1: Overview 12 2

3 Tutologies De Morgn s lws Tutology: oolen expression tht hs vlue for every possile truth ssignment. Exmples: or (not ) not ( nd (not )) ( nd ) or ((not ) or (not )) De Morgn s Lws: Tutologies (not ( or )) = ((not ) nd (not )) (not ( nd )) = ((not ) or (not )) More tutologies: ( nd ( or c)) = (( nd ) or ( nd c)) ( or ( nd c)) = (( or ) nd ( or c)) Intro. to Progrmming, lecture 1: Overview 13 Intro. to Progrmming, lecture 1: Overview 16 Contrdictions Contrdiction: oolen expression tht hs vlue for every possile truth ssignment. Exmples: nd (not ) Stisfile: for t lest one truth ssignment the expression yields. Any tutology is stisfile. No contrdiction is stisfile. Binding Order of inding (strting with tightest inding): not, nd, or, implies (to e introduced), =. nd nd or re ssocitive: nd ( nd c) = ( nd ) nd c or ( or c) = ( or ) or c Style rules: When writing oolen expression, drop the prentheses: Around the expressions of ech side of = if whole expression is n equivlence. Around successive elementry terms if they re seprted y the sme ssocitive opertors. Intro. to Progrmming, lecture 1: Overview 14 Intro. to Progrmming, lecture 1: Overview 17 Equivlence (=) = = opertor is commuttive ( = hs sme vlue s = ). = opertor is reflexive ( = is tutology for ny ). Sustitution: For ny expressions u, v nd e, if u = v is tutology nd e is the expression otined from e y replcing every occurrence of u y v, then e = e is tutology. Impliction (implies) implies implies, for ny nd, is the vlue of (not ) or In implies : is ntecedent, consequent Impliction principle: An impliction hs vlue except if its ntecedent hs vlue nd its consequent hs vlue In prticulr, lwys if ntecedent is Intro. to Progrmming, lecture 1: Overview 15 Intro. to Progrmming, lecture 1: Overview 18 3

4 Impliction in ordinry lnguge implies in ordinry lnguge often mens custion, s in if then Reversing implictions (2) Correct: implies = (not ) implies (not ) If the wether stys like this, skiing will e gret this week-end. If you put this stuff in your hnd luggge, they won t let you through. Exmple: All the people who live ner the lke re rich. She is not rich, so she cn t e living in Küsncht live_ner_lke implies rich = (not rich) implies (not live_ner_lke ) Intro. to Progrmming, lecture 1: Overview 19 Intro. to Progrmming, lecture 1: Overview 22 Misunderstnding implictions Impliction Whenever is, implies is, regrdless of : If tody is Wednesdy, 2+2=5. If 2+2=5, tody is Wednesdy. Both of the ove implictions re. Cses in which is tell us nothing out the truth of the consequent. Intro. to Progrmming, lecture 1: Overview 20 Intro. to Progrmming, lecture 1: Overview 23 Reversing implictions (1) It is not generlly true tht implies = (not ) implies (not ) Exmple (wrong!): All the people in Zurich who live ner the lke re rich. I do not live ner the lke, so I m not rich. live_ner_lke implies rich [1] Semistrict oolen opertors (1) Exmple oolen-vlued expression (x is n integer): x +7 x >1 for x <= -7 Undefined for x = 0 (not live_ner_lke ) implies (not rich ) [2] Intro. to Progrmming, lecture 1: Overview 21 Intro. to Progrmming, lecture 1: Overview 24 4

5 Semistrict oolen opertors (2) Ordinry vs. non-strict oolen opertors BUT: Division y zero: x must not e 0. (x /= 0) nd ( x +7 > 0 ) x Use Ordinry oolen opertors (nd nd or) if you cn gurntee tht oth opernds re defined. nd then, if condition only mkes sense when nother is true. or else, if condition only mkes sense when nother is flse. Exmple: If you re not single, then your spouse must sign the contrct. is_single or else spouse_must_sign Intro. to Progrmming, lecture 1: Overview 25 Intro. to Progrmming, lecture 1: Overview 28 Semistrict oolen opertors (3) Non-strict impliction BUT: nd is commuttive (progrm would crsh). We need non-commuttive version of nd (nd or): Non-strict oolen opertors. Exmple: If you re not single, then your spouse must sign the contrct. (not is_single) implies spouse_must_sign Definition of implies: in our cse, lwys non-strict! implies = (not ) or else Intro. to Progrmming, lecture 1: Overview 26 Intro. to Progrmming, lecture 1: Overview 29 Non-strict opertors (nd then, or else) Eiffel keywords nd mthemticl symols nd then : hs sme vlue s nd if nd re defined, nd hs whenever hs vlue. or else : hs sme vlue s or if nd re defined, nd hs whenever hs vlue. (x /= 0) nd then ( x +7 ) x >0 Non-strict opertors llow us to define n order of expression evlution (left to right). Importnt for progrmming when undefined ojects my cuse progrm crshes. Eiffel keyword not or nd = implies Common mthemticl symol ~ or Intro. to Progrmming, lecture 1: Overview 27 Intro. to Progrmming, lecture 1: Overview 30 5

6 Propositionl nd predicte clculus Existentilly quntified expression Propositionl clculus: property p holds for single oject Predicte clculus: property p holds for severl ojects Boolen expression: s : SOME_SET s.some_property if nd only if t lest one memer of SOME_SET stisfies property some_property Proving : Find one element of SOME_SET tht stisfies the property : Prove tht no element of SOME_SET stisfies the property (test ll elements) Intro. to Progrmming, lecture 1: Overview 31 Intro. to Progrmming, lecture 1: Overview 34 Generlizing or Universlly quntified expression G : group of ojects, p : property or: Does t lest one of the ojects in G stisfy p? Is t lest one sttion of Line 8 n exchnge? Sttion_Blrd.is_exchnge or Sttion_Lourmel.is_exchnge or Sttion_Boucicut.is_exchnge or (ll sttions of Line 8) Existentil quntifier: exists, or s : Sttions_8 s.is_exchnge There exists n s in Sttions_8 such tht s.is_exchnge is true Boolen expression: s: SOME_SET s.some_property if nd only if every memer of SOME_SET stisfies property some_property wht out empty sets? not ( s: SOME_SET not s.some_property) Proving : Prove tht every element of SOME_SET stisfies the property (test ll elements) : Find one element of SOME_SET tht does not stisfies the property Intro. to Progrmming, lecture 1: Overview 32 Intro. to Progrmming, lecture 1: Overview 35 Generlizing nd Dulity nd: Does every oject in G stisfy p? Are ll sttions of Trm 8 exchnges? Sttion_Blrd.is_exchnge nd Sttion_Lourmel.is_exchnge nd Sttion_Boucicut.is_exchnge nd (ll sttions of Line 8) Generliztion of DeMorgn s lws: not ( s : SOME_SET P ) = s : SOME_SET not P not ( s : SOME_SET P ) = s : SOME_SET not P Universl quntifier: for_ll, or s: Sttions_8 s.is_exchnge For ll s in Sttions8 s.is_exchnge is true Intro. to Progrmming, lecture 1: Overview 33 Intro. to Progrmming, lecture 1: Overview 36 6

7 Empty sets s : SOME_SET some_property with SOME_SET empty lwys s : SOME_SET some_property with SOME_SET empty lwys Intro. to Progrmming, lecture 1: Overview 37 Key concepts Logic s tool for resoning Boolen opertors: truth tles Properties of oolen opertors: don t use truth tles! Predicte clculus: to tlk out logicl properties of sets Non-strict oolen opertors Intro. to Progrmming, lecture 1: Overview 38 End of lecture 5 Intro. to Progrmming, lecture 1: Overview 39 7

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