of Prolog An Overview 1.1 An example program: defining family relations

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1 An Overview f Prlg This chaptereviews basic mechanisms f Prlg thrugh an example prgram. Althugh the treatment is largely infrmal many imprtant cncepts are intrduced. 1.1 An example prgram: defining family relatins Prlg is a prgramming language fr symblic, nn-numeric cmputatin. It is specially well suited fr slving prblems that invlve bjects and relatins between bjects. Figure 1.1 shws an example: a family relatin. The fact that Tm is a parent f Bb can be written in Prlg as: parent( tm, bb). Here we chse parent as the name f a relatin; tm and bb are its argu- Figure l.l A family tree.

2 4 PROLOG PROGRAMMING FOR ARTIFICIAL INTELLIGENCE ments. Fr reasns that will becme clear later we write names like tm with an initial lwer-case letter. The whle family tree f Figure 1.1 is defined by the fllwing Prlg prgram: parent( pam, bb). parent( tm, bb). parent( tm, liz). parent( bb, ann). parent( bb, pat). parent( pat, jim). This prgram cnsists f six clauses. Each f these clauses declares ne fact abut the parent relatin. When this prgram has been cmmunicated t the Prlg system, Prlg can be psed sme questins abut the parent relatin. Fr example, Is Bb a parent f Pat? This questin can be cmmunicated t the Prlg system by typing int the terminal:?- parent( bb, pat). Having fund this as an asserted fact in the prgram, Prlg will answer: yes A further query can be:?- parent( liz, pat). Prlg answers n because the prgram des nt mentin anything abut Liz being a parent f Pat. It als answers 'n't the questin?- parent( tm, ben). because the prgram has nt even heard f the name Ben. Mre interesting questins can als be asked. Fr example: Wh is Liz's parent??- parent( X, liz). Prlg's answer will nt be just 'yes' r'n' this time. Prlg will tell us what is the (yet unknwn) value f X such that the abve statement is true. S the

3 AN OVERVIEW OF PROLOG answer ls: X: tm The questin Wh are Bb's children? can be cmmunicated t Prlg as:?- parent( bb, X). This time there is mre than just ne pssible answer. Prlg first answers with ne slutin: X:ann We may nw want t see ther slutins. We can say that t Prlg (in mst Prlg implementatins by typing a semicln), and Prlg will find ther answers: ;g - pat If we request mre slutins again, Prlg will answer 'n' because all the slutins have been exhausted. Our prgram can be asked an even brader questin: Wh is a parent f whm? Anther frmulatin f this questin is: Find X and Y such that X is a parent f Y. This is expressed in Prlg by:?- parent{ X, Y). Prlg nw finds all the parent-child pairs ne after anther. The slutins will be displayed ne at a time as lng as we tell Prlg w want mre slutins, until all the slutins have been fund. The answers ar utput as: X-pam Y : bb; X:tm Y : bb; X:tm Y : liz; We can stp the stream f slutins by typing, fr example, a perid instead f a semicln (this depends n the implementatin f Prlg).

4 PROLOG PROGRAMMING FOR ARTIFICIAL INTELLIGENCE Figure 1.2 The grandparent relatin expressed as a cmpsitin f tw parent relatins. Our example prgram can be asked still mre cmplicated questins like: Wh is a grandparent f Jim? As ur prgram des nt directly knw the grandparent relatin this query has t be brken dwn int tw steps, as illustrated by Figure 1.2. (1) Wh is a parent f Jim? Assume that this is sme Y. (2) Wh is a parent f Y? Assume that this is sme X. Such a cmpsed query is written in Prlg as a sequence f tw simple nes:?- parent( Y, jim), parent( X, Y). The answer will be: X:bb Y-pat Our cmpsed query can be read: Find such X and Y that satisfy the fllwing tw requirements: parent( Y, jim) and parent( X, Y) If we change the rder f the tw requirements the lgical meaning remains the same: parent( X, Y) and parent( Y, jim) We can indeed d this in ur Prlg prgram and the query?- parent( X, Y), parent( Y, jim). will prduce the same result. In a similar way we can ask: Wh are Tm's grandchildren??- parent( tmn X), parent( X, Y).

5 AN OVERVIEW OF PROLOG Prlg's answers are: X:bb Y : ann; X:bb Y-pat Yet anther questin culd be: D Ann and Pat have a cmmn parent? This can be expressed again in tw steps: (1) Wh is a parent, X, f Ann? (?) Is (this same) X a parent f Pat? The crrespnding questin t Prlg is then:?- parent( X, ann), parent( X, pat). The answer is: X:bb Our example prgram has helped t illustrate sme imprtant pints: It is easy in Prlg t define a relatin, such as the parent relatin, by stating the n-tuples f bjects that satisfy the relatin. The user can easily query the Prlg system abut relatins defined in the prgram. A Prlg prgram cnsists f clauses. Each clause terminates with a full stp. The arguments f relatins can (amng ther things) be: cncrete bjects, r cnstants (such as tm and ann), r general bjects such as X and Y. Objects f the first kind in ur prgram are called atms. Objects f the secnd kind are called variables.. Questins t the system cnsist f ne r mre gals. A sequence f gals, such as parent( X, ann), parent( X, pat) means the cnjunctin f the gals: X is a parent f Ann, and X is a parent f Pat. The wrd 'gals' is used because Prlg accepts questins as gals that are t be satisfied. An answer t a questin can be either psitive r negative, depending n

6 PROLOG PROGRAMMING FOR ARTIFICIAL INTELLIGENCE whether the crrespndingal can be satisfied r nt. In the case f a psitive answer we say that the crrespnding gal was satisfiable and that the galsucceeded. Otherwise the gbalwas unsatisfiable anditfailed. If several answers satisfy the questin then Prlg will find as many f them as desired by the user. Exercises 1.1 Assuming the parent relatin as defined in this sectin (see Figure 1.1), what will be Prlg's answers t the flrwing questins? (a) (b) (') (d) parent( jim, X). parent( X, jim). parent( paffi, X), parent( X, pat). parent( paffi, X), parent( X, Y), parent( Y, jim). 1.2 Frmulate in Prlg the fllwing questins abut the parent relatin: (a) Wh is Pat's parent? (b) Des Liz have a child? (c) Wh is Pat's grandparent? 1.2 Extending the example prgram by rules Our example prgram can be easily extended in many interesting ways. Let us first add the infrmatin n the sex f the peple that ccur in the parent relatin. This can be dne by simply adding the fllwing facts t ur prgram: female( pam). male( tm). male( bb). female( liz). female( pat). female( ann). male( jim). The relatins intrduced here are male and female. These relatins are unary (r ne-place) relatins. A binary relatin like parent defines a relatin between pairs f bjects; n the ther hand, unary relatins can be used t declare simple yes/n prperties f bjects. The first unary clause abve can be read: Pam is a female. We culd cnvey the same infrmatin declared in the tw unary relatins with ne binary relatin, sex, instead. An alternative piece

7 AN OVERVIEW OF PROLOG f prgram wuld then be: sex( pam, feminine). sex( tm, masculine). sex( bb, masculine). As ur next extensin t the prgram let us intrduce the ffspring relatin as the inverse f the parent relatin. We culd define ffspring in a similar way as the parent relatin; that is, by simply prviding a list f simple facts abut the ffspring relatin, each fact mentining ne pair f peple such that ne is an ffspring f the ther. Fr example: ffspring( liz, tm). Hwever, the ffspring relatin can be defined much mre elegantly by making use f the fact that it is the inverse f parent, and that parent has already been defined. This alternative way can be based n the fllwing lgical statement: Fr all X and Y, Y is an ffspring f X if X is a parent f Y. This frmulatin is already clse t the frmalism f Prlg. The crrespnding Prlg clause which has the same meaning is: ffspring( Y, X) :- parent( X, Y). This clause can als be read as: Fr all X and Y, if X is a parent f Y then Y is an ffspring f X. Prlg clauses such as ffspring( Y, X) :- parent( X, Y). are called rules. There is an imprtant difference between facts and rules. A fact like parent( tm, liz). is smething that is always, uncnditinally, true. On the ther hand, rules specify things that may be true if sme cnditin is satisfied. Therefre we say that rules have: a cnditin part (the right-hand side f the rule) and

8 10 PROLOG PROGRAMMING FOR ARTIFICIAL INTELLIGENCE a cnclusin part (the left-hand side f the rule). The cnclusin part is als called the headf a clause and the cnditin part the bdy f a clause. Fr example: ffspring( y, X) :- parent( X, y). head bdy If the cnditin parent( X, Y) is true then a lgical cnsequence f this is ffspring( Y, X). Hw rules are actually used by Prlg is illustrated by the fllwing example. Let us ask ur prgram whether Liz is an ffspring bt rm:?- ffspring( liz, tm). There is n fact abut ffsprings in the prgram, therefre the nly way t cnsider this questin is t apply the rule abut ffsprings. The rule ii genlral in the sense that it is applicable t any bjects X and Y; therefre it can als be applied t such particular bjects asliz and tm. T apply the rule t liz and tm, Y has t be substituted with liz, and X with tm. We sav that the variables X and Y becme instantiated t: X=tm and Y:liz After the instantiatin we have btained a special case f ur general rule. The special case is: ffspring( liz, tm) :- parent( tm, liz). The cnditin part has becme parent( tm, Iiz) Nw Prlg tries t find ut whether the cnditin part is true. S the initial gal ffspring( liz, tm) has been replaced with the subgal parent( tm, liz) This (new) gal happens t be trivial as it can be fund as a fact in ur prgram. This means that the cnclusin part f the rule is als true, and Prlg will answer the questin with yes. Let us nw add mre family relatins t ur example prgram. The

9 parent I 9.:'"' r I mther AN OVERVIEW OF PROLOG 11.*"8' -.-rb grandparent Figure 1.3 Definitin graphs fr the relatins ffspring, mther and grandparent in terms f ther relatins. specificatin f the mther relatin can be based n the fllwing lgical statement: Fr all X and Y. X is the mther f Y if X is a parent f Y and X is a female. This is translated int Prlg as the fllwing rule: mther( X, Y) :- parent( X, Y), female( X). A cmma between tw cnditins indicates the cnjunctin f the cnditins, meaning that bth cnditins have t be true. Relatins such as parent, ffspring and mther can be illustrated by diagram such as thse in Figure 1.3. These diagrams cnfrm t the fllwing cnventins. Ndes in the graphs crrespnd t bjects - that is, arguments f relatins. Arcs between ndes crrespnd t binary (r tw-place) relatins. The arcs are riented s as t pint frm the first argument f the relatin t the secnd argument. Unary relatins are indicated in the diagrams by simply marking the crrespnding bjects with the name f the relatin. The relatins that are being defined are represented by dashed arcs. S each diagram shuld be understd as fllws: if relatins shwn by slid arcs hld, then the relatin shwn by a dashed arc als hlds. The grandparent relatin can be, accrding t Figure 1.3, immediately written in Prlg as: grandparent(x, Z) :- parent( X, Y), parent( Y, Z). At this pint it will be useful t make a cmment n the layut f ur prgrams. Prlg gives us almst full freedm in chsing the layut f the prgram. S we can insert spaces and new lines as it best suits ur taste. In general we want t make ur prgrams lk nice and tidy, and, abve all, easy t read. T this end we will ften chse t write the head f a clause and each

10 12 PROLOG PROGRAMMING FOR ARTIFICIAL INTELLIGENCE gal f the bdy n a separate line. When ding this, we will indent gals in rder t make the difference between the head und th. gals mre visible. Fr example, the grandparent rule wuld be, accrding t this cnventin, written as fllws: grandparent(x, Z) :- parent( X, Y), parent( Y, Z). Figure 1.4 illustrates the sister relatin: Fr any X and Y, X is a sister f Y if (1) bth X and Y have the same parent, ancl (2)Xisafemale. female Figure 1.4 Defining the sister relatin. The graph in Figure L.4 can be translated int Prlg as: sister( X, Y) parent( Z, parent( Z, female( X). Ntice the way in which the requirement'bth X and Y have the same parent' has been expressed. The fllwing lgical frmulatin was used: sme Z must be a parent f X, and this same Z must be a parent f Y. An alternative, but less elegant way wuld be t say: Z1.is a parent f X, andz2is a parent f y, and Zt is equal t 22. We can nw sister( ann, pat). The answer will be as expected (see Figure 1.1). Therefre we might

11 AN OVERVIEW OF PROLOG 13 cnclude that the sister relatin, as defined, wrks crrectly' There is, hwever, a rather subtle flaw in ur prgram which is revealed if we ask the questin Wh is Pat's sister?:?- sister( X, pat). Prlg will find tw answers, ne f which may cme as a surprise: X : ann; X-pat U:/BR S, Pat is a sister t herself?! This is prbably nt what we had in mind when defining the sister relatin. Hwever, accrding t ur rule abut sisters Prlg's answer is perfectly lgical. Our rule abut sisters des nt mentin that X and Y must nt be the same if X is t be a sister f Y. As this is nt required Prlg (rightfully) assumes that X and Y can be the same, and will as a cnsequence find that any female wh has a parent is a sister f herself. T crrect ur rule abut sisters we have t add that X and Y must be different. We will see in later chapters hw this can be dne in several ways, but fr the mment we will assume that a relatin different is already knwn t Prlg, and that different( X, Y) is satisfied if and nly if X and Y are nt equal. An imprved rule fr the sister relatin can then be: sister( X, Y) :- parent( Z, X), parent( Z, Y), female( X), different( X, Y). Sme imprtant pints f this sectin are: O a Prlg prgrams can be extended by simply adding new clauses. Prlg clauses are f three types: facts, rules and questins. Facts declare things that are always, uncnditinally true. Rules declare things that are true depending n a given cnditin. By means f questins the user can ask the prgram what things are true. Prlg clauses cnsist f the headand the bdy.the bdy is a list f" gals separated by cmmas. Cmmas are understd as cnjunctins. Facts are clauses that have the empty bdy. Questins nly have the bdy. Rules have the head and the (nn-empty) bdy.

12 t4 PROLOG PROGRAMMING FOR ARTIFICIAL INTELLIGENCE In the curse f cmputatin, a variable can be substituted by anther bject. we say that a variable becmes instantiated. Variables are assumed t be universally quantified and are read as.fr all'. Alternative readings are, hwever, pssible fr variables that appear nly in the bdy. Fr example hasachild( X) :- parent( X, y). can be read in tw ways: (u) Fr all X and Y, if X is a parent f Y then X has a child. (b) Fr all X, X has a child if there is sme Y such that X is a parent f y. Exercises 1.3 Translate the fllwing statements int prlg rules: (a) Everybdy wh has a child is happy (intrduce a ne-argument relatin happy). (b) Fr all X, if X has a child wh has a sister then X has tw children (intrduce new relatin hastwchildren). 1.4 Define the relatin grandchild using the parent relatin. Hint: It wilt be similar t the grandparent relatin (see Figure 1.3). 1.5 Define the relatin aunt( X, Y) in terms f the relatins parent and sister. As an aid yu can first draw a diagram in the style f Figure 1.3 fr the aunt relatin. 1.3 A recursive rule definitin Let us add ne mre relatin t ur family prgram, the predecessrelatin. This relatin will be defined in terms f the parent relatin. The whle definitin can be expressed with tw rules. The first rule will define the direct (immediate) predecessrs and the secnd rule the indirect predecessrs. We say that sme X is an indirect predecessr f sme Z if. there is a parentship chain f peple between X and Z, as illustrated in Figure In ur example f Figure 1.1, Tm is a direct predecessr f Liz and an indirect predecessr f Pat.

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