Basic Semantics of Prolog. CAS 706 Program m ing Language. Xinjun Wu mcmaster.ca

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1 Basic Semantics of Prolog CAS 706 Program m ing Language Xinjun Wu wux8@ mcmaster.ca Basic Sem antics ofprolog Logic Program m ing Language 1 Preface Logical programming or constraint programming is a program m ing paradigm in which a set ofattributes thata solution should have are specified rather than setof steps to obtain such a solution.a widely used logicalprogramming language is Prolog.Another is M ercury. Basic Sem antics ofprolog Logic Program m ing Language 2

2 Preface (Cont ) The nam e prolog is an acronym for PROgramming in LO Gic.It was created by Alain Colm eraueraround 1970s.It was an attem ptto make a programming language thatenables to express logic instead ofcarefu ly specifying instructions on the com puter. Basic Sem antics ofprolog Logic Program m ing Language 3 Form alsem antics form alsem antics is the field concerned with the rigorous m athem aticalstudy of the meaning ofprogramming languages and m odels ofcom putation. Basic Sem antics ofprolog Logic Program m ing Language 4

3 Form alsem antics (Cont ) Denotationalsem antics Operationalsem antics Axiom atic sem antics Basic Sem antics ofprolog Logic Program m ing Language 5 OperationalSem antics An operationalsem antics for a particular programming language describes how any particular valid program in the language is interpreted as sequences ofcom putationalsteps. These sequences then are the m eaning ofthe program. Basic Sem antics ofprolog Logic Program m ing Language 6

4 Tw o Prerequisites Concepts Horn clause "O n sentences which are true ofdirect unions ofalgebras",alfred Horn SLD -resolution,sld-tree SLD-resolution principle is the core ofprolog Basic Sem antics ofprolog Logic Program m ing Language 7 Horn Clauses Horn clause is a proposition ofthe general type (p and q and...and t) implies u, w here the num ber ofpropositions com bined by ands is as large as we like Horn Clause Predicate Six-step procedure translate a predicate p into a Horn Clause.(Pg 257 text) Basic Sem antics ofprolog Logic Program m ing Language 8

5 SLD-resolution principle Be summarized as the fo low ing inference rule ( A1... Ai 1 Ai Ai Am) (H B 1... B n) ( A... A B... B A... A ) θ 1 i 1 1 n i+ 1 m A1,..., A m and H, B 1,...,B n are atomic formulas 1 i m and n 0 θ is a most general unifier of A i and H Basic Sem antics ofprolog Logic Program m ing Language 9 SLD-tree?-customer_of(P, sinclair) :- owns(p, O), makes(sinclair, O) :-makes(sinclair, pavillion) :-makes(sinclair, zx81) :-makes(sinclair, spectrum) Basic Sem antics ofprolog Logic Program m ing Language 10

6 Constructive Item s Term s: Constants & Variables & Com pound Term Predicates: Relations Clauses: Statem ents ofprogram Fact: A Clause withoutbody Query: A Clause withouthead,and Others. Prolog = Fact+ Rule Basic Sem antics ofprolog Logic Program m ing Language 11 ControlConstraints None ofusualcontrolconstraints Such as: do,w hile,for loop,goto etc. Cut(!) may actlike Goto,butnot perm anent,w e wilstudy the case withoutcut Basic Sem antics ofprolog Logic Program m ing Language 12

7 Syntactic Categories Variable::= Var Functor::= Func Predicate::= Pred Term ::= Term Term ::= Variable+ Functor(Term 1, Termn) Basic Sem antics ofprolog Logic Program m ing Language 13 Syntactic Categories (Cont ) Atom ::= Predicate(Term 1, Termn) Literal::= Atom +! Goal::= Literal* Clause::= Atom :-G oal Basic Sem antics ofprolog Logic Program m ing Language 14

8 Interpreter of O perationalsem antics The operationalsem antics is given by an interpreter thatrepeatedly transform s a state encoding a leftm ost SLD-tree interp : State x Clause* FSubst FSubstis the setoffinite substitution Basic Sem antics ofprolog Logic Program m ing Language 15 Term inology Frame :: = Atom*, Var p A goalto be solved and a setofvariables FrameList :: = nil Frame :: FrameList Stack :: = nil FrameList, Subst, Clause* :: Stack Substis the currentfinite substitution State :: = Stack Basic Sem antics ofprolog Logic Program m ing Language 16

9 OperationalSem antics --1 interp : State x Clause* FSubst interp( nil, P) = nil Execution term inates when the runtime stack becom es em pty Basic Sem antics ofprolog Logic Program m ing Language 17 OperationalSem antics --2 in te rp ( n i l, φ, C :: S t, P ) = φ :: in te rp ( S t, P ) When the goalto be solved in the current fram e becom es em pty,a solution has been found,the current substitution is therefore returned,and execution backtracks to search for other solution Basic Sem antics ofprolog Logic Program m ing Language 18

10 OperationalSem antics --3 in terp ( F, φ, nil :: St, P ) 0 = in terp ( St, P) if F nil 0 If there are no more program clauses to match Againsta non-em pty goal,execution fails and Backtracking takes place Basic Sem antics ofprolog Logic Program m ing Language 19 OperationalSem antics --4 ( L GV F φ H B C St P) interp ::, ::,, ( : ) :: ::, o ' P = interp( F ::F ::F, θφ,p :: St,P), where H : B = rename(( H : B ), dom( φ)); θ = unify( φ( L), H ) ( fail) V = dom( φ); F = B, V ; F = G,V ; St ' 2 1 P 1 P 1 P = L::G,V :: F, φ, C :: St P 0 0 Unification Basic Sem antics ofprolog Logic Program m ing Language 20

11 O perationalsem antics 4 (Cont ) A non-em pty goaland sequence ofclauses,if the head ofthe first clause,after appropriate renam ing ofclause variables, unifies with the leftmostliteral,the fram e list is extended with a fram e describing the subgoalconsisting ofthe body ofthatclause, the originalgoal,together with the untried clauses,saved on the stack so that alternative solution may be found on backtracking Basic Sem antics ofprolog Logic Program m ing Language 21 OperationalSem antics --5 interp( L :: GV, :: F, φ,( H : - B) :: C :: St, P) P =interp( L:: G, V :: F, φ, C :: St, P), where P 0 0 H : B = rename(( H : B ), dom( φ)), and unify( φ( L), H ) = fail 1 Unification Basic Sem antics ofprolog Logic Program m ing Language 22

12 Supplem entary Definition φ: V T, andw V, Given a finite substitution The projection of φ on W,written Is the finite substitution with dom ain V1 = W ( vars( φ( v))) v w φ W Basic Sem antics ofprolog Logic Program m ing Language 23 OperationalSem antics --6 interp( nil, V :: F, φ, C :: St, P ) P 0 =interp( F, φ V, P :: St, P) 0 P Basic Sem antics ofprolog Logic Program m ing Language 24

13 Bibliography Saum ya K.Debray,Prateek M ishra Denotationaland O perationalsem antics for Prolog,The University of Arizona John Malpas, Prolog:A Relationallanguage and its Applications,1987.ISBN: UlfNilsson and Jan Maluszynski, Logic,Programming and Prolog,1990.ISBN: Basic Sem antics ofprolog Logic Program m ing Language 25 Thank you Basic Sem antics ofprolog Logic Program m ing Language 26