PROLOG PROgramming in LOGic

Transcription

1 PROLOG PROgramming in LOGic 1 Knowledge-Based Information Systems (Relational) database systems are very efficient in the handling of data. Information is data together with a suitable interpretation. Together with data (facts), knowledge-based IS store interpretation rules. The information system can deduce additional data based on these rules. To express facts and rules, we use logic: Propositional Logic 1 : Propositions with a truth value and boolean operators. Predicate Logic 2 : Extends the propositional logic with variables, quantors, and functions to describe rules. 1 Aussagenlogik 2 Prädikatenlogik c 2004 Jens Teubner, André Seifert, University of Konstanz 1

2 2 Implementation Idea: 1 Feed the information system with facts and rules. Use logic to describe them. 2 Ask questions concerning that knowledge: Can be yes or no questions, or the search for all variable bindings for which a given expression evaluates to true. PROLOG is an implementation of such a system. Prolog is not actually designed to work with large datasets. DATALOG is a variant to handle them. c 2004 Jens Teubner, André Seifert, University of Konstanz 2

3 Example: ➀ Feed facts:?- [user]. : cow(betty). : animal(x) :- cow(x). % betty is a cow (fact) % rule: cows are animals ➁ Pose queries:?- cow(betty). % is betty a cow? (query) Yes?- animal(betty). % is betty an animal? (query) Yes?- animal(x). % give me all animals X = betty ; % betty is a valid binding No % there are no more bindings c 2004 Jens Teubner, André Seifert, University of Konstanz 3

4 3 The Prolog Interpreter In the computer lab, the SWI-Prolog interpreter from the University of Amsterdam is installed. You can start it from the Linux command line: $ pl Welcome to SWI-Prolog (Version 5.0.1) Copyright (c) University of Amsterdam. SWI-Prolog comes with ABSOLUTELY NO WARRANTY. This is free software, and you are welcome to redistribute it under certain conditions. Please visit for details. For help, use?- help(topic). or?- apropos(word).?- You can exit the Prolog interpreter with ^D (Control-D) or by entering halt., followed by the enter key. c 2004 Jens Teubner, André Seifert, University of Konstanz 4

5 3.1 Finding Help in Prolog Prolog has an extensive online help utility. You can access it with the predicate help:?- help(consult). consult(+file) Read File as a Prolog source file. File may be a list of files, in which case all members are consulted in turn. [...] If you want to use Prolog at home, go to You can find sources or binaries for Linux, MacOS X and Windows. I can t give you support for this, however. c 2004 Jens Teubner, André Seifert, University of Konstanz 5

6 3.2 The Prolog Prompt Prolog can be in either one of the two operating modes query mode (Prompt?- or ) You can ask questions interactively in this mode. Prolog tries to give you an instance for the variables you specified that match your question or answers with Yes or No. consult mode (Prompt :) You can teach Prolog facts and rules in this mode. Rules are specified as Horn Clauses. After startup, Prolog is in query mode. c 2004 Jens Teubner, André Seifert, University of Konstanz 6

7 3.2.1 Consult Mode Prolog is in consult mode while reading Prolog source files.?- [foo]. reads the file foo.pl in the current working directory. There is a special source file user. If you advise Prolog to read this file, the file is not read from your hard disk, but you can enter facts and rules on the Prolog prompt.?- [user]. : nice(weather). : ^D % user compiled 0.00 sec, 304 bytes Yes?- c 2004 Jens Teubner, André Seifert, University of Konstanz 7

8 4 Propositional Logic in Prolog 4.1 Facts Facts are represented in Prolog by identifiers that start with a lowercase letter, e. g. x. it_is_winter. i_like_prolog. Facts can be structured: is_bad(weather). owns(john, computer). research_group(professor(scholl), researchers(grust, seifert, teubner)). Don t put a space in front of the (. The structure can be arbitrarily deep. Predicates of this type can be interpreted as relations. Note that relations are distinguished by their name and their arity; relations with the same name but different number of parameters are completely distinct. In Prolog the closed world assumption holds: All facts you enter are true, everything else is considered false. c 2004 Jens Teubner, André Seifert, University of Konstanz 8

9 4.2 Rules Rules are facts that only hold under a certain prerequisite. In Prolog they are specified as h(x) :- g(x,y). which corresponds to the horn clause g(x,y) h(x) g(x,y) h(x) All variables are implicitly quantified and quantifiers are omitted. Examples: z :- x. it_is_cold :- it_is_winter. is_nice(weather) :- not(is_bad(weather)). go_hiking :- is_nice(weather), have_time. will_lose(stuttgart) :- is_ill(hildebrand). Conjunctions are written as, in Prolog. Disjunctions are written as ;. c 2004 Jens Teubner, André Seifert, University of Konstanz 9

10 4.3 Queries After providing Prolog with a fact base, you can formulate queries on the knowledge represented by the given facts and rules. A query is formulated the same way as a fact. The system answers either Yes or No, depending on whether the query statement can be reduced to the facts and rules in the fact base. Prolog uses resolution to make its decision.?- it_is_cold. Yes.?- is_nice(weather). No. c 2004 Jens Teubner, André Seifert, University of Konstanz 10

11 4.4 Data Types Prolog does not employ data types in the usual way. We may rather speak about Prolog lexical elements or data objects instead of data types. We can differentiate between two types of data objects: Simple data objects Atoms, Numbers, and Variables. Complex or constructed data objects Terms and Lists. c 2004 Jens Teubner, André Seifert, University of Konstanz 11

12 4.4.1 Atoms Syntax: An atom is a sequence consisting of letters, numbers and underscores, which begins with a lower-case letter (e.g., ab3_, _5C). Some sequences of special characters are already defined (e.g., :-, +,... ). Strings are usually written as a sequence of characters surrounded by quotes (e.g., PrettyWoman ). They are often internally represented as lists of ASCII codes Numbers Many Prolog implementations don t distinguish integers (e.g., 3, -15) from real numbers (e.g., ). SWI Prolog, however, distinguishes between both types. c 2004 Jens Teubner, André Seifert, University of Konstanz 12

13 4.4.3 Variables Syntax: Variables are denoted by a string consisting of letters, numbers and underscore characters, and begin with an upper-case letter (e.g., X). Variables: are not typed, are not declared, are bound (i.e., have a value) either explicitly via matching or implicitly via goal matching or unbound. Variables used only once in a clause do not need a name. Unnamed variables are called anonymous variables. Syntax: An anonymous variable is written as a single underscore ( ). c 2004 Jens Teubner, André Seifert, University of Konstanz 13

14 Values are not reported for anonymous variables appearing in queries.?- [user]. : weather( , cold). Yes?- weather(a,_). A = ; No? Variables in Queries Parts of a query can be replaced by a variable. The interpreter tries to find variable bindings that make the query statement true and prints them to the screen.?- animal(x). X = betty c 2004 Jens Teubner, André Seifert, University of Konstanz 14

15 By typing a semicolon (;) you can ask the system for more solutions. If no more are found, the system prints No. In queries the anonymous variable ( ) only asks for existence of a solution. If you use the anonymous variable more than once, they are distinct variables Variables in Rules Variables can be necessary in facts and rules to state correlations, either within a fact or between left and right side of the :-: : equal(x,x). : animal(x) :- cow(x). : sister(x,y) :- father(x,father), father(y,father), female(x), female(y). : likes(_, mary). c 2004 Jens Teubner, André Seifert, University of Konstanz 15

16 The anonymous variable can be used as an every or don t care variable.?- [user]. : nice(weather(monday)). : not_nice(weather(tuesday)). : go_hiking(day) :- nice(weather(day)). : ^D % user compiled 0.00 sec, 828 bytes Yes?- go_hiking(monday). Yes?- go_hiking(tuesday). No? Terms Terms are the only way Prolog can represent complex data. A term consists of a head, also called functor and parameters (e.g., parentof(hans, michael), orphan(bernd)). c 2004 Jens Teubner, André Seifert, University of Konstanz 16

17 The functor must be an atom and parameters can be structures (e.g. person(name(teubner, jens), ). The number of parameters, so called arity of term, is significant. A term is identified by its head and arity, usually written as functor/arity (e.g., parentof/2). Terms can be pictured as trees. person name jens.teubner@uni-konstanz.de teubner jens c 2004 Jens Teubner, André Seifert, University of Konstanz 17

18 4.4.5 Lists A list is a collection in which the elements come in order there is a first, second, third element, etc., repetition is allowed, i.e., the same element can occur more than once. Syntax: Lists are enclosed in square brackets ([]) and entries are separated by comma which can be any Prolog structure whatsoever. An empty list is denoted by []. Every non-empty list has a head H and a tail T. H is the first element of the list and may or may not be a list. T is what s left from the list after H has been removed and T is always a list. c 2004 Jens Teubner, André Seifert, University of Konstanz 18

19 List Head Tail Number of List Elements [1,2,3] 1 [2,3] 3 [1,[2,3]] 1 [[2,3]] 2 [[[]]] [[]] [] 1 [[[1,2],[3],2,3]] [[1,2],[3],2,3] [] 1 Lists are recursive constructions. The list [1,2,3] is internally represented as.(1,.(2,.(3,[]))). A syntactic shortcut is [H T], which is mostly used to construct rules. Lists can be processed by processing the first element, and then the rest of the list in a recursive manner Useful List Predicates The goal member(x,l) succeeds if X is one of the elements of the list L. c 2004 Jens Teubner, André Seifert, University of Konstanz 19

20 ?- member(x,[x _]). % X is a member if it is the head X = _G154 ; X = _G154 ;... : member(x,[_,t] :- member(x,t). % X is a member if it is in the tail.?- member(x,[_ T]). X = _G154 T = [_G154 _G226] ; X = _G154 T = [_G225, _G154 _G229];... The goal append(l1, L2, L3) succeeds if the list L3 is constructed from lists L1 and L2 by joining them together with L1 first and L2 second.?- append([1,2],[2,3],x). X = [1, 2, 2, 3]; No?- c 2004 Jens Teubner, André Seifert, University of Konstanz 20

21 The goal reverse(l1,l2) reverses the order of the elements in list L1 and appends the result upon the list L2. : reverse([],[]). : reverse([h T], RevList) :- reverse(t, RevTail), append(revtail, [H], RevList). The first fact says what happens to []. The second clause handles non-empty lists and the body is a conjunction with: the first literal being a recursive call using the original list s tail and the second literal being a goal with constructs the final answer from H and the result of the reversed tail of the original list.?- reverse([1,2,3],a). A = [3, 2, 1] c 2004 Jens Teubner, André Seifert, University of Konstanz 21

22 4.4.6 Matching Most important operation on data objects is matching. Matching is invoked in two different ways: Explicitly via the built-in predicate = (e.g., X = 10). Implicitly when Prolog tries to match a goal against the head of a clause in the knowledge base (e.g. matching the query parentof(x,jens) against the knowledge base parentof(bernd,jens), parentof(sabine,jens) succeeds since X can be bound to bernd or sabine). When studying matching in general, i.e., Prolog tries to unify term1 = term2, the following cases can be differentiated: If term1 and term2 are atoms or number values then they will match when they are of the same atom or same number and type of number. c 2004 Jens Teubner, André Seifert, University of Konstanz 22

23 ?- 34 = 34. % 34 matches 34 Yes?- 34 = 34.0 % 34 does not match 34.0 No?- jens = jens. % jens matches jens Yes?- jens = jens. % jens does not match jens No If term1 is some arbitrary term (e.g., atom, number, functor, etc.) and term2 is an unbound variable then term2 will become bound to that value.?- X = parentof(bernd,jens). X = parentof(bernd, jens) If term1 and term2 are variables and both variables are unbound then they share their eventual binding.?- X = Y. X = _G154 Y = _G154 c 2004 Jens Teubner, André Seifert, University of Konstanz 23

24 If term1 and term2 are variables and one of them is bound then the other unbound variable will be bound to the term of the bound variable.?- X = Y, X = X = 4711 Y = 4711 If term1 and term2 are structures then they will match when they have the same functor name, the same arity, and each of the arguments match.?- abc(a,d,e(x,y)) = abc(d,d,e(f,g)). A = d X = f Y = g?- 5+3 = +(5,3). Yes?- abc(a,d,e(x,y)) = abc(d,d,e(f,g,h)). No c 2004 Jens Teubner, André Seifert, University of Konstanz 24

25 4.4.7 Operators In Prolog and for programmers convenience some built-in functors can (but don t have to) be written in infix notation. You can write 5+3 instead of +(5,3). In Prolog each operator belongs to a priority class. When you write a myop1 b myop2 c, the system knows whether it is: myop1(a,myop2(b,c)) or myop2(myop1(a,b),c). If operators belong to the same priority class, its associativity determines how expressions are evaluated. Assume that operator myop is left-associative. When you write a myop b myop c, the system treats this expression as: myop(myop(a, b), c). c 2004 Jens Teubner, André Seifert, University of Konstanz 25

26 Operators in Prolog can be classified into: built-in operators and user-defined operators. Some built-in operators and predicates are: Operator / Predicate term1 = term2 term1 \= term2 term1 == term2 term1 \== term2 term1 =@= term2 term1 \=@= term2 term2 term2 term2 term2 Description unification of term1 and term2; succeeds if the unification succeeds. term1 and term2 must not match term1 and term2 reference the same object negation of term1 == term2 term1 is structurally equal to term2; structural equivalence is weaker than equivalence (==), but stronger than unification (=) negation of term1 =@= term2 term1 is before term2 in the standard order of terms both terms are equal (==) or term1 is before term2 in the standard order of terms term1 is after term2 in the standard order of terms both terms are equal (==) or term1 is after term2 in the standard order of terms c 2004 Jens Teubner, André Seifert, University of Konstanz 26

27 Operators cont d Operator / Predicate Description R := expr R is the result of the computation of expr R is expr the same meaning as R := expr expr1 =:= expr2 succeeds when expr1 evaluates to a number equal to expr2 expr1 =\= expr2 succeeds when expr1 evaluates to a number non-equal to expr2 expr1 < expr2 the value of expr1 is to be smaller than the value of expr2 expr1 > expr2 the value of expr1 is to be larger than the value of expr2 expr1 =< expr2 the value of expr1 is to be equal to or smaller than the value of expr2 expr1 >= expr2 the value of expr1 is to be larger than or equal to the value of expr addition subtraction... *... multiplication... /... division X mod Y X modulo Y -X unary minus +X unary plus c 2004 Jens Teubner, André Seifert, University of Konstanz 27

28 4.4.8 Arithmetic in Prolog { } { } expr1 := Syntax: variable is expr2.?- X is X = 8?- 7 is 5 + X. [Error in arithmetic expression]?- 8 is yes?- rest(x,y,r) :- integer(x), integer(y), R is X mod Y.?- rest(10,7,r). R = 3?- c 2004 Jens Teubner, André Seifert, University of Konstanz 28

29 5 Prolog and Relational Databases When we told you about the Tuple Relational Calculus (TRC), you already heard something like the TRC is the 1st order predicate calculus. Prolog is an implementation of this calculus. Relations are predicates with as many parameters as the relation s attribute number. The database content is stored as facts. actor(1, Damon, Matt, male). actor(2, Potente, Franka, female). actor(3, Cruise, Tom, male). actor(4, Grant, Hugh, male). act_film(1, 3, Jason Bourne ). act_film(2, 3, Marie Kreutz ). The names of all actors? {t a : actor(a) t(name) = a(name)} actor(, Name, ). c 2004 Jens Teubner, André Seifert, University of Konstanz 29

30 All actors with actor id smaller than 3? {t a : actor(a) t(actor id) < 3 t(actor id) = a(actor id) t(name) = a(name) t(sex) = a(sex)} actor(actor id, Name, Sex), Actor id < Joins and Views Joins are expressed like in the TRC: {t a, b : actor(a) act film(b) a(actor id) = b(actor id) t(name) = a(name) t(role) = b(role)} actor(id, Name, ), act film(id,, Role). Views in SQL correspond to rules in Prolog: CREATE VIEW actor_names AS SELECT Name From actor actor_names(name) :- actor(_, Name, _). c 2004 Jens Teubner, André Seifert, University of Konstanz 30

31 5.2 Recursive Queries With Prolog, it is also possible to express recursive queries. : son(otto, franz). : son(franz, heinz). : son(heinz, hugo). : descendant(d, A) :- son(d, A). : descendant(d, A) :- son(d, X), descendant(x, A).?- descendant(x, heinz). X = otto ; X = franz ; No?- c 2004 Jens Teubner, André Seifert, University of Konstanz 31