159.331 Programming Languages & Algorithms

159.331 Programming Languages & Algorithms Lecture 25 - Logic Programming Languages - Part 1 Prog Lang & Alg

Logic languages • More ideas along the trend of letting the programmer specify the problem and the computer is tasked with figuring out how to solve it. • Logic programming based around the formalism of Horn Logic that can specify problems and can also be executed automatically. Prog Lang & Alg

Unfortunately Horn logic does not meet the ultimate goal of declarative programming - namely to avoid having to have a programmer at all! • To achieve reasonable efficiency on real problems the programmer still needs to be aware the machine executes the program and put appropriate extra controls in place. • Horn Logic is the basis of prolog - a system that is the most important logic programming language. Nevertheless there are others… • Also Prolog is a system that has some non-logic features and is no more a pure logic language than say Lisp is a pure functional language. Prog Lang & Alg

Various concepts we will look at: • Horn clauses • Logical variables • Relations • Data structures • Search order • Some examples and Prolog snippets • Some directions for the future… Prog Lang & Alg

Horn Clauses • A Horn clause expresses that a certain condition is true if zero or more other conditions are true. For example: • if a person is old and wise then that person is happy • if X is the father of Y and Y is the father of Z then X is the grandfather of Z • There can be zero or more conditions in the if-part and exactly one conclusion in the then-part • A Horn clause with no conditions expresses afact Prog Lang & Alg

Logic programming involves the programmer giving a list of if-then rules so that the system can infer automatically whether or not a given condition is true. • The system does so by starting with the given condition and working its way back to the facts. • If it can derive a valid way back to the facts then the condition is true and it has proved it. • This is very different from the if-statements we use in imperative programming • We have no flow of control concept in logic programming Prog Lang & Alg

A Horn clause in general can be written as: G0  G1, G2, … , Gn. • The declarative meaning of the Horn clause is: • If G1 until Gn are all true then G0 is true. • The left hand side of the rule is called the head; the right hand side is called the body • G0 until Gn are called goals • Goals in the body are called subgoals • The commas should be read as logical AND symbols • Note also that the  means just “if” - it does not mean “if and only if” - there may be other rules for G0 • Special case n=0 we have a single fact G0 is always true Prog Lang & Alg

Each goal consists of a predicate name and zero or more arguments • Take arguments to be variables or constants - by convention use upper case for variables ie X • We might have a predicate lucky(jim) saying that the constant jim is somehow lucky • A Horn clause example: • old(confucius). (1) • wise(confucius). (2) • happy(X)  old(X), wise(X). (3) • 1 and 2 are facts, the 3rd says generally that old wise people are happy Prog Lang & Alg

In the above, the facts are about a single person - constants • The 3rd is about a variable X and thus holds for all X • We say that variables in facts and rules are universally quantified (sometimes see this written  X, meaning “for all values of X,…”) • Once the system “knows” all the facts we can ask it questions or “pose queries” • ?- happy(charlie). Is a query asking if the constant charlie is happy Prog Lang & Alg

The system will answer our query with either a Yes or a no • (incidentally the ?- is a Prolog syntax convention for a query) • We can also ask ?-happy(X). • If the system finds any X’s that satisfy the goal it will list them. • Variables in a initial goal are said to be existentially qualified - sometimes we write •  X ?… Essentially asking if any X exist for the stated criteria • The system will assign any X’s that match to the variable and we can match or retrieve/print them out Prog Lang & Alg

Executing Horn Clauses • The Horn clauses are executed automatically - by the system • How this is implemented is of course the trick to making an efficient implementation • There might be several rules for solving the initial goal of ?-happy(charlie) happy(X)  rich(X), famous(X). happy(X)  young(X),in_love(X). happy(X)  old(X), wise(X). happy(X)  enrolled_in_159331(X). Prog Lang & Alg

Each one of these four rules is sufficient to proving the initial goal. • Key idea is that the system tries ALL possibilities in whatever order it sees fit until one succeeds (or it has exhausted all possibilities) • If all clauses fail, the whole query fails and the system answers “no” • During search, the system has to prove subgoals. • For instance we might have: • Famous(X)  moviestar(X). • Famous(X)  popstar(X). Prog Lang & Alg

If we had the facts: moviestar(charlie). and rich(charlie). We could prove that charlie is happy. • We have to be careful however not to write infinite loop rules that never terminate. • The pure logic model allows the system a lot of freedom in how it searches its rule base • It can choose which order to try alternative clauses • And what order to try different subgoals • The search order is non-deterministic (we the programmer have not specified it) • It turns out Prolog uses a fixed (known) search order Prog Lang & Alg

The notion of the system just trying another subgoal when it fails to prove another is called backtracking • We can illustrate the search strategy with the tree diagram in 5.1 • The root of the tree is drawn as the initial goal • Below the root are the alternatives for solving the initial goal • At the next level we have the subgoals • The system searches all possible solutions and it will always find a solution if one exists - known as the completeness-of-search property Prog Lang & Alg

Enrolled on 159331 Prog Lang & Alg

Closed Worlds • If system fails to find a answer proving that charlie is happy it will return a no • We take this to mean that as far as the system can tell charlie is not happy and the goal is not true • This assumes that goals that can be proven are true and goals that cannot are false - known as the closed world assumption • (ie we ignore facts outside our system that we do not know about) • So note that our query answers are only as good as our knowledge-base allows Prog Lang & Alg

The above is a simplification - the goals can in fact have arguments so an actual implementation is more complex • Horn logic is a subset of a more general logic known as predicate logic - which allows multiple conclusions • Restricting ourselves to Horn logic makes it possible to execute clauses automatically Prog Lang & Alg

Logical variables • Used for storing state information and for passing arguments between goals happy(X)  old(X), wise(X). • Initially a logical variable does not contain a value - we say it is unbound • It can contain a value and become bound if it is matched against a value or another variable • After it is bound it cannot be changed (except during backtracking - as we describe later) Prog Lang & Alg

Recall that a clause containing variable X is a general rule about any unspecified X • Within the rule, X must refer to the same object consistently • While executing the clause we can thus make a decision about Xexactly once • So logical variables are bound once only and are thus closer to the concept of mathematical variables rather than the state variables we are used to in imperative programming systems. • Most logic languages are typeless - we do not need to declare X - it is done implicitly Prog Lang & Alg

Executing Queries with Predicates coauthor(X,Y)  author(X,Book),author(Y,Book). author(ullman, database_book). author(ullman, awk_book). author(kernighan, awk_book). author(aho, compiler_book). author(sethi, compiler_book). author(ullman, compiler_book). • First rule means that two people are coauthors if there exists a book of which both are an author. • The other clauses are facts about some books & authors. Prog Lang & Alg

We want to answer a query like: ?- coauthor(ullman,aho). • Which asks if Ullman and Aho are coauthors of any book. • The system needs to find a clause whose head matches the query and whose subgoals can be solved. • The matching rule can be stated: • A head and a goal match if they are equal or if they can be made equal by binding one or more variables in the head or goal. • This is called unification of the head and the goal Prog Lang & Alg

For the head and the goal to be equal they must have the same predicate name and the same number of arguments • Additionally, each argument in the head must be equal (or made equal) to the corresponding argument in the goal. • If two arguments denote the same value they are equal • If one of the two arguments is unbound, equality will arise if we bind the unbound to the other argument • If they are both unbound, they are bound to each other • If one later becomes bound then so (automatically) does the other - to the same value Prog Lang & Alg

In our example, only the first rule can possibly match the query coauthor(ullman,aho). • The head of the rule coauthor(X,Y), can be made equal to the initial goal by binding X to ullman and Y to aho. • The query can thus be answered by solving the subgoals in the body of the clause, after replacing X and Y with the values to which they were just bound: author(ullman,Book), author(aho,Book). Prog Lang & Alg

System might try to solve author(ullman,Book) first • Tries to bind Book to the constant database_book • But cannot solve author(aho,database_book) • System then backtracks over its last decision • Unbinds Book and look for another fact that matches the goal • author(ullman,compiler_book) is tried, binding Book to compiler_book • New attempt to match the subgoal author(aho,compiler_book) succeeds and so the whole query succeeds. • Same search process as in our simpler example but the unification of variables complicates the matching rules. Prog Lang & Alg

Logic Programming Part 1 - Summary • Horn Clauses • Simple facts and queries • Logical Variables • Unification of head and goal variables • See Bal & Grune Chapter 5, Sebesta Chapter 16 • Next - Logic relations and Data Structures Prog Lang & Alg

159.331 Programming Languages & Algorithms