1. CS 330 Programming Languages 11 / 25 / 2008 Instructor: Michael Eckmann

2. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Today’s Topics Questions / comments? Logic Languages / Chap. 16

3. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Logic (or Declarative) languages –Express programs in a form of symbolic logic –Produce results via logical inferences Logic programs –specify the desired results –instead of creating the procedures (or functions) to produce them Since we are writing no procedures, the syntax of logic languages is quite different from imperative (e.g. C, C++, Java) or even Functional (e.g. LISP, Scheme)‏

4. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Prolog is the only widely used Logic Programming Language Logic languages are based on formal logic –Like the axiomatic semantics we saw when we dealt with semantic and correctness proofs –Recall the weakest precondition computations we did

5. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Symbolic logic can be used for the basic needs of formal logic: –express propositions (defined on next slide)‏ –express relationships between propositions –describe how new propositions can be inferred from other propositions Particular form of symbolic logic used for logic programming called predicate calculus

6. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Some terminology Proposition --- logical stmt that may or may not be true The statement consists of objects and relations between terms. –Terms are either constants or variables An atomic proposition consists of compound terms –Compound term – one element of a mathematical relation - -- looks like mathematical function notation with names and parentheses. Two parts are: –Functor –Ordered list of parameters

7. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Compound term –The compound terms are also called tuples. 1-tuples, 2- tuples, 3-tuples, 4-tuples... Depending on the number of parameters Compound term examples: student(jon)‏ like(seth, OSX)‏ like(nick, windows)‏ like(jim, linux)‏ The meaning of the relations are up to the programmer.

8. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages The meaning of the relations are up to the programmer. For instance –like(jim, linux) Could mean jim likes linux. Or it could mean jim is similar to linux. Or it could mean something else entirely. There is no intrinsic semantics. If the relations are named well though, the programs obviously become more readable.

9. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logical connectors a implies b b implies a a  b a  b  implication a is equivalent to b a  b  equivalence a or b a  b  disjunction a and b a  b  conjunction not a  a  negation MeaningExampleSymbolName

10. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Logical operators –Conjunction (and) is like intersection among sets. –Disjunction (or) is like union among sets. Implication –a implies b can be read as "if a, then b"

11. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Precedence: Negation Conjunction, Disjunction, Equivalence Implication

12. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages So far we've seen only constants in these propositions To use a variable in a proposition we need to preface it with either “for all” or “there exists” These are quantifiers –For all is represented by an upside-down A –There exists is represented by a backwards E Predicate calculus consists of the logical connectors we just saw in addition to the two quantifiers above.

13. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages For Predicate Calculus, (that is using the logical connectors and the quantifiers) it should be obvious that there are many ways to state the same thing Anyone want to give an example?

14. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages That is why we use a standard form for propositions Clausal form: B 1  B 2  …  B n  A 1  A 2  …  A m means if all the As are true, then at least one B is true Antecedent: right side Consequent: left side

15. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages The A's and B's in the clausal form can be variables or constants. When they are variables, the “for all” quantifier is implied. Note: the meaning of variable here is different than the meaning we associate with variables in imperative languages. We can use clausal form exclusively because: –All predicate calculus propositions can be algorithmically converted to clausal form. So, we do not lose generality by restriction to this form of propositions.

16. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Some examples: C  A  B ForAll X.(mammal(X)  dog(X))‏ ThereExists X.(father(bob,X)  female(X))‏ –Taken separately, read each of the above in English.

17. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Logic Languages Some examples: C  A  B if both A and B are true, then C is true. ForAll X.(mammal(X)  dog(X))‏ All dogs are mammals ThereExists X.(father(bob,X)  female(X))‏ bob has a daughter

18. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Predicate Calc and proving theorems A use of propositions is to discover new theorems that can be inferred from known axioms and theorems Resolution: an inference principle that allows inferred propositions to be computed from given propositions A simple example: older(joanne, jake)  mother(joanne, jake)‏ wiser(joanne, jake)  older(joanne, jake)‏ A new propositions can be constructed from resolution: wiser(joanne, jake)  mother(joanne, jake)‏

19. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Predicate Calc and proving theorems The presence of variables in propositions requires resolution to find values for those variables that allow the matching process to succeed. This is called unification. Instantiation is the temporary assigning of values to variables. During some more complex resolutions, assigning some temporary values might cause the matching process to fail, at which time, backtracking must occur to instantiate a different value to the variable.

20. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Predicate Calc and proving theorems One possible way to prove a theorem is by contradiction. Make a proposition that is the theorem negated. The theorem is then attempted to be proved by finding an inconsistency. This is a theoretically useful way to prove theorems, but in practice the time it takes might be too long. Theorem proving is the basis for logic programming so we need a faster way for it to be able to do it.

21. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Predicate Calc and proving theorems A restricted kind of clausal form is used when propositions are used for resolution. These special kinds of propositions are Horn clauses –Only 2 forms –A single atomic proposition on left side –Or –Empty left side Left side is called the head. If left side is not empty the Horn clause is called a headed Horn clause. Anyone recall what an atomic proposition is?

22. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Predicate Calc and proving theorems An atomic propostion is a functor and a set of parameters in parentheses. e.g. mother(joanne, jake)‏ A headed Horn clause example: older(joanne, jake)  mother(joanne, jake)‏ these state relationships among relations What do you think the purpose of headless Horn clauses might be? e.g. mother(joanne, jake)‏ Not all (but most) propositions can be stated as/converted to Horn clauses.

23. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Overview of Logic Programming Logic programming semantics is Declarative semantics –There is a simple way to determine the meaning of each statement –Simpler than the semantics of imperative languages In imperative the semantics of a simple assignment statement is complex as it depends on scoping rules, determining types and values of variables, etc. --- sometimes depends on run-time context. –In logic programming the semantics of a statement is quite simple. The statement itself contains all that is needed to be known.

24. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Overview of Logic Programming Logical programming is nonprocedural –Programs do not state how a result is to be computed, but rather the form of the result We don't tell the computer how to get a result. Instead we state the form of the result and assume the computer can determine how to get that result. We provide the relevant information via the predicate calculus. And the computer infers the results using resolution.

25. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Overview of Logic Programming Describe the characteristics of a sorted list, not the process of rearranging a list sort(old_list, new_list)  permute (old_list, new_list)  sorted (new_list)‏ sorted (list)  ForAll j such that 1  j < n, list(j)  list (j+1)‏ This says that if a list is in non-decreasing order, it is sorted. Also, if the old_list is a permutation of the new_list and the new_list is sorted, then that's what it means to sort an old_list to be the new_list

26. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Our text uses the Edinburgh Syntax version of Prolog Term: a constant, variable, or structure Constant: an atom or an integer Atom: symbolic value of Prolog Atom consists of either: –a string of letters, digits, and underscores beginning with a lowercase letter –a string of printable ASCII characters delimited by apostrophes

27. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Variable: any string of letters, digits, and underscores beginning with an uppercase letter Instantiation: binding of a variable to a value –Lasts only as long as it takes to satisfy one complete goal Structure: represents atomic proposition functor(parameter list)‏

28. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Fact statements –Headless Horn clauses student(jonathan). brother(tyler, cj). Rule statements –Headed Horn clauses –Right side: antecedent (if part)‏ May be single term or conjunction –Left side: consequent (then part)‏ Must be single term Conjunction: multiple terms separated by logical AND operations (implied)‏

29. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog parent(kim,kathy):- mother(kim,kathy). Can use variables (universal objects) to generalize meaning: parent(X,Y):- mother(X,Y). sibling(X,Y):- mother(M,X), mother(M,Y), father(F,X), father(F,Y).

30. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog GOAL STATEMENTS For theorem proving, a theorem is in the form of a proposition that we want the system to prove or disprove Same format as headless Horn student(james)‏ this asks if james is a student, it either fails (james is not a student) or it doesn't (james is indeed a student)‏ Conjunctive propositions and propositions with variables are also legal goals father(X,joe)‏ this asks for all fathers of joe, if any, otherwise it fails.

31. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog INFERENCING Queries are called goals If a goal is a compound proposition, each of the facts is a subgoal To prove a goal is true, the system must find a chain of inference rules and/or facts. For goal Q: B :- A C :- B … Q :- P Process of proving a subgoal is called matching, satisfying, or resolution

32. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog POSSIBLE WAYS TO INFER Bottom-up resolution, forward chaining –Begin with facts and rules of database and attempt to find sequence that leads to goal Top-down resolution, backward chaining –begin with goal and attempt to find sequence that leads to set of facts in database Prolog implementations use backward chaining

33. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog POSSIBLE WAYS TO INFER When goal has more than one subgoal, can use either –Depth-first search: find a complete proof for the first subgoal before working on others –Breadth-first search: work on all subgoals in parallel Prolog uses depth-first search –Can be done with fewer resources so it is preferred for use in Prolog

34. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Backtrack during inference For a goal with multiple subgoals, if fail to show truth of one of subgoals, reconsider previous subgoal to find an alternative solution: backtracking Begin search where previous search left off Caveat – in worst case, can take lots of time and space because may find all possible proofs to every subgoal

35. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Recap: When trying to prove a goal, Prolog uses –backward chaining (starting at goal) to try to find facts that satisfy it –depth first search when satisfying a chain of subgoals –backtracking during inference to possibly get an alternative solution to an earlier subgoal

36. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Prolog also supports integer variables and integer arithmetic is operator: takes an arithmetic expression as right operand and variable as left operand A is B / 10 + C Not the same as an assignment statement B & C must have been instantiated and A must not have been instantiated for that expression to have A take on the value of the expression. But if A is instantiated prior to this expression or B or C are not instantiated, then the clause is not satisfied and no instantiation can take place. Example: Sum is Sum + Number.

37. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog speed(ford,100). speed(chevy,105). speed(dodge,95). time(ford,20). time(chevy,21). time(dodge,24). distance(X,Y) :- speed(X,Speed), time(X,Time), Y is Speed * Time. Here, the facts are the first six statements and the last is a rule stating the relationship between time, speed and distance. Example query: distance(chevy, Chevy_Distance)‏ Chevy_Distance is instantiated with the value 2205

38. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog TRACE Tracing model of execution - four events: –Call (beginning of attempt to satisfy goal)‏ –Exit (when a goal has been satisfied)‏ –Redo (when backtrack occurs)‏ –Fail (when goal fails)‏

39. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog likes(jake,chocolate). likes(jake,apricots). likes(darcie,licorice). likes(darcie,apricots). trace. likes(jake,X), likes(darcie,X). This is asking, what is X such that both jake and darcie like X Let's trace it on the board.

40. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog List Structures are another basic data structure in Prolog besides atomic propositions. A List is a sequence of any number of elements Elements can be atoms, atomic propositions, or other terms (including other lists)‏ [apple, prune, grape, kumquat] [] ( empty list)‏ [X | Y] ( head X and tail Y)‏

41. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog [X | Y] ( head X and tail Y)‏ Head X is like Scheme's car of a list Tail Y is like Scheme's cdr of a list | can be used to dismantle a list into head and tail in query mode | can be used to construct a list from a head and a tail

42. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog [X | Y] ( head X and tail Y)‏ –my_list([apple, prune, grape]). –my_list([apricot, peach, pear, lime]). These signify that the lists are new elements of the relation my_list. Just like parent(jane). stated that jane is an element of the relation parent. my_list is not a variable --- it is a relation. Example query: –my_list([ My_head | My_tail ] )‏ –Results in my_head instantiated with apple, and my_tail instantiated with [prune, grape]

43. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog [X | Y] ( head X and tail Y)‏ –my_list([apple, prune, grape]). –my_list([apricot, peach, pear, lime]). –my_list([ My_head | My_tail ] )‏ If this query was part of a goal and it backtracked, the next value for my_head would be: apricot and the next value for my_tail would be: [peach, pear, lime]

44. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog [X | Y] ( head X and tail Y)‏ –also used for constructing lists Construction example: –[apricot, peach | [pear, lime]]

45. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Append –The first two parameters to append are appended together to produce the result which parameter 3 gets instantiated with. –Note: we will “call” append with only the third parameter being a variable! append([], List, List). append([Head | List_1], List_2, [Head | List_3])‏ :- append (List_1, List_2, List_3). Order matters! The termination step (the headless clause) is first (just like in Scheme.) The recursion step is second. Prolog will try to match the propositions in order because it uses depth-first search. How do we read these propositions?

46. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog append([], List, List). append([Head | List_1], List_2, [Head | List_3])‏ :- append (List_1, List_2, List_3). So it says, if the first parameter is the empty list, the resulting list (parameter 3) is instantiated to the value of the second parameter. Otherwise, list3 is list1 and list2 appended together if the head of the first parameter is the head of parameter 3 (the result) and the tail of the 3 rd parameter is the tail of the 1 st parameter appended to the second parameter. Or in other words: appending [Head | List_1] to List_2 results in [Head | List_3] only if, List_3 is List_1 appened to List_2. It'll become clearer with an example.

47. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Similar to scheme's: (cons (car list1) (append (cdr list1) list2))‏ append([], List, List). append([Head | List_1], List_2, [Head | List_3])‏ :- append (List_1, List_2, List_3). append( [bob, jo], [jake, darcie], Family). Let's trace through on the board how Prolog will instantiate Family. (page 705.)‏

48. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog As stated earlier, Prolog uses depth-first search and left-to-right evaluation. So, resolution problems can occur with recursion that is not tail recursion. ORDER OF PROPOSITIONS AND SUBGOALS MATTER Example: ancestor(X, X). ancestor(X, Y) :- ancestor(Z, Y), parent(X, Z). Problem because second proposition has 1 st subgoal same as proposition (left-recursion). Suppose it is given the query ancestor(mike, jane). It goes to ancestor(Z, jane) and instantiates Z with a value. Then it does the proposition again to determine if it is true... Just like the problem a recursive descent parser has (remember?)‏ How to solve this?

49. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Example: member(Element, [Element | _ ] ). member(Element, [ _ | List] ) :- member(Element, List). /* _ is anonymous variable, use when we won't use the variable in the proposition. */ These member propositions state that if first parameter is the head of the second parameter, then member is satisfied. If not, then Element is a member of the List if Element is in the tail of the List.

50. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Cut (!) to prevent backtracking (to earlier subgoals before the cut during a resolution.)‏ Recall, that prolog backtracks when a goal (or subgoal) fails. Let's say we have something like this on the right hand side of a proposition: –member(X, Lis), do_something(X, Lis, Y). If member is satisfied and then do_something is attempted and fails, backtracking will occur and cause member to be reattempted. Inside member's implementation, it will pick up where it left off and possibly not find another match. To make sure it doesn't backtrack, we can use the Cut. member(Element, [Element | _ ] ) :- !. member(Element, [ _ | List] ) :- member(Element, List). Now, Prolog cannot backtrack to reevaluate member once Element is found in the list.

51. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Explain what cut does.

52. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Negation problem parent(bill, jake). parent(bill, shelly). sibling( X, Y) :- parent(M, X), parent(M, Y). goal sibling(X, Y). results in: X = jake Y = jake Because it goes through the facts first with parent(M, X) and instantiates them to bill and jake. Then parent(M,Y) starts at the beginning of the facts and tries to find a fact with bill as parent. It finds bill and jake again.

53. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Negation problem solution parent(bill, jake). parent(bill, shelly). sibling( X, Y) :- parent(M, X), parent(M, Y), X <> Y. goal sibling(X, Y). results in: X = jake Y = shelly Because it goes through the facts first with parent(M, X) and instantiates them to bill and jake. Then parent(M,Y) starts at the beginning of the facts and tries to find a fact with bill as parent. It finds bill and jake again. But then X <> Y becomes jake <> jake which fails, so backtracking occurs to parent(M, Y). Prolog remembers where it left off in the list of parent facts, so it gets the next one (bill, shelly) that's good, so then X <> Y becomes jake <> shelly which is satisfied. Done.

54. Michael Eckmann - Skidmore College - CS 330 - Fall 2008 Prolog Fail Can be used (as a subgoal) to force a goal to fail, thus forcing backtracking. Only makes sense as the last subgoal of a goal (because any subgoals after it would never get tested.)‏