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Signature: parent (Parent, Child) /2 Purpose: Parent is a parent of Child

Logic Programming: Introduction. A procedure: a set of axioms (rules and facts) with identical signature ( predicate symbol and arity ). A logic program : a set of procedures (predicates) , defining relations in the program domain. Signature: parent (Parent, Child) /2

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Signature: parent (Parent, Child) /2 Purpose: Parent is a parent of Child

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  1. Logic Programming: Introduction • A procedure: a set of axioms (rules and facts) with identical signature (predicate symbol and arity). • A logic program: a set of procedures (predicates), defining relations in the program domain. • Signature: parent (Parent, Child) /2 • Purpose: Parent is a parent of Child • 1. parent (erik, jonas). • 2. parent(lena, jonas). • Signature: male(Person) /1 • Purpose: Person is a male • 1. male (erik). • Signature: father (Dad, Child) /2 • Purpose: Dad is father of Child • 1. father (Dad, Child) :- parent(Dad, Child) , male (Dad). arity procedure predicate program fact axioms rule • if • and • Dad is male • The relation fatherholds between Dad and Child • parentholds

  2. Logic Programming: Introduction • The prolog interpreter operates in a read-eval-print loop. Given a query,it attempts to prove it based on the program: • If it fails, it answers false. • Else, if the query has no variables, it answers true. • Else, it outputs all possible variables assignmentsfound during proof process. query • % Signature: parent (Parent, Child) /2 • % Purpose: Parent is a parent of Child • parent (erik, jonas). • parent (lena, jonas). • % Signature: male(Person) /1 • % Purpose: Person is a male • male (erik). • % Signature: father (Dad, Child) /2 • % Purpose: Dad is father of Child • father (Dad, Child) :- parent(Dad, Child), male (Dad). ? - father (X,Y). X=erik, Y=jonas ? - parent (X,jonas). X=erik ; X=lena • Next possible assignment

  3. Power Power Connection point Resistor(symmetric) Transistor(asymmetric) Ground Ground Logic Programming: Example 1 – logic circuits • An electronic logic circuit combines: • Logic gates: resistor, transistor, implement simple logic functions. • Connection points: connects one logic gateto another, or to poweror ground. N1 N2 N3 • A program that models electronic logic circuits: • Connection points: are individual constants. • Logic gates: are relations on the constants. N4 N5 % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2 resistor(power, n2). 3 resistor(n1, power). 4 resistor(n2, power). % Signature: transistor (Gate, Source, Drain)/3 % Purpose: … 1 transistor(n2,ground,n1). 2 transistor(n3,n4,n2). 3 transistor(n5,ground,n4). • Note:In contrast resistor, the order of arguments in transistor is important. Each has a different role. 

  4. Logic Programming: Example 1 – logic circuits % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2 resistor(power, n2). 3 resistor(n1, power). 4 resistor(n2, power). %Signature:resistor(End1,End2)/2 %Purpose End1,End2 1 resistor(power,n1). 2 resistor(power, n2). 3 resistor(n1, power). • Reminders… • A procedure begins with a contract. • Constants start with lowercase characters. • A predicatename is also a constant,which definesa relation between its arguments. • variablesstart with uppercase characters (‘_’ for wildcard). • Atomic formulas are either true, false or of the formpredicate(t1 , ..., tn), ti is a term. • A rule is a formula defining a relation that depends on certain conditions. • A fact is an atomic formula which is unconditionally true. terms

  5. Logic Programming: Example 1 – logic circuits % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2 resistor(power, n2). 3 resistor(n1, power). 4 resistor(n2, power). %Signature:resistor(End1,End2)/2 %Purpose End1,End2 1 resistor(power,n1). 2 resistor(power, n2). 3 resistor(n1, power). • Reminders… • A query is a sequence of atomic formulas: ?- resistor(power, n1),resistor(n2, power). true; false No more answers “Does an Xexists such that the resistor relation holds for (power, X)?” • ?- resistor(power, X). • X = n1 ; • X = n2; • false Is there another answer?

  6. power ground Logic Programming: Example 1 – logic circuits • Combining logic gates to create a “not” logic circuit: • The resistor and transistor relations are based on facts. • The relations can be combined by a rule to determine whether or not the not_circuitrelation stands for Input and Output. resistor not circuit transistor Output Input

  7. power ground Logic Programming: Example 1 – logic circuits Combining logic gates to create a NOT logic circuit: % Signature: not_circuit(Input, Output)/2 % Purpose: not logic circuit. 1. not_circuit(Input, Output) :- transistor(Input, ground, Output) , 2 . resistor(power, Output). “and” Rule head:an atomic formula with variables Rule body resistor not circuit ?- not_circuit(X,Y). X=n2, Y=n1; false transistor Output Input • “For all Input and Output: • (Input, Output) stands in the relation not_circuitif: • (Input, ground, Output) stand in the relation transistorand • (power, Output) stand in the relation resistor.”

  8. Power Power Connection point Ground Ground Logic Programming: Example 1 – logic circuits Combining logic gates to create a NAND logic circuit: % Signature: nand_circuit(Input1, Input2, Output)/3 % Purpose: nand logic circuit nand_circuit(Input1, Input2, Output) :- transistor(Input1, X, Output), transistor(Input2, ground, X), resistor(power, Output). N1 N2 N3 N4 N5 ?- not_circuit(X, Y), nand_circuit(In1, In2, X). X = n2, Y = n1, In1 = n3, In2 = n5; false nand circuit

  9. Semantics: Unification algorithm • A program execution is triggered by a queryin attempt to prove its goals: • To find a possible proof, the answer-queryalgorithm is used. • It makes multiple attempts to apply rules on a selected goal. • This is done by applying a unification algorithm, Unify, to the rule head and the goal.

  10. Semantics: Unification algorithm Definitions: binding:a non-circular expression, X=t, where X is a variableand t is a termnot including X. Substitution:a function from a finite set of variables to a finite set of terms (or bindings). Application of a sub. S to an atomic formula A, replaces vars in A with corresponding terms in S:S = {I=X} A = not_circuit(I, I) , A º S = not_circuit(X, X)B = not_circuit(X,Y) , Bº S = not_circuit(X, Y)the result is an instance of A Unifier:a substitution S is called a unifier of formulas A and B if A º S = B º S. For example:S = {I=X} º {X=Y} = {I=Y, X=Y} A º S = not_circuit(Y, Y)B º S = not_circuit(Y, Y) The algorithm Unify receives two atomic formulas and returns their most general unifier. S = {J=5, X=5, Y=5} is a unifier for A and B, but not the most general one.

  11. Semantics: Proof trees • Executing answer-query: • The interpreter searches for a proof for a given query (a conjunction of goals). • The search is done by building and traversing a proof tree where all possibilities are examined. • The possible outcome is one of the following: • The algorithm finishes, and possible values of the query variables are given. • The algorithm finishes, but there is no proof for the query (false). • The proof attempt loops and never ends.

  12. Semantics: Proof trees • The tree structure depends on Prolog's goal selection and rule selection policies: • Query goals (Q1,…,Qn) are at the root of the proof tree. • Choose current goal (atomic formula). Prolog's policy: the leftmost goal. • Choose current rule to prove current goal. (top to bottom program order). • Rename the variables in the rule and unify the rule head with the goal.If unification succeeds: • A new child node is created. • The query for this node is the rule body. • A leaf may be created if the goal list is empty (success), or if the goal cannot be proven (failure). • Backtracking: When a leaf is reached, the search travels up to the first parent node where another rule can be matched. Q = ?- Q1, ..., Qn

  13. Semantics: Example 2 – A generated proof tree ?- nand_circuit(In1, In2, Out). nand_circuit(Input1,Input2,Output) :- transistor(Input1,X,Output), transistor(Input2,ground,X), resistor(power, Output). nand_circuit(In1, In2, Out) mgu { Input1_1 = In1, Input2_1 = In2, Output_1 = Out } Rule 1 transistor (Gate, Source, Drain) :- transistor(n2,ground,n1). transistor(n3,n4,n2). transistor(n5,ground,n4). transistor(In1, X_1, Out), transistor(In2, ground, X_1),resistor(power, Out) { In1=n5, X_1=ground, Out=n4} Fact 2 – transistor { In1=n2, X_1=ground, Out=n1} Fact 1 – transistor resistor(End1,End2) :- resistor(power, n1). resistor(power, n2). resistor(n1, power). resistor(n2, power). {In1=n3, X_1=n4, Out=n2} Fact 2 – transistor transistor(In2,ground,ground),resistor(power, n1) transistor(In2,ground,n4),resistor(power, n2) transistor(In2,ground,ground),resistor(power, n2) { In2=n5} Fact 3 – transistor {In2=n2} Fact 1 – transistor { In2=n2} Fact 1 – transistor { In2=n5} Fact 3 – transistor { In2=n3} Fact 2 – transistor • a finite success treewith one success path. { In2=n3} Fact 2 – transistor fail fail fail fail fail resistor(power, n2) true

  14. Semantics: proof trees • Possible types of proof trees: • A success tree has at least one success path in it. • A failure tree is one in which every path is a failure path. • A proof tree isan infinite tree if it contains an infinite path. • Otherwise, it is a finite tree. • Example: • An infinite is generated by repeatedly applying the rule p(X):-p(Y),q(X,Y) (left recursion).To avoid the infinite path, we could rewrite the rule: p(X):-q(X,Y),p(Y) (tail recursion).

  15. Semantics: Example 3 – SQL in Relational Logic Programming An example: Relational logic programming & SQL operations. • Relations may be regarded as tables in a database of facts. • Recall the resistor and transistor relations presented earlier: Table name: resistor Schema: End1, End2 Data:(power, n1), (power, n2), (n1, power), (n2, power). Table name: transistor Schema: Gate, Source, Drain Data: (n2, ground, n1) (n3, n4, n2), (n5, ground, n4). SQL Operation: Natural join • ?-res_join_trans(End1,X,Source,Drain). % Signature: res_join_trans(End1, X, Gate, Source)/4 % Purpose: Join between resistor and transistor % according to End2 of resistor and Gate of transistor. res_join_trans(End1, X, Source, Drain):- resistor(End1,X), transistor(X, Source, Drain). End1 = power, X = n2, Source = ground, Drain = n1 ; false.

  16. Semantics: Example 3 – SQL in Relational Logic Programming An example: Relational logic programming & SQL operations. • Relations may be regarded as tables in a database of facts. • Recall the resistor and transistor relations presented earlier: Table name: resistor Schema: End1, End2 Data:(power, n1), (power, n2), (n1, power), (n2, power). Table name: transistor Schema: Gate, Source, Drain Data: (n2, ground, n1) (n3, n4, n2), (n5, ground, n4). Transitive closure for the resistor relation • ?- res_closure(X, Y). %Signature: res_closure(X, Y)/2 res_closure(X, Y) :- resistor(X, Y). res_closure(X, Y) :- resistor(X, Z), res_closure(Z, Y). X = power, Y = n1 ; X = power, Y = n2 ; X = n1, Y = power ; X = n2, Y = power ; X = power , Y = power ; X = power, Y = n1 ; X = power, Y = n2 ; X = power, Y = power ; X = power, Y = n1; ... The resistor relation is symmetric. Therefore, we get a finite series of answers repeated an infinite number of times.

  17. Logic Programming: An example: Relational logic programming & SQL operations. • Relational logic programming has no ability to describe composite data, only atoms. • Logic programming is equipped with functors to describe composite data. Q: In the following procedure, which symbols are predicates? Which are functors? % Signature: tree_member(Element,Tree)/ 2 % Purpose: Checks if Element is an element of the binary tree Tree. tree_member (X,tree(X,Left,Right)). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Left). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Right). Predicate A Functorapplied to three data items. • Q: how can you tell? by their relative location in the program: • A predicate appears as an identifier of an atomic formula. • A functor is way to construct a term. A term is a part of a formula. • A functor can be nested – a predicate cannot. • NOTE: The same name may be used for both a predicate and a functor!

  18. Logic Programming: Unification with functors An example: Relational logic programming & SQL operations. Unification is more complex with functors. Here is an execution of the Unify algorithm, step by step:Unify(A,B) where A = tree_member (tree (X, 10, f(X)), W) ; B = tree_member (tree (Y, Y, Z), f(Z)). Initially, s={} • Disagreement-set = {X=Y} • X does not occur in Y •  s=s  {X=Y} = {X=Y} As= tree_member (tree (X, 10, f(Y)), W ) Bs= tree_member (tree (Y, Y , Z ), f(Z)) As ≠ Bs As ≠ Bs As ≠ Bs As ≠ Bs As= tree_member (tree (Y, 10, f(Y)), W ) Bs= tree_member (tree (Y, Y ,Z ), f(Z)) • Disagreement-set = {Y=10} •  s=s  {Y=10} = {X=10, Y=10} As= tree_member (tree (10, 10, f(10)), W ) Bs= tree_member (tree (10, 10, Z ), f(Z))  Disagreement-set = {Z=f(10)}  s=s  {Z=f(10)} = {X=10, Y=10, Z=f(10)} As= tree_member (tree (10, 10, f(10)), W ) Bs= tree_member (tree (10, 10, f(10)), f(f(10))) • Disagreement-set = {W=f(f(10))} • s={X=10, Y=10, Z=f(10), W=f(f(10))} As= tree_member (tree (10, 10, f(10)), f(f(10)) ) Bs= tree_member (tree (10, 10, f(10)), f(f(10))) Q:Why do we check for occurence?

  19. Logic Programming: Unification with functors An example: Relational logic programming & SQL operations. A: Consider the following two atomic formulas: A = tree_member (tree (X, Y, f(X)), X ) B = tree_member(tree (Y, Y, Z ), f(Z)) Applying Unify(A,B) will result in a loop: X=Y, Z=f(Y), Y=f(Z)=f(f(Y))… the substitution cannot be successfully solved.

  20. Logic Programming: Example queries An example: Relational logic programming & SQL operations. % Signature: tree_member(Element,Tree)/ 2 % Purpose: Testing tree membership, checks if Element is % an element of the binary tree Tree. tree_member (X,tree(X,Left,Right)). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Left). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Right). ?- tree_member(1, tree(1,nil, nil)). true ?- tree_member(2,tree(1,tree(2,nil,nil), tree(3,nil, nil))). true. • ?- tree_member(1, tree(3,1, 3)). • false. ?- tree_member(X,tree(1,tree(2,nil,nil), tree(3,nil, nil))). X=1; X=2; X=3; false.

  21. Full Logic Programming: An example proof tree ?- tree_member(X, tree(1, tree(2, nil, nil), tree(3, nil, nil))). • tree_member(X, tree(1, tree(2, nil, nil), tree(3, nil, nil))) {X_1 = X, Y_1 = 1, Left_1 = tree(2,nil,nil), Right_1 = tree(3, nil, nil)} {X_1 = X, Y_1 = 1, Left_1 = tree(2,nil,nil), Right_1 = tree(3, nil, nil)} {X_1 = 1, X = 1Left_1 = tree(2,nil,nil), Right_1 = tree(3, nil, nil)} true • tree_member (X, tree(3,nil,nil)) • tree_member (X, tree(2,nil,nil)) {X_2 = 2, X = 2Left_2 = nil,Right_2 = nil} {X_2 = X, Y_2 = 2, Left_2 = nil, Right_2 = nil} {X_2 = X, Y_2 = 2, Left_2 = nil, Right_2 = nil} {X=1} • tree_member (X, nil) • tree_member (X, nil) true fail fail {X=2} {X_2 = 3, X = 3 Left_2 = nil,Right_2 = nil} {X_2 = X, Y_2 = 3, Left_2 = nil, Right_2 = nil} % Signature: tree_member(Element,Tree)/ 2 % …tree_member (X,tree(X,Left,Right)). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Left). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Right). {X_2 = X, Y_2 = 3, Left_2 = nil, Right_2 = nil} • tree_member (X, nil) • tree_member (X, nil) true {X=3} fail fail

  22. Full Logic Programming: An example proof tree % Signature: tree_member(Element,Tree)/ 2 % Purpose: Testing tree membership, checks if Element is % an element of the binary tree Tree. tree_member (X,tree(X,Left,Right)). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Left). tree_member (X,tree(Y,Left,Right)):- tree_member(X,Right). • ?- tree_member(1, T). • T = tree(1, _G445, _G446) ; • T = tree(_G444, tree(1, _G449, _G450), _G446) ; • T = tree(_G444, tree(_G448, tree(1, _G453, _G454), _G450), _G446) ; • ... • Looking for all trees in which 1 is a member, we get an infinite success tree with partially instantiated answers (containing variables). • We use a rule that requires a defined input, but our input is a variable. Possible answers are generated by the proof algorithm. • In this case we call the rule a generator rule.

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