o A procedure: a set of axioms (rules and facts) with identical signature (predicate symbol and arity). o A logic program: a set of procedures defining relations in the program domain. % 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). program rule fact arity procedure predicate The relation father holds between Dad and Child if parent holds and Dad is male 1 Logic Programming: Introduction

2 The prolog interpreter operates in a read-eval-print loop. Given a query, it attempts to prove it based on the program: o If it fails, it answers false. o Else, if the query has no variables, it answers true. o Else, it outputs all possible variables assignments found during proof process. Logic Programming: Introduction % 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 query

3 Logic Programming: Example 1 – logic circuits Connection point Resistor Ground Transistor Power Ground N1 N2 N3 N4 N5 % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2resistor(power, n2). 3resistor(n1, power). 4resistor(n2, power). % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2resistor(power, n2). 3resistor(n1, power). 4resistor(n2, power). % Signature: transistor (Gate, Source, Drain)/3 % Purpose: … 1transistor(n2,ground,n1). 2transistor(n3,n4,n2). 3transistor(n5,ground,n4). % Signature: transistor (Gate, Source, Drain)/3 % Purpose: … 1transistor(n2,ground,n1). 2transistor(n3,n4,n2). 3transistor(n5,ground,n4). Note: In contrast to resistor, the arguments order in transistor is important. Each has a different role. n1 Tran n5 n3 n4 n2 Res Tran Res Power Ground Source Drain Gate Source Drain Gate Source Drain Gate

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

5 Logic Programming: Example 1 – logic circuits % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2resistor(power, n2). 3resistor(n1, power). 4resistor(n2, power). % Signature: resistor(End1,End2)/2 % Purpose: A resistor component connects two ends 1 resistor(power, n1). 2resistor(power, n2). 3resistor(n1, power). 4resistor(n2, power). Reminders… o A procedure begins with a contract. o Constants start with lowercase characters. o A predicate name is also a constant, which defines a relation between its arguments. o variables start with uppercase characters (‘_’ for wildcard). o Atomic formulas are either true, false or of the form predicate(t 1,..., t n ), t i is a term. o A rule is a formula defining a relation that depends on certain conditions. o A fact is an atomic formula which is unconditionally true. % Signature: resistor(End1,End2)/2 % Purpose 3resistor(n1, power). 1 resistor(power, n1). 2resistor(power, n2). End1,End2 terms

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

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

% Signature: not_circuit(Input, Output)/2 % Purpose: not logic circuit. not_circuit(Input, Output) :- transistor(Input, ground, Output), resistor(power, Output). % Signature: not_circuit(Input, Output)/2 % Purpose: not logic circuit. not_circuit(Input, Output) :- transistor(Input, ground, Output), resistor(power, Output). Rule head: an atomic formula with variables Rule body “and” 8 power ground transistor resistor Input Output Logic Programming: Example 1 – logic circuits Combining logic gates to create a NOT logic circuit: ?- not_circuit(X,Y). X=n2, Y=n1; false “For all Input and Output: (Input, Output) stands in the relation not_circuit if: (Input, ground, Output) stand in the relation transistor and (power, Output) stand in the relation resistor.”

9 Input2 Output Logic Programming: Example 1 – logic circuits Combining logic gates to create a NAND logic circuit: Input1 % 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). % 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).

10 Logic Programming: Example 1 – logic circuits האם יש יחס and במערכת ? ?- not_circuit(X, Y), nand_circuit(In1, In2, X). X = n2, Y = n1, In1 = n3, In2 = n5; false Connection point Ground Power Ground N1 N2 N3 N4 N5

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

12 Definitions: 1.binding: a non-circular expression, X=t, where X is a variable and t is a term not including X. 2.Substitution: a function from a finite set of variables to a finite set of terms (in other words: a finite set of bindings with no variable repetitions). 3.Application (º) of a substitution S to an atomic formula A replaces variables in A with corresponding terms in S (the result is an instance of A). For example: S = {J=X}A = not_circuit(J, J), A º S = not_circuit(X, X) B = not_circuit(X,Y), B º S = not_circuit(X, Y) 4.Unifier: a substitution S is called a unifier of formulas A and B if A º S = B º S. For example: S = {J=X} º {X=Y} = {J=Y, X=Y}A º S = not_circuit(Y, Y) B º S = not_circuit(Y, Y) Semantics: Unification algorithm 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.

Combinations of substitutions (shown in class): In the following example: We combine the substitution S1={X=Y, Z=3, U=V} with the substitution S2= {Y=4, W=5, V=U, Z=X}. This is denote by: S1 º S2. (2 nd step) Variables in S2 that have a binding in S1 are removed from S2. So far : S1={ X=4, Z=3, U=U }, S2 remains the same So far : S1={ X=4, Z=3, U=U }, S2= { Y=4, W=5, V=U} (4 rd step) Identity bindings are removed. The result : S1={ X=4, Z=3, Y=4, W=5, V=U} (3 rd step) S2 is added to S1. So far : S1={ X=4, Z=3, U=U, Y=4, W=5, V=U} (1 st step) S2is applied to the terms of S1: { X=Y, Z=3, U=V }, { Y=4, W=5, V=U, Z=X } 13 Semantics: Unification algorithm

14 Semantics: Proof trees Executing answer-query: o The interpreter searches for a proof for a given query (a conjunction of goals). o The search is done by building and traversing a proof tree where all possibilities are examined. o 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.

15 Semantics: Proof trees The tree structure depends on Prolog's goal selection and rule selection policies: 1.Query goals (Q 1,…,Q n ) are at the root of the proof tree. 2.Choose current goal (atomic formula). Prolog's policy: the leftmost goal. 3.Choose current rule to prove current goal. (top to bottom program order). 4.Rename the variables in the rule and unify the rule head with the goal. If unification succeeds: 1.A new child node is created. 2.The query for this node is the rule body. 5.A leaf may be created if the goal list is empty (success), or if the goal cannot be proven (failure). 6.Backtracking: When a leaf is reached, the search travels up to the first parent node where another rule can be matched. Q = ?- Q 1,..., Q n

resistor(power, n1). resistor(power, n2). resistor(n1, power). resistor(n2, power). resistor(power, n1). resistor(power, n2). resistor(n1, power). resistor(n2, power). nand_circuit(In1, In2, Out) transistor(In1, X_1, Out), transistor(In2, ground, X_1), resistor(power, Out) transistor(In2,ground,ground), resistor(power, n1) transistor(In2,ground,n4), resistor(power, n2) nand_circuit(Input1,Input2,Output) :- transistor(Input1,X,Output), transistor(Input2,ground,X), resistor(power, Output). nand_circuit(Input1,Input2,Output) :- transistor(Input1,X,Output), transistor(Input2,ground,X), resistor(power, Output). { In1=n2, X_1=ground, Out=n1} Fact 1 – transistor transistor(n2,ground,n1). transistor(n3,n4,n2). transistor(n5,ground,n4). transistor(n2,ground,n1). transistor(n3,n4,n2). transistor(n5,ground,n4). { In1=n3, X_1=n4, Out=n2} Fact 2 – transistor fail { In2=n3} Fact 2 – transistor fail { In2=n5} Fact 3 – transistor true resistor(power, n2) { In2=n2} Fact 1 – transistor fail { In2=n3} Fact 2 – transistor { In2=n5} Fact 3 – transistor { In2=n2} Fact 1 – transistor fail { Input1_1 = In1, Input2_1 = In2, Output_1 = Out } Rule 1 mgu 16 Semantics: Example 2 – A generated proof tree ?- nand_circuit(In1, In2, Out). transistor(In2,ground,ground), resistor(power, n2) { In1=n5, X_1=ground, Out=n4} Fact 2 – transistor a finite success tree with one success path.

17 Semantics: proof trees Possible types of proof trees: o A success tree has at least one success path in it. o A failure tree is one in which every path is a failure path. o A proof tree is an infinite tree if it contains an infinite path. o 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).

An example: Relational logic programming & SQL operations. 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). % 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). % 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. ?-res_join_trans(End1,X,Source,Drain). SQL Operation: Natural join 18 Semantics: Example 3 – SQL in Relational Logic Programming o Relations may be regarded as tables in a database of facts. o Recall the resistor and transistor relations presented earlier:

19 An example: Relational logic programming & SQL operations. Semantics: Example 3 – SQL in Relational Logic Programming o Relations may be regarded as tables in a database of facts. o 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). 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. Transitive closure for the resistor relation %Signature: res_closure(X, Y)/2 res_closure(X, Y) :- resistor(X, Y). res_closure(X, Y) :- resistor(X, Z), res_closure(Z, 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). ?- res_closure(X, Y).

% 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). % 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). Q: In the following procedure, which symbols are predicates? Which are functors? 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! 20 An example: Relational logic programming & SQL operations. Full Logic Programming o Relational logic programming has no ability to describe composite data, only atoms. o Full logic programming is equipped with functors to describe composite data.

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  Disagreement-set = {Y=10}  s=s  {Y=10} = {X=10, Y=10} A  s= tree_member (tree (Y, 10, f(Y)), W ) B  s= tree_member (tree (Y, Y,Z ), f(Z)) A  s ≠ B  s  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, Z ), f(Z)) A  s ≠ B  s  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)), W ) B  s= tree_member (tree (10, 10, f(10)), f(f(10))) A  s ≠ B  s A  s= tree_member (tree (10, 10, f(10)), f(f(10)) ) B  s= tree_member (tree (10, 10, f(10)), f(f(10))) 21 An example: Relational logic programming & SQL operations. Full Logic Programming: Unification with functors

22 An example: Relational logic programming & SQL operations. Full Logic Programming: Unification with functors 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.

% 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). % 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. 23 Full Logic Programming: Example queries

tree_member(X, tree(1, tree(2, nil, nil), tree(3, nil, nil))) 24 ?- tree_member(X, tree(1, tree(2, nil, nil), tree(3, nil, nil))). Full Logic Programming: An example proof tree % 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). % 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_1 = 1, X = 1 Left_1 = tree(2,nil,nil, Right_1 = tree(3, nil, nil)} true {X=1 } tree_member (X, tree(2,nil,nil)) {X_1 = X, Y_1 = 1, Left_1 = tree(2,nil,nil), Right_1 = tree(3, nil, nil)} true {X=2 } tree_member (X, nil) {X_2 = X, Y_2 = 2, Left_2 = nil, Right_2 = nil} fail tree_member (X, nil) {X_2 = X, Y_2 = 2, Left_2 = nil, Right_2 = nil} fail {X_2 = 2, X = 2 Left_2 = nil, Right_2 = nil} {X_1 = X, Y_1 = 1, Left_1 = tree(2,nil,nil), Right_1 = tree(3, nil, nil)} tree_member (X, tree(3,nil,nil)) true {X=3 } tree_member (X, nil) {X_2 = X, Y_2 = 3, Left_2 = nil, Right_2 = nil} fail tree_member (X, nil) {X_2 = X, Y_2 = 3, Left_2 = nil, Right_2 = nil} fail {X_2 = 3, X = 3 Left_2 = nil, Right_2 = nil}

% 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). % 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) ;... o Looking for all trees in which 1 is a member, we get an infinite success tree with partially instantiated answers (containing variables). o We use a rule that requires a defined input, but our input is a variable. Possible answers are generated by the proof algorithm. o In this case we call the rule a generator rule. 25 Full Logic Programming: An example proof tree