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Foundations of Logic and Constraint Programming Lecturer: Pedro Barahona

Foundations of Logic and Constraint Programming Lecturer: Pedro Barahona Web page: http://ssdi.di.fct.unl.pt/mcl/flcp. Propositional Logic. Logic Programming is based on logic .

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Foundations of Logic and Constraint Programming Lecturer: Pedro Barahona

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  1. Foundations of Logic and Constraint Programming Lecturer: Pedro Barahona Web page: http://ssdi.di.fct.unl.pt/mcl/flcp Foundations of Logic and Constraint Programming

  2. Propositional Logic • Logic Programming is based on logic. • A main objective of logic is to formalise certain forms of reasoning, namely to allow the inference of a certain proposition (conclusion) from other propositions (Theory). • Example: ConclusionC:homelogically follows from theoryT, whereT:{P1, P2, P3}and P1: sunny  cloudy  rainy, P2: rainy  home P3: sunny  cloudy • Symbolically, T |= C • Formal Logic achieves this by means of inference rules. Many systems exist but an important one is based on a single inference rule: resolution. Foundations of Logic and Constraint Programming

  3. Propositional Logic - Resolution • Resolution works with sets of clauses (disjunctions of literals) P1: sunny  cloudy  rainy P2: rainy  home ( rainy  home) P3a: sunny P3b: cloudy • A single inference rule (Resolution):A  P P  B A  B • Inferences correspond to refutations T |== C  T  {C} |==  • Refutation {P1, P2, P3a, P3b, C } |==  ( sunny  cloudy) Foundations of Logic and Constraint Programming

  4. sunny (sunny  cloudy  rainy)  rainy rainy  cloudy (cloudy  rainy) (rainy  home) home home (rainy  home)  rainy (sun  cloudy  rainy) (sun  cloudy) sun cloudy cloudy  Propositional Logic – (Linear) Resolution • Example of a Refutation:{P1, P2, P3a, P3b, C } |==  • There are other forms of organising the resolution of clauses. An important one is arranging them in a linear sequence, starting with the conclusion (negated). Foundations of Logic and Constraint Programming

  5. Propositional Logic - Definite Clauses • Proving that a set of general clauses is contradictory is an NP-complete problem, for which there are no known algorithms that execute in polinomial time. • However, if clauses are definite, their refutation becomes polinomial in the number of clauses. • Definite clauses, also known as Horn clauses, are those where at most one literal may be positive. • Definite clauses come in two “flavours”: Facts and Rules. • A fact is a clause without negative literal. A rule is a clause with negative literals that can be written in an if-then format. • For example: Fact:sunny Rule:rainy  busy  home  home ← rainy, busy Foundations of Logic and Constraint Programming

  6. Resolution with Definite Clauses • A linear deduction through resolution can be presented as a mechanism for answering queries, in theories expressed in definite clauses. • Example: Answering Query Q: happy from Theory P1: happy ← summer, warm P2: warm ← sunny P3: sunny P4: summer P1: happy  summer  warm P2: warm  sunny P3: sunny P4: summer Q0: happy Q1: summer  warmQ0, P1 Q2: warm Q1, P4 Q3: sunny Q2, P2 Q4:  Q3, P3 P1: happy ← summer, warm P2: warm ← sunny P3: sunny P4: summer Q0: ←happy Q1: ←summer, warmQ0, P1 Q2: ←warm Q1, P4 Q3: ←sunny Q2, P2 Q4: ←□ Q3, P3 Foundations of Logic and Constraint Programming

  7. Deduction and Logic Inference • The previous slides illustrate the process of deduction, • Reasoning aims at inferring propositions from those of the theory, by means of the rules of an inference (deduction) system. • A proposition C is deduced from a theory T in a system S, if it can be inferred from the propositions of T by repeated application of the inference rules of S. • Symbolically: T |-- C • In alternative, the focus may be on the the truth values of propositions. • Reasoning aims at finding logical consequences of the theory. • A proposition C is a logical consequence of a theory T, if whenever all the propositions of the theory are true so is C. • Symbolically: T |== C • In general, it is important that these two approaches are equivalent, for a given logic and inference system S. This is guaranteed by means of soundness and completeness theorems. • This is the case with propositional logic and the resolution system. Foundations of Logic and Constraint Programming

  8. Soundness and Completeness of Propositional Logic • Theorem: Resolution is sound and complete for propositional logic. • Soundness: Any proposition that is inferred from a theory by means of resolution is a logical consequence of the theory. • Simbolically: (T |-- C)  (T |== C) • Completeness: Any proposition that is a logical consequence of a theory can be inferred from it by means of resolution on propositions of the theory. • Simbolically: (T |== C)  (T |-- C) • Note 1: Resolution is not the only inferrence system for propositional logic, but is the one where logic programming is based on. • Note 2: Similar results are obtained for First-Order Logic (FOL) restricted to definite clauses (full FOL is only semi-decidable). Foundations of Logic and Constraint Programming

  9. First Order Logic (Horn Clauses) • In first order logic, literals may have “arguments” (variables, constants or more complex terms). • Example: P1a: connect(X,Y) ← direct(X,Y). P1b: connect(X,Y) ← direct(X,Z), connect(Z,Y) P2: direct(lisbon, london). P3: direct(london, sidney). • Deduction is now more complex, since variables (that are assumed to be universally quiantified) can now be substituted for other terms. Q0: ←connect(lisbon,sidney) Q1: ←direct(lisbon,Z),connect(Z,sidney) {X/lisbon, Y/sidney} Q2: ←connect(london,sidney) {Z/london} Q3: ←direct(london,sidney) {X1/london, Y1/sidney} Q4: ←□ Foundations of Logic and Constraint Programming

  10. First Order Logic (Horn Clauses) P1a: connect(X,Y) ← direct(X,Y). P1b: connect(X,Y) ← direct(X,Z), connect(Z,Y) P2: direct(lisbon, london). P3: direct(london, sidney). • Queries may also be more “informative than Yes/No, if they include variables. Q0: ←connect(W,U) Q1: ←direct(X,Z),connect(Z,Y) {W/X, U/Y} Q2: ←connect(london,Y) {X/lisbon, Z/london} Q3: ←direct(london,Y) {X1/london, Y1/Y} Q4: ←□{Y/sidney} • In this case, the answer is not only Yes, but it also allows the naming of the cities that are connected, by inspection of the substitutions that are performed in the deduction, and their composition. {W/X, U/Y}{X/lisbon, Z/london} = {W/lisbon, U/Y, ...} {W/lisbon, U/Y, ...}  {Y/sidney} = {W/lisbon, U/Sidney, ...} i.e. Lisbon (W) is connected to Sidney (U). Foundations of Logic and Constraint Programming

  11. Complex Terms • The fragment of logic that only admits as arguments constants and variables (known as DATALOG) although convenient for its “simplicity” (given a finite set of constants, there is only a finite set of literals), but is also quite limited. • More expressive literals use functions as arguments. These more complex terms are also subject to substitutions. • In the previous theory, the atom connect can be augmented with a third argument, representing the whole path between the connected points. P1a: connect(X,Y,p(Y,Z)) ← direct(X,Y). P1b: connect(X,Y,p(X,P)) ← direct(X,Z), connect(Z,Y,P) P2: direct(lisbon, london). P3: direct(london, sidney). • This new theory allows the answer not only to include initial and final airports, but also the complete path between them. Foundations of Logic and Constraint Programming

  12. First Order Logic (Horn Clauses) P1a: connect(X,Y,p(X,Y)) ← direct(X,Y). P1b: connect(X,Y,p(X,P)) ← direct(X,Z), connect(Z,Y,P) P2: direct(lisbon, london). P3: direct(london, sidney). • The following query reports the whole path in the answer. Q0: ←connect(W,U,Q) Q1: ←direct(X,Z),connect(Z,Y,P) {W/X, U/Y, Q/p(X,P)} Q2: ←connect(london,Y, P) {X/lisbon, Z/london} Q3: ←direct(london,Y) {X1/london, Y1/Y, P/p(london,Y)} Q4: ←□{Y/sidney} • The path Q can now be obtained via the composition of the various substitutions. {W/X, U/Y, Q/p(X,P)}{X/lisbon, Z/london} = {W/lisbon, U/Y, Q/p(lisbon,P), ...} {W/lisbon,U/Y,Q/p(lisbon,P),...}{X1/london,Y1/Y,P/p(london,Y)}= {W/lisbon, U/Y, Q/p(lisbon,p(london,Y)),...} {W/lisbon, U/Y, Q/p(lisbon,p(london,Y)),...} {Y/sidney} = {W/lisbon, U/sidney, Q/p(lisbon,p(london,sidney)),...} Foundations of Logic and Constraint Programming

  13. Logic and Logic Programming • Logic Programming is based in logic, and in particular in the Resolution System. In fact, the theories addressed before, are also logic programs. % A simple logic program connect(X,Y,p(X,Y)):- direct(X,Y). connect(X,Y,p(X,P)):- direct(X,Z), connect(Z,Y,P). direct(lisbon, london). direct(london, sidney). • Logic Programming is declarative. A program specifies the logical relationships between different literals, and not the way in which they are computed and manipulated during the execution of the program. • In principle, the programmer does not have to be concerned with the execution of the program. All that is required from her/him is to guarantee that the facts and rules correctly model her/his problem. • The language “executor” computes the answers to the queries, according to some specific policy of applying the resolution inference rule. ?- connect(A,B,Path). A = lisbon, B = sidney, Path = p(lisbon,p(london,sidney) Foundations of Logic and Constraint Programming

  14. A B C D nil Functions and Data Structures: Lists • Functional terms are the way in which complex data structures may be used in logic programming • An important data structure is the List, for which special syntax exist. • Example of a List: [A, B, C, D] • Internal representation of the list: •(A, •(B, •(C, •(D, nil)))) where• is the functor. • Special Syntax • The list •(Head,Tail) is represented as [Head|Tail] • The nil list is represented as []. • Example: connect(X,Y,[X , Y]):- direct(X,Y). connect(X,Y,[X | P]):- direct(X,Z), connect(Z,Y,P). Foundations of Logic and Constraint Programming

  15. Declarative Languages • Logic Programming is an example of a declarative programming paradigm that should be contrasted with the more conventional imperative programming paradigm. • Imperative Programming Languages • Declaration part defines possible states (of variables) • Statement part defines transformation on states • Close to von Neumann architecture (states  memory) • Description on how results are computed • Example Languages: Pascal, C • Declarative Programming Languages • Abstraction from states and state transformation • Direct formulation of mathematical objects (functions, relations, constraints) • Description of what is computed • Example Languages: Prolog, Haskell, Curry, SICStus Prolog, Eclipse Foundations of Logic and Constraint Programming

  16. Declarative Languages • Logic Programming is one of several declarative programming paradigms. Other paradigms and languages exist with similar, but distinct characteristics. • Logic Programming Languages • Example: Prolog • Functional Programming Languages • Example: Haskell • Integrated (Functional-Logic) Programming Languages • Example: Curry, Mercury • Constraint Logic Programming Languages • Example: SICStus Prolog, Eclipse Foundations of Logic and Constraint Programming

  17. Advantages of Declarative Programming • Specifications are programs. In fact, programs are executable specifications. • The computation mechanism is not part of the language. • Declarative languages require such a (hidden) executor. • Thinking declaratively (what are things) is easier that thinking pocedurally (how things are built). • Declarative Programs are therefore much simpler to understand, develop and prove (correct). • The output of a logic program is a logical consequence of the program. • Logic Programs are flexible. • Arguments can be used in a variety of “modes” (input/output/mixed) Foundations of Logic and Constraint Programming

  18. Flexibility of Logic Programming • The program below defines whether X belongs to a list L. % predicate member member(X,[X|_]). % X is the head of the list member(X,[_|L]):- member(X,L). % or X is member of its tail • Flexibility and expressivity is illustrated by the following queries ?- member(1, [1,2,3,4]) yes ?- member(5, [1,2,3,4]) no ?- member(X, [1,2,3,4]) X = 1?; X = 2?; X = 3?; X = 4?; no. ?- member(1, L). L = [1|_]?; X = [_,1|_]?; ... Foundations of Logic and Constraint Programming

  19. Flexibility of Logic Programming • This flexibility is quite important to use and reuse definitions in different contexts. • Example: Predicate member_both, is valid when X is both a member of lists L1 and L2. • % predicate member_both member_both(X,L1,L2):- member(X,L1), % X is member of L1 member(X,L2), % and X is a member of L2 • Once defined this new predicate, it can be used in different queries. ?- member_both(4, [1,2,3,4], [5,6,7,8]) no ?- member_both(X, [1,2,3,4], [2,3,6,7]) X = 2?; X = 3?; no. • Notice that in this last query, X is an “output” argument in the first “call” to member, but is an “input” argument in the second “call”. Foundations of Logic and Constraint Programming

  20. Comparison with Imperative Programming • Procedures can be defined in imperative languages to achieve similar effects. However, they are much more cumbersome to program. • The following PASCAL procedure returns, in an array c, the common elements of arrays a and b. type list = array[1..n] of integer; procedure member_both(a,b: list; var c: list); var i,j,k: integer; begin k := 1; for i := 1 to n do for j := 1 to n do if a[i] = b[j] then begin c[k] := a[i]; k := k + 1 end end; • However, checking whether an integer belongs to arrays a and b requires inspection of the returned array c ! Foundations of Logic and Constraint Programming

  21. Logic Programming Shortcomings: Termination • As a programming language, (an executor of) Prolog has “built in” a strategy to search among the possible deductions. • Because there is no universally optimal strategy, that used in Prolog may cause problems, namely in program termination. This can be illustrated by a small modification of the previous program P1a: connect(X,Y):- direct(X,Y). P1b: connect(X,Y):- direct(X,Z), connect(Z,Y). P2: direct(lisbon, london). P3: direct(london, sidney). • If the clause P1a and P1b are swapped and rewritten as shown below P1b: connect(X,Y):-connect(Z,Y), direct(X,Z). P1a: connect(X,Y):- direct(X,Y). the query ?- connect (X,Y) would not terminate. • Nor would, the more strict query ?- connect (X,sidney). Foundations of Logic and Constraint Programming

  22. Prolog Handling of Termination: Declarativeness? • Provisions were made in the design of the Prolog language to circumvent some of the problems with termination. • The introduction of the cut (!), is one of the most used of such constructs. The problem is illustrated with the “more efficient” definition of member. % predicate member (modified) member(X,[X|_]):- !. % if X is the head of the list member(X,[_|L]):- member(X,L). % do not search in its tail • In this definition once X is found in the head of the list, it is no more searched in the tail of the list. However, the declarativeness of logic programming is lost, since this only works if X is an “input” argument. ?- member(3, [1,2,3,4]) yes ?- member(5, [1,2,3,4]) no ?- member(X, [1,2,3,4]) X = 1?; no. Foundations of Logic and Constraint Programming

  23. Logic Programming Shortcomings: Arithmetics? • Other pitfalls of Logic Programming are related to the handling of arithmetics. • In Logic Programming, all arguments are considered terms of the Herbrand universe, with no implicit semantics. • Integer operations can be defined in such universe with a base element (0) and a sucessor function s/1. For example, the set of integers may be defined with predicate % predicate is_integer/1 is_integer(0). is_integer(s(X)) :- is_integer(X) • The sum of integers may also be defined with with predicate is_sum/3 % predicate sum/3 sum(0,X,X). sum(s(X),Y,s(Z)) :- sum(X,Y,Z) • Other operations could be defined similarly % predicate is_prod/3 prod(0,X,0). prod(s(X),Y, Z):- prod(X,Y,W), sum(Y,W,Z). Foundations of Logic and Constraint Programming

  24. Logic Programming Shortcomings: Arithmetics? • This logic handling of arithmentics is declarative. For example, operations are reversible ?- sum_2(X, s(0), s(s(s(0)))) % ? X+1 = 3 X = s(s(0)); no. % X = 2 ?- sum_2(X, Y, s(s(s(0)))) % ? X+Y = 3 X = 0, Y = s(s(s(0))); % X = 0; Y = 3; X = s(0); Y = s(s(0)); % X = 1; Y = 2; X = s(s(0)); Y = s(0); % X = 2; Y = 1; X = s(s(s(0))); Y = 0; no.% X = 3; Y = 0; • However, such pure approach is VERY inefficient (and quite cumbersome). • Representing “number” N requires a term of size O(N) • Suming M with N requires O(M) “calls” of predicate is_sum • Multiplying M with N requires M “calls” to predicate sum with N as first argument, i.e. O(M*N) calls of predicate sum • Lack of typing is also a potential problem: sum(s(0),a, s(a)) ? Foundations of Logic and Constraint Programming

  25. Prolog Handling of Arithmetics: Declarativeness? • Prolog assumes some implicit typing for some special terms (integers) to “bypass” such problems . • Hence the sum of two integers (teturning an integer) may be defined by the more “realistic” predicate is_sum_2, using the built_in predicate is/2. % predicate sum_2/3 sum_2(X,Y,Z) :- Z is X+Y. • Such “fix” destroys somehow the declarativeness of the language. The predicate works as expected, when both X and Y are provided as “input” ?- sum_2(3, 5, 8) yes ?- sum_2(3, 5, X) X = 8?; no. but Prolog is not able to handle inverse operations (it generates an error message) ?- sum_2(3, X, 8). ???? ?- sum_2(X, Y, 8). ???? Foundations of Logic and Constraint Programming

  26. Constraints and Logic Programming • Declarativeness may be maintained in “logic programmes” if some variables have certain “types”, i.e. range over certain domains, and the algebra of such domains is considered in the executor. • Assume for example predicates defined over rational numbers and take predicates % predicates over rationals {} lin1(X,Y):- {2*X + 3*Y = 7}. lin2(X,Y):- {3*X – 2*Y = 4}. and take query ?- lin1(X,Y), lin2(X,Y). • Although pure Prolog would not be able to handle such query, constraint logic programming, an extension of logic programming for particular domains, is able to handle this query and report the correct answer X = 2; Y = 1; no. • Other important domains (finite, sets, ...) are usually considered in the CLP(X) scheme. Foundations of Logic and Constraint Programming

  27. Plan of the Course • Foundations of Logic Programming • Logic Programming by example • Unification • SLD Derivations • Interpretations and Declarative Semantics of LP • Search. Execution Trees and Operational Semantics of LP • Negation (by failure) in Logic Programming • Foundations of Constraint Logic Programming • Constraints: Modelling examples • LP and CLP – Rewritting systems and Operational Semanbtics • Boolean Unification: CLP(B) • Algebra of Linear Constraints over Rationals: CLP(Q) • Finite Domains: Modelling, Consistency maintenance (general) • Finite Domains: Global constraints and specialised consistency maintenance. Foundations of Logic and Constraint Programming

  28. Bibliography • Logic Programming • Text Books: • Krzysztof Apt, From Logic Programming to Prolog, Prentic Hall, 1997 • Leon Sterling and Ehud Shapiro, The Art of Prolog, MIT Press, 1986 • (Logic) Constraint Programming • Text Books: • Krzysztof Apt, Principles of Constraint Programming, Cambridge University Press, 2003 • K. Marriot and P. Stuckey, Programming with Constraints – An Introduction, MIT Press, 1998 • Dina Rechter, Constraint Processing, Morgan Kauffman, 2003 • Web Page • http://ssdi.di.fct.unl.pt/mcl/flcp Foundations of Logic and Constraint Programming

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