The Logic Programming Paradigm

Transcription

1 Roadma Tabled Logic Programming and Its Alications Manuel Carro, Pablo Chico de uzmán School of Comuter Science, Technical University of Madrid, Sain IMDEA Software Institute, Sain Prometidos-CM Summer School Facultad de Informática, UCM Setember 9-, 0 Introduction: the Logic Programming aradigm. Pitfalls. Tabling Basics: Memoing. Breaking loos. Analysis of some alications. Advanced Tabling Caabilities: Subsumtive tabling. Negation with tabling. Imlementation issues. Oen research toics. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS / 45 Intro to LP Logic rogramming: declarative, relational style. Origins linked to theorem roving and linguistics. Try to solve roblems / execute rograms with a theorem rover. Can you do that? Yes, but you (initially) end u with a sort of limited theorem rover. Tabling also tries to overcome some of these initial limitations. Aim: clean, exressive, obviously correct rograms. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45

2 Facts, Queries, Rules Execution of a Query erson ( mike ). erson ( alice ). erson ( bob ). erson ( mirna ). food ( vegetable, salad ). likes ( mike, vegetable ). food ( vegetable, beans ). likes (mike, fish ). food ( meat, chicken ). likes ( alice, vegetable ). food (meat, rabbit ). likes (bob, fish ). food (meat, ork ). likes (bob, meat ). food (fish, soul ). likes (bob, vegetable ). food (fish, octous ). likes ( mirna, fish ). likes ( mirna, vegetable ). meal ( Person, Food ): - likes ( Person, Tye ), food (Tye, Food ). same_food (P, P, Food ): - meal (P, Food ), meal (P, Food ).?- meal(alice, What). likes(alice, T), food(t, What) T = vegetables food(vegetables, What) What = salad What = beans Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 5 / 45 Execution of a Query Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 6 / 45 Logical Variables?- meal(who, beans). Who = mike T = vegetable food(vegetable, beans) Who = mike T = fish likes(who, T), food(t, beans) Who = alice T = vegetable food(vegetable, beans) Who = bob T = fish...?- likes (mirna, X), X = meat. no food(fish, beans) food(fish, beans) fail fail Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 7 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 8 / 45

3 Arithmetic Prolog arithmetic is a comromise. X is Ex Aritmetically evaluates Ex Unifies result with X?- X is 3 * srt (). X = ??- X = 3 * srt (). X = 3* srt ()??- is * X. { ERROR : arithmetic :*/, arg - Instantiation Error } Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 9 / 45 Structured Data coule ( married ( mirna, bob )).?- coule (C). C = married ( mirna, bob )?- coule ( married ( mirna, X )). X = bob Lists What Prolog notation Emty list Singleton list a Two-element list a, b List with a head a and a tail T a T List starting with a, b, c and continuing with tail T a, b, c T Interretation of Formulas Rules and facts are stylized formulas: meal ( Person, Food ): - likes ( Person, Tye ), food (Tye, Food ). is a way of writing the formula and a uery?- meal ( mirna, X). P F T likes(p, T ) food(t, F ) meal(p, F ) is an attemt to mechanically and constructively determine whether X.meal(mirna, X ) holds. If so, it is demonstrated by finding an X for which meal(mirna, X ) can be inferred from the rogram. This is what a Prolog system does by alying an automated deduction rocedure. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 0 / 45 Pitfalls Redundant comutations hanoi (N, L): - hanoi (N, A, B, C, L), A = a, B = b, C = c. hanoi (0, _, _, _, ). hanoi (N, A, B, C, M): - N > 0, N is N -, hanoi (N, A, C, B, M), hanoi (N, C, B, A, M), aend (M, move (A, B) M, M). Non termination of otherwise correct rograms ath (A, B): - edge (A, C), ath (C, B). ath (A, B): - edge (A, B). fib (0) := 0. fib () :=. fib (N) := fib (N -) + fib (N -) :- N >. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS / 45

4 Tabling Basics Towards Tabling: Memoing Tabling Basics Remember calls and their answers. Can imrove seed. We will see two examles. Tabling Basics Towards Tabling: Memoing (Cont.) Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Tabling Basics Towards Tabling: Memoing (Cont.) hanoi (0, _A, _B, _C, ). hanoi (N, A, B, C, M): - N > 0, N is N -, hanoi (N, A, C, B, M), hanoi (N, C, B, A, M), aend (M, move (A, B) M, M).?- hanoi (, A, B, C, L). L = move (A,C), move (A,B), move (C,B) Solutions for a given size differ only on name of the egs. There are two (recursive) calls of the same size. Calling with free variables gives the most general answer. Memoing answer and recovering it adjusting variables in the answer to match those in the call avoids recomuting answer. Length of answer in order of comutation stes, so constant seedu exected. Exerimentally, 3-fold seedu by adding :- table hanoi/5. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 5 / 45 fib (0) := 0. fib () :=. fib (N) := fib (N -) + fib (N -) :- N >. Comuting fib(n-) includes the comutation of fib(n-). Memoing answers greatly seeds u comutation. :- table fib/. reduces comlexity from exonential (O(( + 5 ) n )) to linear! Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 6 / 45

5 Tabling Basics Tabling Basics Towards Tabling: Termination Issues Towards Tabling: Termination Issues ath (A, B): - edge (A, C), ath (C, B). ath (A, B): - edge (A, B). a b c ath (A, B): - edge (A, C), ath (C, B). ath (A, B): - edge (A, B). a b c edge (a, b). edge (b, a). edge (b, c). edge (a, b). edge (b, a). edge (b, c). Logical reading: There is a ath from A to B if they are directly connected by edges. There is a ath from A to B if there is an edge from A to C and a connection from A to B. Question: Which nodes are reachable from a??- ath(a, X). Standard Prolog (SLD) strategy leads to loos:?- ath (a, X). edge (a, C ), ath (C, X) edge (a, b), ath (b, X) edge(a, b), edge (b, C ), ath (C, X) edge(a, b), edge (b, a), ath(a, X) Initial reaction: carry around list of visited nodes. But: is it necessary? What is wrong with the rogram? Does not it recisely secify the meaning of being reachable? Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 7 / 45 Tabling Basics Towards Tabling: Bottom-U Evaluation Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 8 / 45 Tabling Basics Towards Tabling: Bottom-U Evaluation Read the rogram as logic: ath (A, B) edge (A, C) ath (C, B). ath (A, B) edge (A, B). a b c ath (A, B) edge (A, C) ath (C, B). ath (A, B) edge (A, B). edge (a, b). edge (b, a). edge (b, c). a b c edge (a, b). edge (b, a). edge (b, c). From {edge(a, b), edge(b, a), edge(b, c)} and ath(a, B) edge(a, B) infer {ath(a, b), ath(b, a), ath(b, c)} using ath(a, B) edge(a, C) ath(c, B) infer {ath(a, a), ath(b, b), ath(a, c)} Nothing more can be inferred. The meaning of the rogram is: {edge(a, b), edge(b, a), edge(b, c), ath(a, b), ath(b, a), ath(b, c), ath(a, a), ath(b, b), ath(a, c)} Note that: We have not changed the rogram (no visited node list). But: we wanted to find out which nodes are reachable from node a and we comuted much more. We actually comuted the least fixoint of the rogram (its standard semantics). Other alications could reuire the greatest fixoint. More on bottom-u evaluation surely in the Deductive Databases talk. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 9 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 0 / 45

6 Tabling Basics Tabling Basics Towards Tabling: Susending and Resuming Tabled Evaluation ath (A, B) :- edge (A, C), ath (C, B). ath (A, B) :- edge (A, B). edge (a, b). edge (b, a). edge (b, c). Challenge: a goal-directed (to-down) strategy which Detects loos. a b c Derives only the conseuences necessary for to-down execution. Idea: when we have :-,, r.. and the call to is reeated (enters loo): Susend the comutation before calling. Use the fact to generate a solution to the original uery. 3 Use this solution to resume the susended comutation. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS / 45 Examle :- table /. (A) :-, (B), A =. (A) :- A =. Subgoal Answers 4. A =. (A) 8. A = 3. Comlete 6. (), A =. 7. A =.. (B), A =. Susension. (A).., (B), A =. 0. (), A =.. A =.. fail 3. A =. Linear tabling uses recomutation instead of susension. 5. A =.? (A). 9. A =. 4. no. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS / 45 Parsing Let us assume inut has been tokenized Arithmetic terms exr exr + term exr term term term * fact term fact fact ( exr ) fact int ( Int ) Examles of strings we want to arse: String Evaluates to * * 4 4 ( + 3) * 4 0 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45

7 Parsing (Cont.) Attemt of a Prolog arser for the revious grammar exr exr + term term term term * fact fact fact ( exr ) int ( Int ) exr (S0,S) :- exr (S0,S), S = + S, term (S,S). exr (S0,S) :- term (S0,S). term (S0,S) :- term (S0,S), S = * S, fact (S,S). term (S0,S) :- fact (S0,S). fact (S0,S) :- S0 = ( S, exr (S,S), S = ) S. fact (S0,S) :- S0 = N S, integer (N). exr(i, R) means: R is a suffix of I, and the refix of I w.r.t. R is a valid exression. Comlete arsing:?- exr(inut, ).?- exr (3, +, 4, *, R). R = * R = +,4,* Parsing (Cont.) Parsing in action exr (S0,S) :- exr (S0,S), S = + S, term (S,S). exr (S0,S) :- term (S0,S). term (S0,S) :- term (S0,S), S = * S, fact (S,S). term (S0,S) :- fact (S0,S). fact (S0,S) :- S0 = ( S, exr (S,S), S = ) S. fact (S0,S) :- S0 = N S, integer (N).?- exr(3, ). (wait forever) What is haening??- exr (3, ). exr(3, S ), S = + S, term (S, ) exr(3, S ), S = + S, term (S, S ), S = + S, Prolog rovides a secific syntax to write arsers, closer to the notation. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 5 / 45 Parsing (Cont.) A left-recursive rule exr (S0,S) :- exr (S0,S), S = + S, term (S,S). generates a reeated comutation and a loo. Tabling left-recursive rules amends this roblem: :- table exr /, term /. Then:?- exr (3, +, 4, *, ). no?- exr (3, +, 4, *, 7, ). yes?- exr ( (, 3, +, 4, ), *, 7, ). yes?- E = _,_,_,_,_,_,_,_, exr (E, ). no Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 6 / 45 Tabling and Parsing Tabling + a recursive descendant arser (like the one we have resented) imlements an Earley arsing: The Earley arser is a tye of chart arser mainly used for arsing in comutational linguistics, named after its inventor, Jay Earley. The algorithm uses dynamic rogramming. Wikiedia Earley arsers arse all context-free grammars: Comlexity O(n 3 ) for ambiguous grammars. Comlexity O(n ) for unambiguous grammars Linear for a large class of grammars. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 7 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 8 / 45

8 For Comleteness: Earley Parser f u n c t i o n EARLEY PARSE( words, grammar ) ENQUEUE( ( γ S, 0, 0 ), c h a r t 0 ) f o r i from 0 to LENTH( words ) do f o r each s t a t e i n c h a r t i do i f INCOMPLETE?( s t a t e ) then i f NEXT CAT( s t a t e ) i s a n o n t e r m i n a l then PREDICTOR( s t a t e ) // non t e r m i n a l e l s e do SCANNER( s t a t e ) // t e r m i n a l e l s e do COMPLETER( s t a t e ) end end r e t u r n c h a r t r o c e d u r e PREDICTOR ( (A α B, i, j ) ), f o r each (B γ ) i n RAMMAR RULES FOR(B, grammar ) do ENQUEUE( (B γ, j, j ), c h a r t j ) end r o c e d u r e SCANNER( (A α B, i, j ) ), i f B PARTS OF SPEECH( word j ) then ENQUEUE( (B word j, j, j + ), c h a r t j + ) end r o c e d u r e COMPLETER( (B γ, j, k ) ), f o r each (A α Bβ, i, j ) i n c h a r t ( j ) do ENQUEUE( (A αb β, i, k ), c h a r t k ) end Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 9 / 45 Automatic Dynamic Programming Automatic Dynamic Programming Kna-sack roblem: given n items of integer size k i ( i n), and a kna-sack size K: Determine whether there is a subset of the items that sums to K. Find such a subset we are not maximizing / minimizing. Program: try all items (n, n,..., ) and decide, for each item, whether to include it or not. ks (0,0). ks(i,k) :- I >0, %% Ski item Ith I is I -, ks(i,k). ks(i,k) :- I >0, %% Include item Ith item_size (I,Ki), K is K-Ki, K >= 0, I is I -, ks(i,k ). item_size (,). item_size (,3). item_size (3,5). item_size (4,6). Worst-case comlexity is n. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 30 / 45 Knasack and (again) rah Traversal A tabling solution to the knasack roblem: :- table ks/. ks (0,0). ks(i,k) :- I >0, I is I -, ks(i,k). ks(i,k) :- I >0, item_size (I,Ki), K is K-Ki, K >= 0, I is I -, ks(i,k ). item_size (,). item_size (,3). item_size (3,5). item_size (4,6). Worst-case comlexity is O(nK). Intuitition: for every n and every number smaller than or eual to K, we check only once whether it succeeds or fails. add I I 0 I I I n I n add add I add add I n add Decision at every node vs. incrementally building a transitive closure. In fact, Knasack solution very similar to grah traversal. Many roblems very similar to grah traversals or directly grah traversal. E.g., model checking. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 3 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 3 / 45

9 For Comleteness: Udi Manber Procedural Version Tabling: Reca and Big Picture Tabling terminates and is comlete for all calls with the bounded-term-deth roerty (those for which there is a bound on the size of the terms which can be generated). Tabling comutes a fix oint (the least fix oint) for tabled redicates in a goal-driven way. Imroves efficiency (automatic dynamic rogramming). Imroves termination. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 33 / 45 Tabling: Reca and Big Picture Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 34 / 45 It is not comlete (naturally) for rograms with an infinite least fixoint: n (0). n(s(x)): - n(x).?- n(x), X = a. It terminates for rograms which loo due to reeated calls. Advanced Tabling Caabilities n(s(x)): -?- n(x). n(x). It may not terminate for rograms without bound term deth roerty: (X): - (s(x )).?- (X). Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 35 / 45

10 Advanced Tabling Caabilities Advanced Tabling Caabilities Variant vs. Subsumtion Tabling Variant vs. Subsumtion Tabling (Cont.) (f(y),x,) and (f(z),u,) are variants as one can be made to look like the other by a renaming of the variables. Let us have t : (f(y),x,) and t : (f(z),z,). We can rename variables in t to become t, but not the other way around. They are not variants. t subsumes (is more general than) t. Call Variance vs Call Subsumtion: Either Variance or Subsumtion can be used to check if we are executing a new tabled call. Answer Variance vs Answer Subsumtion: Either Variance or Subsumtion can be used to check if we have comuted a new answer for a tabled redicate. Variant Tabling: More efficient table look-u. Better for side-effects. Useful for goal-directed ueries and meta-interreters. Subsumtion Tabling: Can avoid more recomutation (catches more similarities). Better termination: :- table s / as subsumtive. (X) :- (s(x )). Can save sace in internal tables. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 37 / 45 Advanced Tabling Caabilities Variant vs. Subsumtion Tabling (Cont.) Subsumtion can be arametric It is natural to use syntactical subsumtion. However, from a more semantic / knowledge reasoning oint of view, terms may have an interretation from which subsumtion can be inferred. E.g., the term X > 3 subsumes X = 4 in the arithmetic domain. is(some animal, dog) is subsumed by belongs(some animal, mammal). Lattice for subsumtion check. Also in abstract interretation: relations in lattice of abstract domain encoded in subsumtion lattice. Remember tabling comutes a fixoint. Basically, what an abstract interreter does! (and the mechanism is not very different) Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 38 / 45 Advanced Tabling Caabilities Negation and Well-Founded Semantics ives semantics for all normal logic rograms. It allows logic rograms to adeuately handle inconsistencies and araconsistencies, for examle: The village barber shaves everyone in the village who does not shave himself. shaves ( barber, Person ): - villager ( Person ), not shaves ( Person, Person ). shaves ( doctor, doctor ). % Play with this villager ( barber ). villager ( doctor ). villager ( mayor ).?- shaves (X, Y). Alications: verification, Flora-, medical informatics... Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 39 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 40 / 45

11 Imlementation Break Imlementation Break X Imlementation Break Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Imlementation Break CHAT AREA Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45

12 Imlementation Break Imlementation Break X CHAT AREA 3 CHAT AREA C r s Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Imlementation Break Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Imlementation Break 3 CHAT AREA X CHAT AREA C r C 3 r C 3 s s Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45

13 Imlementation Break Imlementation Break CHAT AREA C 3 CHAT AREA r C 3 r C 3 s s Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 4 / 45 Oen Research Toics Oen Research Toics Oen Research Toics Pruning of Answer-On-Demand Tabling. Subsumtive tabling with constraints. Pruning of revious subsumed calls. Call abstraction. Parallelism. Table comression. Side-effects. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 44 / 45

14 Oen Research Toics Thanks Parts of this resentation draw from material from: Terrance Swift (CENTRIA, Universidade Nova de Lisboa) David S. Warren (NYU at Stony Brook) to whom we thank. Carro, Chico de uzmán (IMDEA,UPM) Tabling and alications Prometidos SS 45 / 45