How does CFP deal with stratifiable databases (with recursion)? layer 2. layer 1

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1 CFP-Semantics for Stratifiable Databases (1) How does CFP deal with stratifiable databases (with recursion)? R s(x) t(x,y), not p(x,y). layer p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). layer 2 layer 1 r(1,2,). r(2,,4). r(,1,2). t(1,4). F 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 1

2 Example Treated by Iterated Fixpoint Computation (a) Just to remind you how F* had been computed previously using layers and local fixpoints: R s(x) t(x,y), not p(x,y). p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). R# R# 2 R# 1 r(1,2,). r(2,,4). r(,1,2). t(1,4). F T neg * [R# 1] r(1,2,). q(1,2). r(2,,4). q(2,). r(,1,2). q(,1). F 1 T neg * [R# 1] r(1,2,). r(2,,4). r(,1,2). F t(1,4). t(1,4) Prof. Dr. Rainer Manthey Intelligent Information Systems 2

3 Example Treated by Iterated Fixpoint Computation (b) R s(x) t(x,y), not p(x,y). p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). R# R# 2 R# 1 r(1,2,). r(2,,4). r(,1,2). t(1,4). F F 2 T neg * [R# 2]... p(1,2). p(,2). p(2,2). p(,). p(1,1).... p(1,2). p(,2). p(1,). p(2,1). T neg * [R# 2] T neg * [R# 2]... q(1,2). q(2,). q(,1). p(1,2). p(2,). p(,1). T neg * [R# 2] r(1,2,). q(1,2). r(2,,4). q(2,). r(,1,2). q(,1). t(1,4). F Prof. Dr. Rainer Manthey Intelligent Information Systems

4 Example Treated by Iterated Fixpoint Computation (c) R s(x) t(x,y), not p(x,y). p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). R# R# 2 R# 1 r(1,2,). r(2,,4). r(,1,2). t(1,4). F F r(1,2,). q(1,2).... r(2,,4). q(2,). r(,1,2). q(,1). r(1,2,). q(1,2). p(,2). r(2,,4). q(2,). p(1,). r(,1,2). q(,1). p(2,1). F 2 t(1,4). p(1,2). p(2,). s(1). p(2,2). T neg * [R# ] t(1,4). p(1,2). p(1,1). p(2,). p(2,2). p(,1). p(,) Prof. Dr. Rainer Manthey Intelligent Information Systems 4

5 CFP-Semantics for Stratifiable Databases (1) How does CFP deal with stratifiable databases (with recursion)? R s(x) t(x,y), not p(x,y). p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). r(1,2,). r(2,,4). r(,1,2). t(1,4). F Expansion phase? 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 5

6 CFP-Semantics for Stratifiable Databases (1) How does CFP deal with stratifiable databases (with recursion)? R s(x) t(x,y), not p(x,y). layer p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). layer 2 layer 1 r(1,2,). r(2,,4). r(,1,2). t(1,4). F CFP does not need any layers/strata, but applies the expansion phase to all rules simultaneously, however without touching negation (true-bodies are omitted here): r(1,2,). r(2,,4). r(,1,2). t(1,4).... q(1,2). q(2,). q(,1). s(1) not p(1,4).... p(1,2) not q(2,1).... p(,2) not q(2,1).... p(2,2) not q(2,1) Prof. Dr. Rainer Manthey Intelligent Information Systems 6

7 CFP-Semantics for Stratifiable Databases (2) R s(x) t(x,y), not p(x,y). p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). r(1,2,). r(2,,4). r(,1,2). t(1,4). F F cond * r(1,2,). r(2,,4). r(,1,2). s(1) not p(1,4). Reduction phase? t(1,4). p(1,2) not q(2,1). p(,2) not q(2,1). q(1,2). p(2,2) not q(2,1). q(2,).... q(,1) Prof. Dr. Rainer Manthey Intelligent Information Systems 7

8 CFP-Semantics for Stratifiable Databases (2) R s(x) t(x,y), not p(x,y). p(x,y) q(x,y), not q(y,x). p(x,y) q(x,z), p(z,y). q(x,y) r(x,y,z). r(1,2,). r(2,,4). r(,1,2). t(1,4). F F cond * F red * r(1,2,). r(2,,4). r(,1,2). s(1) not p(1,4). r(1,2,). r(2,,4). r(,1,2). s(1). t(1,4). p(1,2) not q(2,1). p(,2) not q(2,1). q(1,2). p(2,2) not q(2,1). q(2,).... q(,1). Reduction eliminates all negative literals t(1,4). p(1,2). p(,2). q(1,2). p(2,2). q(2,).... q(,1) Prof. Dr. Rainer Manthey Intelligent Information Systems 8

9 Semantics for Different Classes of Deductive Databases: Overview In both stratifiable examples (recursive and non-recursive) the CFP-method always delivered the same result as iterated fixpoint computation (IFP) based on a stratification, even though in a different manner. This observation holds for each database! CFP- and IFP-semantics coincide for stratifiable databases. CFP extends IFP conservatively for unstratifiable databases. This is true in particular for (semi-)positive DBs, too, where all three methods deliver the same results as simple fixpoint iteration. Proofs for these claims can be found in the original research papers introducing the resp. methods (which are not at all trivial!). CFP-method Unstratifiable Stratifiable Semi-positive Iterated FP-method "Simple" FPI 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 9

10 Which Databases are Reasonable"? A database can be regarded as meaningful without any doubt, if (even in presence of unstratifiable rules) no fact is assigned the truth value undefined. Certainly not meaningful are only such DBs, for which all facts have to be assigned undefined. Undefined facts only No undefined facts Stratifiable Semi-positive Unstratifiable Individual undefined facts 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 10

11 Appendix: Formalization of CFP Expansion Phase In phase 1 (expansion), rules are applied to conditional facts. When doing so, only positive body literals are replaced, however, and conditions in the bodies of the substituted conditional facts are transferred. First, we define once again an operator formalizing the application of individual rules: R i A B 1,..., B n,, not C 1,..., not C m. Name of this operator: T ' cond Attention! The following definition is not yet final! It still contains a serious error in reasoning (thus, a ' with the T)! Transferred conditions Not yet "evaluated" negative body literals T ' cond [R i ] ( CF ) = def {(A D 1,..., D n, not C 1,..., not C m ) σ σ is a consistent variable substitution, such that 1 j n : (B j D j ) σ CF holds } Conjunctions of literals 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 11

12 T' cond in the "Normal Case" A B 1, not C 1. p(x) q(x), not r(x). q(x) s(x), not t(x). r(x) s(x), not w(x). R 1 B j D j q(a) not t(a). q(b) not t(b). r(a) not w(a). r(b) not w(b). T ' cond [R 1 ] A D j, not C 1. p(a) not t(a), not r(a). p(b) not t(b), not r(b). T ' cond [R i ] ( CF ) = def {(A D 1,..., D n, not C 1,..., not C m ) σ σ is a consistent variable substitution, such that 1 j n : (B j D j ) σ CF holds } 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 12

13 1 st Problem with T' cond : Redundant true-literals in Conditions R p(x) q(x), not r(x). q(x) s(x), not t(x). r(x) s(x), not w(x). R 2 F cond = CF 0 s(a) true. s(b) true. t(b) true. w(b) true. w(c) true. T ' cond [R 2 ] T ' cond [R 2 ] (CF 0 ) q(a) true, not t(a). q(b) true, not t(b). If conditional facts with "artificial" bodies are used for replacing positive literals, logically redundant true-literals will occur which will have to be absorbed by the "real" literals, if any, due to a simple law of propositional logic: true and A A 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 1

14 2 nd Problem with T' cond : Duplicates in Conditions In our running example, this second problem does not occur, but a fitting example can be easily constructed: p(x) q(x), v(x), not r(x). q(a) not t(a). v(a) not s(c,a), not t(a). p(a) not t(a), not s(c,a), not t(a), not r(a). Double occurrence of the same literal According to the Law of Idempotence in logic, duplicates can be eliminated and this should be done, because the multiplicity of literals may grow and grow due to recursion: A and A A 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 14

15 Conditional T-Operator with "Pruning" of Conditions "Pruning" is a notion from gardening, where it means cutting of shrubs or other plants. We will introduce an operator 'prune', which cuts a given conjunction of literals in such a way that no duplicates do occur anymore and each occurrence of true is eliminated (until at least one literal remains). For 'prune' we don t present any formal definition its meaning should be evident. Thus, we are able to present an improved version of the operator for single rules applied to conditional facts as follows: Now unprimed! T cond [R i ] ( CF ) = def { A σ prune [(D 1,..., D n, not C 1,..., not C m ) σ ] σ is a consistent variable substitution, such that 1 j n : (B j D j ) σ CF holds } 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 15

16 Formalization of the Expansion Phase: Summary The expansion phase is organized overall like a quite "normal" fixpoint iteration similar to the way we organized it for positive rules. All rules are applied to a set of conditional facts using a collective T cond : T cond [R](CF) = def R i R T cond [R i ] (CF) Iteration can be done if we "carry along" all results previously obtained. For doing so, a cumulative T cond *-Operator is used: T cond * [R] (CF) = def T cond [R] (CF) CF For a given transformed set of base facts F cond, the result of the first phase of the CFPapproach is again the least fixpoint of T cond * [R] containing all of F cond : F cond * = def lim T cond *[R] i (F cond ) i 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 16

17 Formalization of the Reduction Phase (1) The reduction phase requires two basic operators: One operator for eliminating negative literals from conditions, and another operator for eliminating entire conditional facts from the input set. Negative literals can be regarded as satisfied over a given set of conditional facts, if no head of any conditional fact corresponds to the positive part of that literal. As a useful tool for formalizing this step (and others to come), we introduce an operator for extracting all head literals from a set of conditional facts CF: heads (CF) = def { A (A B) CF } If for a negative literal not L the positive part L is not in heads (CF), then the resp. negative literal can be replaced by true and subsequently eliminated by "prune". An operator performing this step will be called Red true (for "reduce") Prof. Dr. Rainer Manthey Intelligent Information Systems 17

18 Formalization of the Reduction Phase (2) Red true can be formalized as follows: Red true (CF) = def { A prune [C red 1,...,C red n ] (A not C 1,..., not C n ) CF } with for 1 i n C red i = def true, if C i heads(cf) not C i else No longer negation as failure! For better understanding of this (non-trivial) construct, consider the corresponding natural language version of this definition again: If for a negative literal not L the positive part L is not in heads (CF), then this negative literal can be replaced by true and subsequently eliminated by "prune" Prof. Dr. Rainer Manthey Intelligent Information Systems 18

19 Example for the First Reduction Operator CF q(a) not t(a). q(b) not t(b). r(a) not w(a). r(b) not w(b). p(a) not t(a), not r(a). p(b) not t(b), not r(b). s(a) true. s(b) true. w(b) true. w(c) true. t(c) true. Red true (CF) q(a) true. q(b) true. r(a) true. r(b) not w(b). p(a) true, not r(a). p(b) true, not r(b). s(a) true. s(b) true. w(b) true. w(c) true. t(c) true. prune heads(cf) = {q(a), q(b), r(a), r(b), p(a), p(b), s(a), s(b), t(c), w(b), w(c)} 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 19

20 Formalization of the Reduction Phase () If for a negative literal not L it is already clear that the positive part L is satisfied (because there is an "unconditional fact" L true), we can replace it by false. Conditional facts the body of which contains false can never evaluate to true and thus should be eliminated from the current set of conditional facts. These considerations motivate a second reduction operator Red false : Red false (CF) = def CF CF false where CF false = def {A not C 1,..., not C n) (A not C 1,..., not C n ) CF and 1 i n: C i true CF } Both reduction operators have to be applied sequentially: Red (CF) = def Red false (Red true (CF)) 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 20

21 Example for the Second Reduction Operator CF q(a) true. q(b) true. r(a) true. r(b) not w(b). p(a) not r(a). p(b) not r(b). s(a) true. s(b) true. w(b) true. w(c) true. t(c) true. Conditional facts to be eliminated: CF false CF CF false r(b) not w(b). p(a) not r(a). q(a) true. q(b) true. r(a) true. p(b) not r(b). s(a) true. s(b) true. w(b) true. w(c) true. t(c) true. Will be eliminated during the next iteration by the first operator Prof. Dr. Rainer Manthey Intelligent Information Systems 21

22 Formalization of the Entire CFP-Semantics In the reduction phase all reduction steps will be applied in alternation as long as the set of facts under reduction does not shrink anymore: Least fixpoint of Red F red * = def lim Red i (F cond *) i Reduction starts from the result of the expansion phase The semantics of a deductive database D = (R,F) is based on three truth values true, false, and undefined, determined from the final result of the "double iteration" as follows: F pos * = def { A A true F red * } F undef * = def heads(f red *) F pos * F neg * = def H D heads(f red *) Heads of all finally "unconditional" facts Heads of all "surviving" conditional facts Complement of both sets 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 22

23 Important Original Sources Concerning Semantics of Deductive Databases Main source of CFP method van Emden/Kowalski: "The semantics of predicate logic as a programming language", Journal of the ACM 2(4):7-742, 1976 Apt/Blair/Walker: "Towards a theory of declarative knowledge", in: J. Minker (Hrsg.): "Foundations of Deductive Databases and Logic Programming":89-148, Morgan Kaufmann, 1988 (Preprint 1986 as workshop proceedings) van Gelder/Ross/Schlipf: "Unfounded Sets and Well-Founded Semantics for General Logic Programs, PODS 1988:221-20, (long version: "The well-founded semantics for general logic programs", Journal of the ACM 8(): , 1991) Gelfond/Lifschitz: "The stable model semantics for logic programming", ICLP 1988: van Gelder: "The alternating fixpoint of logic programs with negation", PODS 1989: 1-10 (long version: Journal of the ACM 47: , 199) Bry: "Logic programming as Constructivism: A Formalization and its Application in Databases", PODS 1989:4-50 Bry: "Negation in logic programming: A formalization in constructive logic, Proc. IS&AI, 1990:0-46 Dung/Kanchansut: "A Fixpoint Approach to Declarative Semantics of Logic Programs", NACLP 1989: Brass/Zukowski/Freitag: "Transformation-based Bottom-Up Computation of the Well- Founded Model", NMELP 1996, LNCS 1216: Prof. Dr. Rainer Manthey Intelligent Information Systems 2

24 A More "Realistic" Unstratifiable, but Meaningful Example (1) Finally: Example of a rule set for which unstratifiability arises quite "naturally" and which might appear in many real applications. Intuitive semantics: Quality control of complex objects (Air planes, computers etc.) is_ok(t): Part T is in an acceptable state has_been_tested(t): Part T has been tested explicitly component_of(x,y): Part Y is a component of part X is_complex_part(t): Part T is composed from components has_bad_component(t): Part T has at least one component, which is not ok. R is_ok(t) has_been_tested(t). is_ok(t) is_complex_part(t), not has_bad_component(t). has_bad_component(t) component_of(t, T'), not is_ok(t'). is_complex_part(t) component_of(t, _). is_ok not not has_bad_component 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 24

25 A More "Realistic" Unstratifiable, but Meaningful Example (2) F R Component_of: is_ok(t) has_been_tested(t). is_ok(t) is_complex_part(t), not has_bad_component(t). has_bad_component(t) component_of(t, T'), not is_ok(t'). is_complex_part(t) component_of(t, _). p 1 p 2 p p 4 tested (thus, ok) ok recursively bad component p 5 p 6 p 7 p 8 p 9 p Prof. Dr. Rainer Manthey Intelligent Information Systems 25

26 Example in Compact Form: Facts in Datalog and Graphically R: ok(t) t(t). F: ok(t) cx(t), not bc(t). bc(t) cp(t, T'), not ok(t'). cx(t) cp(t, _). cp(1,2). cp(4,7). t(2). cp(1,). cp(4,8). t(7). cp(1,4). cp(8,9). t(9). cp(2,5). cp(8,10). t(10). cp(2,6). Component_of: p 1 p 2 p p 4 tested (thus, ok) ok recursively bad component p 5 p 6 p 7 p 8 p 9 p Prof. Dr. Rainer Manthey Intelligent Information Systems 26

27 Treating this Example by CFP: Expansion Phase? R: ok(t) t(t). F: ok(t) cx(t), not bc(t). bc(t) cp(t, T'), not ok(t'). cx(t) cp(t, _). cp(1,2). cp(4,7). t(2). cp(1,). cp(4,8). t(7). cp(1,4). cp(8,9). t(9). cp(2,5). cp(8,10). t(10). cp(2,6). This big example will be discussed and performed In detail in the exercises next Wednesday (Dec. 21, 2016). The slides with the detailed solution are already included In the lecture part because of the general technique applied Here (which is recommendable for CFP in general) Prof. Dr. Rainer Manthey Intelligent Information Systems 27

28 Treating this Example by CFP: Expansion Phase! R: ok(t) t(t). F: ok(t) cx(t), not bc(t). bc(t) cp(t, T'), not ok(t'). cx(t) cp(t, _). cp(1,2). cp(4,7). t(2). cp(1,). cp(4,8). t(7). cp(1,4). cp(8,9). t(9). cp(2,5). cp(8,10). t(10). cp(2,6). Expansion phase (without condition parts for definite facts, only new facts per iteration): 1: ok(2). ok(7). ok(9). ok(10). bc(1) not ok(2). bc(1) not ok(). bc(1) not ok(4). bc(2) not ok(5). bc(2) not ok(6). bc(4) not ok(7). bc(4) not ok(8). bc(8) not ok(9). bc(8) not ok(10). cx(1). cx(2). cx(4). cx(8). 2: ok(1) not bc(1). ok(2) not bc(2). ok(4) not bc(4). ok(8) not bc(8). : --- Fixpoint reached! 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 28

29 CFP in this Example: Reduction Phase A Recommendable Bookkeeping Scheme For each round of the fixpoint iteration two steps are performed: a) Eliminate negative conditions (Red true )! b) Eliminate conditional facts (Red false )! Use result of a) as input! Round 1 2 Step a) b)... n Ø Ø Fixpoint reached! (If both steps without new result for the first time.) At the beginning of each step: Determine newly discovered definitely false (before applying Red true ) resp. definitely true facts (before applying Red false ) Prof. Dr. Rainer Manthey Intelligent Information Systems 29

30 Reduction Phase: At the Beginning Result of expansion phase before step a) of reduction phase, 1 st round of iteration: ok(2). ok(7). ok(9). ok(10). cx(1). cx(2). cx(4). cx(8). bc(1) not ok(2). bc(1) not ok(). bc(1) not ok(4). bc(2) not ok(5). bc(2) not ok(6). bc(4) not ok(7). bc(4) not ok(8). bc(8) not ok(9). bc(8) not ok(10). ok(1) not bc(1). ok(2) not bc(2). ok(4) not bc(4). ok(8) not bc(8). Definitely false: bc(), bc(5), bc(6), bc(7), bc(9), bc(10) Also definitely false: ok(), ok(5), ok(6) Herbrand Universe: {1,2,,., 10} 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 0

31 Reduction Phase: 1a Reduction phase: ok(2). ok(7). ok(9). ok(10). 1 st round step a, eliminating negative literals (based on false facts) cx(1). cx(2). cx(4). cx(8). bc(1) not ok(2). bc(1) not ok(). bc(1) not ok(4). bc(2) not ok(5). bc(2) not ok(6). bc(4) not ok(7). bc(4) not ok(8). bc(8) not ok(9). bc(8) not ok(10). just afterwards: Eliminating subsumed conditional facts (copies, less general facts) bc(1) not ok(2). bc(1). bc(1) not ok(4). bc(2). bc(2). bc(4) not ok(7). bc(4) not ok(8). bc(8) not ok(9). bc(8) not ok(10). ok(1) not bc(1). ok(2) not bc(2). ok(4) not bc(4). ok(8) not bc(8). false: {ok(), ok(5), ok(6)} ok(1) not bc(1). ok(2) not bc(2). ok(4) not bc(4). ok(8) not bc(8) Prof. Dr. Rainer Manthey Intelligent Information IIS 2011 Systems 1

32 Reduction Phase: After 1a, Performing 1b Reduction phase: 1 st round after step a), assessment of true facts (facts rearranged a bit) cx(1). cx(2). cx(4). cx(8). bc(1). bc(2). bc(4) not ok(7). bc(4) not ok(8). bc(8) not ok(9). bc(8) not ok(10). ok(2). ok(7). ok(9). ok(10). ok(1) not bc(1). ok(4) not bc(4). ok(8) not bc(8). Both true facts have been discovered during step a) in the same iteration: Can be used immediately in step b)!! 1 st round step b, eliminating conditional facts (based on true facts) bc(4): still open bc(8): now false bc(4) not ok(7). bc(4) not ok(8). bc(8) not ok(9). bc(8) not ok(10). ok(1) not bc(1). ok(4) not bc(4). ok(8) not bc(8). ok(1): also discovered false 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 2

33 Reduction Phase: After 1b, Performing 2a Reduction phase: 2 nd round after 1b, before 2a, new assessment of false facts cx(1). cx(2). cx(4). cx(8). bc(1). bc(2). bc(4) not ok(8). ok(2). ok(7). ok(9). ok(10). ok(4) not bc(4). ok(8) not bc(8). false: {bc(8), ok(1), ok(), ok(5), ok(6)} Reduction phase: 2 nd round step a), eliminating conditions cx(1). cx(2). cx(4). cx(8). bc(1). bc(2). bc(4) not ok(8). ok(2). ok(7). ok(9). ok(10). ok(4) not bc(4). ok(8) not bc(8). After 2a): one new true fact discovered! 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems

34 Reduction Phase: After 1b, Performing 2a Reduction phase: 2 nd round after 2a, before 2b, new assessment of true facts cx(1). cx(2). cx(4). cx(8). bc(1). bc(2). bc(4) not ok(8). ok(2). ok(7). ok(8). ok(9). ok(10). ok(4) not bc(4). Reduction phase: 2 nd round step b), eliminating conditional facts bc(4) not ok(8). false: {bc(4), bc(8), ok(1), ok(), ok(5), ok(6)} ok(4) not bc(4). Reduction phase: rd round step a), eliminating negative literals After a): A final new true fact discovered! ok(4) not bc(4). Reduction phase: rd round step b), eliminating conditional facts Nothing to be done anymore, as no conditional facts are left! 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 4

35 Reduction Phase: 4 and Truth Value Determination Reduction phase: 4 th round steps a) and b) cx(1). cx(2). cx(4). cx(8). bc(1). bc(2). ok(2). ok(4) ok(7). ok(8). ok(9). ok(10). At least formally, a final round has to be performed for confirming the fixpoint! (Practically, reduction can stop as soon as no CFs are present anymore.) Finally: Determination of truth values! cx(1). cx(2). cx(4). cx(8). true: bc(1). bc(2). ok(2). ok(4) ok(7). ok(8). ok(9). ok(10). false: {bc(4), bc(8), ok(1), ok(), ok(5), ok(6)} No conditional facts left in the fixpoint of the reduction phase: no undefined facts! 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 5

36 Reduction Phase Summary of the Steps and Rounds of the Iteration Scheme For each round two steps: a) Eliminate negative conditions (Red true )! b) Eliminate conditional facts (Red false )! Use result of a) as input! Round Step 1 bc(1). bc(2). a) new true facts b) new false facts ok(1). bc(8). 2 ok(8). bc(4). ok(4). 4 Ø Ø Ø Fixpoint reached! 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 6

37 Comparing CFP Result and Intuitive Solution CFP result: ok(2). ok(7). ok(9). ok(10). cx(1). cx(2). cx(4). cx(8). bc(1). bc(2). ok(4). ok(8). Component_of: p 1 p 2 p p 4 tested (thus, ok) ok rekursively bad component p 5 p 6 p 7 p 8 p 9 p Prof. Dr. Rainer Manthey Intelligent Information Systems 7

38 Last Minute Addition: Semantics for Rules with Comparison Operators Test literals resemble negative literals in as much as their evaluation is unable to produce any variable bindings. Test literals are just able to test bindings produced elsewhere! On the other hand, satisfaction of comparisons cannot be tested by accessing the fact base of the database, but has to be done by externally programmed test procedures. Formalizing this observation can be achieved by introducing a Boolean auxiliary predicate 'is_satisfied/1', which if applied to a ground test literal returns true in case the resp. comparison operator is satisfied for the two (ground) parameters, false else. All fixpoint operators for single rules introduced till now can thus be extended in order to handle test literals, e.g. (here in case of positive Datalog, analogously for negation): R i A B 1,...., B n,v 1,..,V m V k test literals T [R i ] ( F ) = def {Aσ... 1 j n : B j σ F and 1 k m : is_satisfied(v k σ) } 2016 Prof. Dr. Rainer Manthey Intelligent Information Systems 8

39 Last Minute Addition: Semantics for Rules with Functions If rules contain functional terms (in test literals), all such terms have to be evaluated by means of 'is_satisfied' before evaluating these literals. Term evaluation is performed via external code as well, the formalization of which has to be done on an abstract level. In order to do so, we introduce an auxiliary function 'eval/1', which if applied to a ground test literal replaces all functional terms occuring in the parameters of the literal by their resp. values (leaving all other parameters identical). Calls to 'eval' have to be added to the extended form of the fixpoint operators in order to cover the full range of concepts in Datalog, e.g.: T [R i ] ( F ) = def {Aσ... 1 k m : is_satisfied( eval(v k σ) ) } Already addressed earlier: If recursion and term construction interleave (similar to the interleaving of recursion and negation in case of unstratifiability), infinite term construction will occur cases like this have to be excluded Prof. Dr. Rainer Manthey Intelligent Information Systems 9