Reasoning with Logic Programming

1
Reasoning with Logic Programming

• Luís Moniz Pereira
• José Júlio Alferes

2
Aim

• To provide a logic programming integrated
  framework and working system, for representing
  knowledge and reason about classical AI topics
  and applications, such as
• taxonomies
• hypothetical reasoning
• abduction
• paraconsistent reasoning
• revision
• diagnosis

3
Topics

• Overview of Logic Programs semantics
• Explicit negation semantics
• Paraconsistency
• Methodology for KR in LP
• Proof procedures
• Belief revision and abduction

4
LP forKnowledge Representation

• Due to its declarative nature, LP has become a
  prime candidate for Knowledge Representation and
  Reasoning
• This has been more noticeable since its relations
  to other NMR formalisms were established
• For this usage of LP, a precise declarative
  semantics was in order

5
Language

• A Normal Logic Programs P is a set of rules
• H ??A1, , An, not B1, not Bm (n,m ? 0)
• where H, Ai and Bj are atoms
• Literal not Bj are called default literals
• When no rule in P has default literal, P is
  called definite
• The Herbrand base HP is the set of all
  instantiated atoms from program P.
• We will consider programs as possibly infinite
  sets of instantiated rules.

6
Declarative Programming

• A logic program can be an executable
  specification of a problem

member(X,XY). member(X,YL) ? member(X,L).

• Easier to program, compact code
• Adequate for building prototypes
• Given efficient implementations, why not use it
  to program directly?

7
LP and Deductive Databases

• In a database, tables are viewed as sets of
  facts
• Flight(lisbon,adam)
• Flight(lisbon,london)
• Flight(lisbon, munich)
• Flight(munich,dresden)
• Other relations are represented with rules
• Connection(A,B) if flight(A,B).
• Connection(A,B) if flight(A,C), connection(C,B).
• chooseAnother(A,B) if not connection(A,B).

8
Default Rules

• The representation of default rules, such as
• All birds fly
• can be done via the non-monotonic operator not
• Flies(A) if bird(A), not abnormal(A).
• Bird(P) if penguin(P)
• Abnormal(P) if penguin(P)
• Bird(a)
• Penguin(p).

9
The need for a semantics

• In all the previous examples, classical logic is
  not an appropriate semantics
• In the 1st, it does not derive not
  member(3,1,2)
• In the 2nd, it never concludes choosing another
  company
• In the 3rd, all abnormalities must be expressed
• The precise definition of a declarative semantics
  for LPs is recognized as an important issue for
  its use in KRR.

10
2-valued Interpretations

• A 2-valued interpretation I of P is a subset of
  HP
• A is true in I (ie. I(A) 1) iff A ? I
• Otherwise, A is false in I (ie. I(A) 0)

11
3-valued Interpretations

• A 3-valued interpretation I of P is a set
• I T U not F
• where T and F are disjoint subsets of HP
• A is true in I iff A ? T
• A is false in I iff A ? F
• Otherwise, A is undefined (I(A) 1/2)
• 2-valued interpretations are a special case,
  where
• HP T U F

12
Models

• Models can be defined via an evaluation function
  Î
• For an atom A, Î(A) I(A)
• For a formula F, Î(not F) 1 - Î(F)
• For formulas F and G
• Î((F,G)) min(Î(F), Î(G))
• Î(F ? G) 1 if Î(G) ? Î(F), and 0 otherwise

13
Minimal Models Semantics

• The idea of this semantics is to minimize
  positive information. What is implied as true by
  the program is true everything else is
  false. ableMathematician(X) if
  physicist(X) physicist(einstein) president(sam
  paio)

• pr(s),pr(e),ph(s),ph(e),aM(s),aM(e) is a model
• Lack of information that sampaio is a physicist,
  should indicate that he isnt
• The minimal model is pr(s),ph(e),aM(e)

14
Minimal Models Semantics

• Truth ordering For interpretations I and J, I ?
  J iff for all atom A, I(A) ? J(A), i.e.
• TI ? TJ and FI ? FJ

15
TP operator

• The minimal models of a definite P can be
  computed (bottom-up) via operator TP

16
On Minimal Models

• SLD can be used as a proof procedure for the
  minimal models semantics
• If the is a SLD-derivation for A, then A is true
• Otherwise, A is false
• The semantics does not apply to normal programs
• p ? not q has two minimal models
• p and q
• There is no least model !

17
Stable Models definition
18
Cumulativity
19
Relevance
20
Problems with SMs

• Dont provide a meaning to every program
• P a ? not a has no stable models
• Its non-cumulative and non-relevant

• The only SM is not a, c, b

a ? not b c ? not a b ? not a c ? not c

• However b is not true in P U c (non-cumulative)
• P U c has 2 SMs not a, b, c and a, not b,
  c
• b is not true in Relb(P) (non-relevance)
• The rules in Relb(P) are the 2 on the left
• Relb(P) has 2 SMs not a, b and a, not b

21
Problems with SMs (cont)

• Its computation is NP-Complete
• The intersection of SMs is non-supported

a ? not b c ? a b ? not a c ? b
22
Well Founded Semantics

• Defined in GRS90, generalizes SMs to 3-valued
  models.
• Note that
• there are programs with no fixpoints of ?
• but all have fixpoints of ?2

P a ? not a

• ?(a) and ?() a
• There are no stable models

23
Partial Stable Models
P a ? not a, has a single PSM
This program has 3 PSMs , a, not b and c,
b, not a The 3rd corresponds to the single SM
a ? not b c ? not a b ? not a c ? not c
24
WFS definition
25
Well Founded Semantics

• Let M be the well founded model of P
• A is true in P iff A ? M
• A is false in P iff not A ? M
• Otherwise (i.e. A ? M and not A ? M) A is
  undefined in P

26
WFS Properties

• Every program is assigned a meaning
• Every PSM extends one SM
• If WFM is total it coincides with the single SM
• It is sound wrt to the SMs semantics
• If P has stable models and A is true (resp.
  false) in the WFM, it is also true (resp. false)
  in the intersection of SMs

27
More WFS Properties

• The WFM is supported
• WFS is cumulative and relevant
• Its computation is polynomial (on the number of
  instantiated rules of P)
• There are top-down proof-procedures, and sound
  implementations
• these are mentioned in the sequel

28
Extended LPs

• In Normal LPs all the negative information is
  implicit. Though thats desired in some cases
  (e.g. the database with flight connections),
  sometimes an explicit form of negation is needed
  for Knowledge Representation
• Penguins dont fly could be noFly(X) ?
  penguin(X)
• This does not relate fly(X) and noFly(X) in
• fly(X) ? bird(X)
• noFly(X) ? penguin(X)

29
Extended LP motivation

• is also needed in bodies
• Someone is guilty if is not innocent
• cannot be represented by guilty(X) ? not
  innocent(X)
• This would imply guilty in the absence of
  information about innocent
• Instead, guilty(X) ? innocent(X) only implies
  guilty(X) if X is proven not to be innocent
• The difference between not p and p is essential
  whenever the information about p cannot be
  assumed to be complete

30
ELP motivation (cont)

• allows for greater expressivity
• If youre not sure that someone is not innocent,
  then further investigation is needed
• Can be represented by
• investigate(X) ? not innocent(X)
• extends the relation of LP to other NMR
  formalisms. E.g
• it can represent default rules with negative
  conclusions and pre-requisites, and positive
  justifications
• it can represent normal default rules

31
Explicit versus Classical

• Classical complies with the excluded middle
  principle (i.e. F v F is tautological)
• This makes sense in mathematics
• What about in common sense knowledge?
• A is the the opposite of A.
• The excluded middle leaves no room for
  undefinedness

The excluded middle implies that every X is
either hired or rejected It leaves no room for
those about whom further information is need to
determine if they are qualified
hire(X) ? qualified(X) reject(X) ? qualified(X)
32
ELP Language

• An Extended Logic Program P is a set of rules
• L0 ??L1, , Lm, not Lm1, not Bn (n,m ? 0)
• where the Li are objective literals
• An objective literal is an atoms A or its
  explicit negation A
• Literals not Lj are called default literals
• The Extended Herbrand base HP is the set of all
  instantiated objective literals from program P
• We will consider programs as possibly infinite
  sets of instantiated rules.

33
ELP Interpretations

• An interpretation I of P is a set
• I T U not F
• where T and F are disjoint subsets of HP and
• L ? T ? L ? F (Coherence Principle)
• i.e. if L is explicitly false, it must be
  assumed false by default
• I is total iff HP T U F
• I is consistent iff ? L L, L ? T
• In total consistent interpretations the Coherence
  Principle is trivially satisfied

34
The coherence principle

• Generalizing WFS in the same way yields
  unintuitive results

pacifist(X) ? not hawk(X) hawk(X) ? not
pacifist(X) pacifist(a)

• Using the same method the WFS is pacifist(a)
• Though it is explicitly stated that a is
  non-pacifist, not pacifist(a) is not assumed, and
  so hawk(a) cannot be concluded.
• Coherence is not satisfied...
• Coherence must be imposed

35
Imposing Coherence

• Coherence is L ? T ? L ? F, for objective L
• According to the WFS definition, everything is
  false that doesnt belong to ?(T)
• To impose coherence, when applying ?(T) simply
  delete all rules for the objective complement of
  literals in T

If L is explicitly true then when computing
undefined literals forget all rules with head L
36
WFSX definition
37
WFSX example
Ps pacifist(X) ? not hawk(X), not
pacifist(X) hawk(X) ? not pacifist(X ), not
hawk(X) pacifist(a) ? not pacifist(a)
pacifist(b) ? not pacifist(b)
T0 ?s(T0) p(a),p(a),h(a),p(b),h(b) T1
p(a) ?s(T1) p(a),h(a),p(b),h(b) T2
p(a),h(a) T3 T2
P pacifist(X) ? not hawk(X) hawk(X) ? not
pacifist(X) pacifist(a) pacifist(b)
38
Properties of WFSX

• Complies with the coherence principle
• Coincides with WFS in normal programs
• It is supported, cumulative, and relevant
• Its computation is polynomial
• It has sound implementations (cf. below)

39
Inconsistent programs

• Some ELPs have no WFM. E.g. a ???a ?
• What to do in these cases?
• Explosive approach everything follows from
  contradiction
• taken by answer-sets
• gives no information in the presence of
  contradiction
• Belief revision approach remove contradiction by
  revising P
• computationally expensive
• Paraconsistent approach isolate contradiction
• efficient
• allows to reason about the non-contradictory part

40
WFSXp definition

• The paraconsistent version of WFSX is obtained by
  dropping the requirement that T and F are
  disjoint, i.e. dropping T ? ?Ps(T)

41
SLXProof procedure for WFSXp

• SLX (SL with eXplicit negation) is a top-down
  procedure for WFSXp
• Here we only present an AND-trees
  characterization
• The proof procedure details are in AP96
• Is similar to SLDNF
• Nodes are either successful or failed
• Resolution with program rules and resolution of
  default literals by failure of their complements
  are as in SLDNF
• In SLX, failure doesnt mean falsity. It simply
  means non-verity (i.e. false or undefined)

42
Success and failure

• A finite tree is successful if its root is
  successful, and failed if its root is failed
• The status of a node is determined by
• A leaf labeled with an objective literal is
  failed
• A leaf with true is successful
• An intermediate node is successful if all its
  children are successful, and failed otherwise
  (i.e. at least one of its children is failed)

43
Negation as Failure?

• As in SLS, to solve infinite positive recursion,
  infinite trees are (by definition) failed
• Can a NAF rule be used?

YES

• This is the basis of SLX. It defines
• T-Trees for proving truth
• TU-Trees for proving truth or undefinedness

44
T and TU-trees

• They differ in that literals involved in
  recursion through negation, and so undefined in
  WFSXp, are failed in T-Trees and successful in
  TU-Trees

a ? not b b ? not a
45
Explicit negation in SLX

• -literals are treated as atoms
• To impose coherence, the semi-normal program is
  used in TU-trees

?
?
a ? not b b ? not a a
?
?
?
46
Explicit negation in SLX (2)

• In TU-trees L also fails if L succeeds true
• I.e. if not L fails as true-or-undefined

a
not a
?
?
?
c ? not c b ? not c b a ? b
?
?
?
?
?
?
?
47
Paraconsistent example
b ? not a a a
48
T and TU-trees definition
49
Success and Failure
50
Properties of SLX

• SLX is sound and (theoretically) complete wrt
  WFSXp.
• Even ignoring rules concerning explicit negation,
  SLX is sound and (theoretically) complete wrt
  WFS.
• See AP96 for the definition of a refutation
  procedure based on the AND-trees
  characterization, and for all proofs and details

51
Infinite trees example
?
?
?
52
Negative recursion example
53
Guaranteeing termination

• The method is not effective, because of loops
• To guarantee termination in ground programs
• Local ancestors of node n are literals in the
  path from n to the root, exclusive of n
• Global ancestors are assigned to trees
• the root tree has no global ancestors
• the global ancestors of T, a subsidiary tree of
  leaf n of T, are the global ancestors of T plus
  the local ancestors of n
• global ancestors are divided into those occurring
  in T-trees and those occurring in TU-trees

54
Pruning rules

• For cyclic positive recursion

55
Pruning rules (2)
56
Other sound rules
57
Pruning examples
a ? not b b ? not a a
?
?
b
a
c ? not c b ? not c b a ? b
not a
?
?
?
?
?
58
Simple diagnosis example
inv(G,I,0) ? node(I,1), not ab(G). inv(G,I,1) ?
node(I,0), not ab(G). node(b,V) ?
inv(g1,a,V). node(a,1). node(b,0). Fault
model inv(G,I,0) ? node(I,0), ab(G). inv(G,I,1) ?
node(I,1), ab(G).
The only revision is P U ab(g1) ? u It does
not conclude node(b,1).
59
2-valued Revision

• In diagnosis one often wants the IC
• ab(X) v not ab(X) ?
• With these ICs (that are not denials), 3-valued
  revision is not enough.
• A two valued revision is obtained by adding facts
  for revisables, in order to remove contradiction.
• For 2-valued revision the algorithm no longer
  works

60
Example
? ? p. ? ? a. ? ? b, not c. p ? not a, not b.
The only support is not a, not b. Removals are
not a and not b.

• In 2-valued revision
• some removals must be deleted
• the process must be iterated.

61
Algorithm for 2-valued revision

• Let Revs
• For every element R of Revs
• Add it to the program and compute removal sets.
• Remove R from Revs
• For each removal set RS
• Add R U not RS to Revs
• Remove non-minimal sets from Revs
• Repeat 2 and 3 until reaching a fixed point of
  Revs.
• The revisions are the elements of the final Revs.

62
Example of 2-valued revision
? ? p. ? ? a. ? ? b, not c. p ? not a, not b.
Rev0
Rev1 a, b
Rev2 b
Rev3 b,c
Rev4

• Choose . Removal sets of P U are not a
  and not b. Add them to Rev.

• Choose a. P U a has no removal sets.

• Choose b. The removal set of P U b is not
  c. Add b, c to Rev.

• Choose b,c. The removal set of P U b,c is
  . Add b, c to Rev.

• The fixed point had been reached. P U b,c is
  the only revision.

63
REVISE

• The algorithm with various optimizations had been
  implemented - REVISE system.
• Revise also has
• The possibility of defining a partial ordering of
  preferences among revisables.
• Associate probabilities to revisables.
• Also computes revisions with minimal cardinality.

• Optimizations include
• Pruning of the search space.
• Making use of preferences and probabilities.

64
Revision and Diagnosis

• In model based diagnosis one has
• a program P with the model of a system (the
  correct and, possibly, incorrect behaviors)
• a set of observations O inconsistent with P (or
  not explained by P).
• The diagnoses of the system are the revisions of
  P U O.
• Weve shown how to perform mixed consistency and
  abduction based diagnosis.

65
Diagnosis Example
66
Diagnosis Program
Observables obs(out(inpt0, c1),
0). obs(out(inpt0, c2), 0). obs(out(inpt0, c3),
0). obs(out(inpt0, c6), 0). obs(out(inpt0, c7),
0). obs(out(nand, g22), 0). obs(out(nand, g23),
1).
Connections conn(in(nand, g10, 1), out(inpt0,
c1)). conn(in(nand, g10, 2), out(inpt0,
c3)). conn(in(nand, g23, 1), out(nand,
g16)). conn(in(nand, g23, 2), out(nand, g19)).
Predicted and observed values cannot be
different ?? obs(out(G, N), V1), val(out(G, N),
V2), V1 ? V2.
Value propagation val( in(T,N,Nr), V ) ? conn(
in(T,N,Nr), out(T2,N2) ), val( out(T2,N2), V
). val( out(inpt0, N), V ) ? obs( out(inpt0, N),
V ).
Normal behavior val( out(nand,N), V ) ? not
ab(N), val( in(nand,N,1), W1), val( in(nand,N,2),
W2), nand_table(W1,W2,V).
Abnormal behavior val( out(nand,N), V ) ?
ab(N), val( in(nand,N,1), W1), val( in(nand,N,2),
W2), and_table(W1,W2,V).
67
Diagnosis Example
, and ab(g16),ab(g22)
Revision are
ab(g23)
, ab(g19)
68
Abduction as Revision

• For abductive queries
• Declare as revisable all the abducibles
• If the abductive query is Q, add the IC
• ? ? not Q
• The revision of the program are the abductive
  solutions of Q.

69
Bibliography (1)

• Most of the work covered by this course is
  detailed in
• AP96 J.J. Alferes and L.M. Pereira. Reasoning
  with Logic Programming. Springer LNAI 1111, 1996
• See references there for the original definitions
  
• For more on semantics of normal LPs see
• PP90 H. Przymusinska and T. Przymusinski.
  Semantic issues in deductive databases and logic
  programs. In R. Banerji ed, Formal Techniques in
  AI, a Sourcebook. North Holland, 1990
• For more on derivation procedures
• AB94 K. Apt and R. Bol. Logic Programming and
  Negation A survey. In Journal of L.P., 1994

70
Bibliography (2)

• For more on LP for KRR
• BG94 C. Baral and M. Gelfond. Logic Programming
  and Knowledge Representation. In Journal of L.P.,
  1994
• For relation with NMR e structural properties
• BDK97 G. Brewka, J. Dix and K. Konolige.
  Nonmonotonic Reasoning an overview. CSLI
  Publications, 1997
• More on our LP belief revision approach in
• ADP95 J. J. Alferes and C. V. Damásio and L. M.
  Pereira, A Logic Programming System for
  Non-monotonic Reasoning, Journal of Automated
  Reasoning (14), 1995

71
Bibliography (3)

• For more on LP update
• MT96 V. Marek and M. Truszczynski. Revision
  specifications by means of programs. In JELIA94,
  Springer LNAI 838, 1994
• ALPPP98 J.J. Alferes, J. Leite, L.M. Pereira,
  H. Przymusinska and T. Przymusinski. Dynamic
  Logic Programming. In KR98, Morgan Kaufmann,
  1998
• Another good entry point for these questions is
• DPP97 J. Dix, L.M. Pereira, T. Przymusinski.
  Prolegomena to Logic Programming and
  Non-Monotonic Reasoning. In Non-Monotonic
  Extensions of Logic Programming. Springer LNAI
  1216, 1997