Logic Program Semantics Background

1
Logic Program Semantics Background

• Luís Moniz Pereira

AI Centre, Universidade Nova de Lisboa
U.I. at Jakarta, Jan/Feb 2006
2
Language

• A Normal Logic Programs P is a set of rules
• H A1, , An, not B1, not Bm (n,m ³ 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.

3
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?

4
LP and Deductive Databases

• In a database, tables are viewed as sets of facts

• Other relations are represented with rules

5
LP and Deductive DBs (cont)

• LP allows to store, besides relations, rules for
  deducing other relations
• Note that default negation cannot be classical
  negation in

• A form of Closed World Assumption (CWA) is needed
  for inferring non-availability of connections

6
Default Rules

• The representation of default rules, such as
• All birds fly
• can be done via the non-monotonic operator not

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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.

8
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 Î I
• Otherwise, A is false in I (ie. I(A) 0)

• Interpretations can be viewed as representing
  possible states of knowledge.
• If knowledge is incomplete, there might be in
  some states atoms that are neither true nor false

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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 Î T
• A is false in I iff A Î F
• Otherwise, A is undefined (I(A) 1/2)
• 2-valued interpretations are a special case,
  where
• HP T U F

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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

• I is a model of P iff, for all rule H B of P
• Î(H B) 1

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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.

• 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)

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Minimal Models Semantics

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

• Every definite logic program has a least (truth
  ordering) model.

• minimal models semantics An atom A is true in
  (definite) P iff A belongs to its least model.
  Otherwise, A is false in P.

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TP operator

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

• TP Let I be an interpretation of definite P.
• TP(I) H (H Body) Î P and Body Í I

• If P is definite, TP is monotone and continuous.
  Its minimal fixpoint can be built by
• I0 and In TP(In-1)

• The least model of definite P is TPw()

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Stable Models Idea

• The identification of models can be done by
  guessing a possible model, processing it into P
  and checking if its least model coincides with
  the guess.
• This can be applied to non-stratified programs.

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Stable Models Idea (cont)

• Guessing a model corresponds to assuming
  default negations not. This type of reasoning is
  usual in NMR
• Assume some default literals
• Check in P the consequences of such assumptions
• If the consequences completely corroborate the
  assumptions, they form a stable model
• The stable models semantics is defined as the
  intersection of all the stable models (i.e. what
  follows, no matter what stable assumptions)

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SMs preliminary example
a not b c a p not q b not a c b
q not r r

• Assume, e.g., not r and not p as true, and all
  others as false. By processing this into P

a false c a p false b false c b
q true r

• Its least model is not a, not b, not c, not p,
  q, r
• So, it isnt a stable model
• By assuming not r, r becomes true
• not a is not assumed and a becomes false

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SMs example (cont)
a not b c a p not q b not a c b
q not r r

• Now assume, e.g., not b and not q as true, and
  all others as false. By processing this into P

a true c a p true b false c b q
false r

• Its least model is a, not b, c, p, not q, r
• I is a stable model
• The other one is not a, b, c, p, not q, r
• According to Stable Model Semantics
• c, r and p are true and q is false.
• a and b are undefined

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Stable Models definition

• Let I be a (2-valued) interpretation of P. The
  definite program P/I is obtained from P by
• deleting all rules whose body has not A, and A Î
  I
• deleting from the body all the remaining default
  literals
• GP(I) least(P/I)

• M is a stable model of P iff M GP(M).
• A is true in P iff A belongs to all SMs of P
• A is false in P iff A doesnt belongs to any SMs
  of P (i.e. not A belongs to all SMs of P).

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Properties of SMs

• Stable models are minimal models
• Stable models are supported
• If P is locally stratified then its single stable
  model is the perfect model
• Stable models semantics assign meaning to (some)
  non-stratified programs
• E.g. the one in the example before

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Importance of Stable Models

• Stable Models were an important contribution
• Introduced the notion of default negation (versus
  negation as failure)
• Allowed important connections to NMR. Started the
  area of LPNMR
• Allowed for a better understanding of the use of
  LPs in Knowledge Representation
• It is considered as THE semantics of LPs by a
  significant part of the community.

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LP representing a static world

• The work on LP allows the (non-monotonic)
  addition of new knowledge.
• But
• Much of the work does not consider this evolution
  of knowledge
• LPs represent a static knowledge of a given world
  in a given situation.
• The issues of how to add new information to a
  logic program are less studied.

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Knowledge Evolution

• In real situations knowledge evolves by
• completing it with new information (revision)
• changing it according to the changes in the world
  itself (updates)

• I know that I have a flight booked for London
  (either for Heathrow or for Gatwick).
• I learn that it is not for Heathrow (revision)
• I conclude my flight is for Gatwick
• I learn that flights for Heathrow were canceled
  (update)
• Either I have a flight for Gatwick or no flight
  at all

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Model Updates

• Updates are usually performed model by model.
• Marek and Truszczynski defined a language for
  defining updates
• in(A0) out(A0) ? in(A1), , out(An)
• Given an update program and a model of the
  current situation, produce model(s) of the new
  situation.
• If several models of the current situation exist,
  one has to proceed model by model.

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Updates of Logic Programs
P

• Weve defined a program transformation to
  directly obtain Pu

• Weve generalized MTs approach to the 3-valued
  case

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Updates of LPs by LPs

• When updating LPs, doing it model by model is not
  desired. It loses the directional information of
  the LP arrow.

P sleep ? not tv_on. watch ? tv_on. tv_on.
M tv,w
Mu pf,w vs
pf,s
U not tv_on ? p_failure. p_failure.
U2 not p_failure.
Mu2 w vs
tv,w

• Bodies are evaluated in the last state.

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Generalized LPs

• A generalized logic program P is a set of
  propositional Horn clauses
• L ? L1 ,, Ln
• where L and Li are atoms from LK , i.e. of
  the form A or not A.
• Program P is normal if no head of the clause in P
  has form not A.

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Generalized LP semantics

• A set M is an interpretation of LK if for every
  atom A in K exactly one of A and not A is in M.
• Definition
• An interpretation M of LK is a stable model of a
  generalized logic program P if M is the least
  model of the Horn theory P ? not A A ? M.

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Generalized LPs example

• Example K a,b,c,d,e
• P a ? not b
• c ? b
• e ? not d
• not d ? a, not c
• d ? not e
• this program has exactly one stable model
  
• M Least(P ? not b, c, d) a, e, not b,
  not c, not d
• N not a, not e, b, c, d is not a stable
  model since
• N ? Least(P ? not a, not e)

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Relation to stable models

• Proposition
• An interpretation M of LK is a stable model of
  a generalized logic program P iff for every A?
  LK
• if P/M - A then A? M
• if A? K ? M then P/M - A
• where P/M denotes Gelfond-Lifschitz transform of
  P wrt M
• Conclusion
• The class of stable models of generalized logic
  programs extends the class of stable models of
  normal programs.

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Drawbacks of Interpretation Updates

• How to update a logic program P by a logic
  program U obtaining as a result a new, updated
  logic program P ? U.
• Interpretation update approach (H.Katsuno and
  A.Mendelzon, M.Winslett) models of DB
  updated models of DB
• Drawbacks of this approach
• all the models of DB have to be computed and
  updated
• separately
• no natural way to compute DB (DB may not
  exist)
• produces counter-intuitive results when
  intensional part of DB
• is allowed to be updated.

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Update Example

• Example
• P sleep ? not tv_on
• watch_tv ? tv_on
• tv_on ?
• the only stable model is M tv_on, watch_tv
• U not tv_on ? power_failure
• power_failure ?
• the only update is MU power_failure,
  watch_tv
• the intended model is MI power_failure,
  sleep
• U2 not power_failure ?

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Update Example (2)

• Example
• P innocent ? not found_guilty
• the only stable model is M innocent
• U found_guilty ?
• the only update is MU innocent, found_guilty
• the intended model is MI found_guilty

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Dynamic Program Updates

• Program P is semantically equivalent to the
  program
• P innocent ?
• the model MU innocent, found_guilty is the
  only reasonable model of the update of P by
  U.
• DB depends not only on semantics of DB and
  update U (interpretation updates) but also on
  their syntax.
• We propose a new approach to the problem of
  updating knowledge bases represented by logic
  programs that attempts to eliminate the drawbacks
  of the previous approaches

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Dynamic Program Updates

• How to update a logic program with another
• A ? B1 , , Bm , not C1, , not Cn
• not A ? B1 , , Bm , not C1, ,
  not Cn

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Program Update

• Definition Let P and U be generalized logic
  programs in the language L. By the update of P by
  U we mean the generalized logic program P ? U,
  consisting of the clauses
• (RP) Rewritten original program clauses
• AP ? B1 , , Bm , C1, , Cn
  
• AP ? B1 , , Bm , C1, ,
  Cn
• (RU) Rewritten updating program clauses
• AU ? B1 , , Bm , C1, , Cn
• AU ? B1 , , Bm , C1, ,
  Cn

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Translation into LP

• (UR) Update rules
• A ? AU and not A ? AU
• (IR) Inheritance rules
• A ? AP , not AU and A ? AP ,
  not AU
• (DR) Default rules
• A ? not AP , not AU and not
  A ? A

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Example

• Example
• P sleep ? not tv_on
• watch_tv ? tv_on
• tv_on ?
• U not tv_on ? power_failure
• power_failure ?
• P ? U (RP) ? (RU) ? (UR) ? (IR) ? (DR)
• RP sleepP ? tv_on RU tv_onU ?
  power_failure
• watch_tvP ? tv_on power_failureU ?
• tv_onP ?
• M power_failure, sleep is the only stable
  model of P ? U

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Semantic characterization

• Definition Let M be a model of the program U
  in the language L.
• Def M not A M ? Body, ?(A? Body) ? P ?
  U
• Rej M
• A? Body ? P ? (not A ? Body) ? U
  and M Body
• ? not A? Body? P ? (A ? Body) ? U
  and M Body
• Res M P ? U Rej M.

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Equivalence to LP translation

• Theorem
• An interpretation N is a stable model of the
  update program P ? U iff N is an extension of a
  model M of U such that
• M Least(Res M ? Def M)
• Conclusion
• If N is a stable model of P ? U then its
  restriction M to the language L is a stable
  model of Res M.

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Properties

• Proposition
• If M is a stable model of the union P ? U of
  programs P and U , then its extension N is a
  stable model of the update program P ? U.
• Thus, the semantics of the program P ? U is
  always weaker than or equal to the semantics of P
  ? U.
• If either P or U is empty, or if both P and U
  are normal programs, then semantics of P ? U and
  P ? U coincide.

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Dynamic Program Updates

• Definition
• Let P Ps s? S be a finite or
  infinite sequence of generalized logic programs.
  The dynamic program update over the sequence of
  programs P and at the state s? S is a logic
  program ?s P resulting from the successive
  updates.

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Dynamic LP example

• Example P P1, P2, P3
• P1 sleep ? not tv_on
• watch_tv ? tv_on
• tv_on ?
• P2 not tv_on ? power_failure
• power_failure ?
• P3 not power_failure ?
• M1 tv_on, watch_tv is the unique stable
  model of program ?1 P

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Dynamic LP example (2)

• M2 sleep, power_failure is the unique
  stable model of the program ?2P.
• M3 tv_on, watch_tv is the unique stable
  model of the program ?3P.
• Program ?2P is semantically equivalent to
• P1 ?P2.

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Dynamic LP example (3)

• Example P P1, P2, P3, P4
• P1 not fly(X) ? animal(X) P4
  animal(X) ? bird(X)
• P2 fly(X) ? bird(X) bird(X) ?
  penguin(X)
• P3 not fly(X) ? penguin(X)
  animal(pluto) ?
• 
  bird(duffy) ?
• 
  penguin(tweety) ?
• Program ? 4 P has a unique stable model in
  which fly(duffy) is true and both fly(pluto) and
  fly(tweety) are false.