RuleBased Reasoning

1
Rule-Based Reasoning

• Jacques Robin

2
Outline

• Rules as a knowledge representation formalism
• Common characteristics of rule-based systems
• Diversity of rules and rule-based reasoning
• Advantages and limitations
• History of rule-based systems
• Constraint Handling Rules (CHR)
• Abstract syntax
• Declarative logical semantics
• High-level operational semantics
• Example rule bases
• Comparison with rewrite and production rules
• Extension CHRV
• Practical applications

3
Rules as a Knowledge Representation Formalism

• What is a rule?
• A statement that specifies that
• If a determined logical combination of
  conditions is satisfied,
• over the set of perceptions and/or the fluent
  facts in an agents Knowledge Base (KB),
• that represents the current, past and/or
  hypothetical future of its environment model, its
  goals and/or its preferences,
• then a logico-temporal combination of actions
  can or must be executed by the agent,
• on directly on its environment (through
  actuators) or on its fluent facts.
• A KB agent such that the persistent part of its
  KB consists entirely of such rules is called a
  rule-base agent
• In such case, the inference engine used by the KB
  agent is an interpreter or a compiler for a
  specific rule language.

4
Rule-Based Agent
Environment
Sensors
Ask
Tell
Retract

• Rule Engine
• Problem class independent
• Only dependent on rule language
• Declarative code interpreter or compiler

Ask

• Rule Base
• Persistant intentional knowledge
• Dependent on problem class, not instance
• Declarative code

Effectors
5
Rule Languages Common Characteristics

• Syntax generally
• Extends a host programming language and/or
• Restricts a formal logic and/or
• Uses a semi-natural language with
• closed keyword set expressing logical
  connectives and actions classes,
• and an open keyword set to refer to the entities
  and relations appearing in the agents fact base
• Some systems provide 3 distinct syntax layers
  for different users with automated tools to
  translate a rule across the various layers
• Declarative semantics generally based on some
  formal logic
• Operational semantics
• Generally based on transition systems, automata
  or similar procedural formalisms
• Formalizes the essence of the rule interpreter
  algorithm.

6
Rule Languages General Advantages

• Human experts in many domains (medicine, law,
  finance, marketing, administration, design,
  engineering, equipment maintenance, games,
  sports, etc.) find it intuitive and easy to
  formalize their knowledge as a rule base
• Facilitates knowledge acquisition
• Rules can be easily paraphrased in semi-natural
  language syntax, friendlier to experts averse to
  computational languages
• Facilitates knowledge acquisition
• Rule bases easy to formalize as logical formulas
  (conjunctions of equivalences and/or
  implications)
• Facilitates building rule engine that perform
  sound, logic-based inference
• Each rule largely independent of others in the
  base (but to precisely what degree depends highly
  of the rule engine algorithm)
• Can thus be viewed as an encapsulated
  declarative piece of knowledge
• Facilitates life cycle evolution and composition
  of knowledge bases
• Very sophisticated, mature rule base compilation
  techniques available
• Allows scalable inference in practice
• Some engines for simple rule languages can
  efficiently handle millions of rules

7
Rule Languages General Limitations

• Subtle interactions between rules hard to debug
  without
• sophisticated rule explanation GUI
• detailed knowledge of the rule engines
  algorithm
• Especially serious with
• Object-oriented rule languages for combining
  rule-based deduction or abduction with
  class-based inheritance
• Probabilistic rule languages for combining
  logical and Bayesian inference
• But purely logical relational rule language do
  not naturally
• Embed within mainstream object-oriented modeling
  and programming languages
• Represent inherently procedural knowledge,
  taxonomic and uncertain knowledge
• Current research challenge
• User-friendly reasoning engine for probabilistic
  object-oriented rules

8
Roadmap of Rule-Based Languages
9
Constraint Handling Rules (CHR)Key Ideas

• Originally a logical rule-based language to
  declaratively program specialized constraint
  solvers on top of a host programming language
  (Prolog, Haskell, Java)
• Since evolved in a general purpose first-order
  knowledge representation language and
  Turing-complete programming language
• Fact base contains both extensional and
  intentional knowledge in the form of a
  conjunction of constraints
• Rule base integrates and generalizes
• Event-Condition-Action rules (themselves
  generalizing production rules) for constraint
  propagation
• Conditional rewrite rules for constraint
  simplification
• Relies on forward chaining and rule LHS matching
• Extended variant CHRV adds backtracking search
  and thus generalizes Prolog rules as well

10
CHR Syntax Overview
CHR Base

• Simplification rule sh1(X,a), sh2(b,Y)
  ltgt g1(X,Y),
  g2(a,b,c) b1(X,c), b2(Y,c).
• Propagation rule ph1(X,Y), ph2(d)
  gt g3(X), g4(d,Y)
  b3(X,d), b4(X,Y).
• Simpagation rule ph3(X), ph4(Y,Z) \
  sh3(X,U), sh4(Y,V) ltgt g5(X,Z), g6(Z,Y)
  b5(X), b6(Y,Z).
• Simplification rules are conditional rewrite
  rules (condition is the guard)
• Propagation rules are event-condition-action
  rules (event is the guard)
• Simpagation rules heads are hybrid syntactic
  sugar, each can be replaced by a semantically
  equivalent simplification rule, ex, p, r \ s, t
  ltgt g, h b, c. is equivalent to p, r, s, t ltgt
  g, h p, r, b, c.

Body Rule-Defined and Built-In Constraints
Guard Built-In Constraints (from host
language)
Head Rule-Defined Constraints
11
CHR Complete Abstract Syntax
body
guard
0..1
Logical Formula
CHR Rule
2..
simplified head
CHR Base

0..1
And Formula
propagated head
0..1
non-overlapping, complete
Atomic Formula
Simpagation Rule
Simplification Rule
Propagation Rule
12
CHR Derivation Data Structures
body
guard
0..1
CHR Logical Formula
CHR Rule
2..
simplified head
CHR Base

0..1
And Formula
propagated head
0..1
arg
Atomic Formula

Simpagation Rule
Simplification Rule
Propagation Rule
Term
Built-In Constraint
Rule Defined Constraint
13
CHR Declarative Semantics inClassical
First-Order Logic (CFOL)

• Simplification rule sh1, ... , shi ltgt g1, ...,
  gj b1, ..., bk.
• where X1, ..., Xn vars(sh1 ? ... ? shi ? g1
  ? ... ? gj) and Y1, ... , Ym vars(b1 ? ... ?
  bk) \ X1, ..., Xn?X1, ..., Xn g1 ? ... ? gj
  ? (sh1 ? ... ? shi ? ?Y1, ... , Ym b1 ? ... ? bk)
• Propagation rule ph1, ... , phi gt g1, ..., gj
  b1, ..., bk.
• where X1, ..., Xn vars(ph1 ? ... ? phi ? g1
  ? ... ? gj) and Y1, ... , Ym vars(b1 ? ... ?
  bk) \ X1, ..., Xn?X1, ..., Xn g1 ? ... ? gj
  ? (ph1 ? ... ? phi ? ?Y1, ... , Ym b1 ? ... ? bk)

14
CHR Operational Semantics
15
CHR Rule Priority Heuristics

• No standard, implementation dependent
• Generally
• Simplification before simpagation before
  propagation
• Rule whose head matches the highest number of
  RDCS constraints first
• Rule matching yet untried for the longest time
  first
• Writing order

16
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

17
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

18
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

19
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

20
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

21
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

22
CHR Base Example Is West Criminal?Goal-Directed
Version with Rewrite Rules

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(P,N,W) ? N nono ? P west owns(N,W) ?
  missile(W)
• hostile(N) ? M america enemy(N,M)
• weapon(W) ? missile(W)
• owns(N,W) ? missile(W) ? american(P) ? nation(N)
  ? enemy(N,M) ? N nono ? W m1 ? P west ? M
  america true

23
CHR Base Example Is West Criminal?CHR are
matched not unified with constraint store

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(west,nono,W) ? owns(nono,W) ? missile(W)
• hostile(N) ? enemy(N,america)
• weapon(W) ? missile(W)
• owns(nono,m1) ? missile(m1) ? american(west) ?
  nation(nono) ? enemy(nono,america) ? true

24
CHR Base Example Is West Criminal?CHR are
matched not unified with constraint store

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(west,nono,W) ? owns(nono,W) ? missile(W)
• hostile(N) ? enemy(N,america)
• weapon(W) ? missile(W)
• owns(nono,m1) ? missile(m1) ? american(west) ?
  nation(nono) ? enemy(nono,america) ? true

25
CHR Base Example Is West Criminal?CHR are
matched not unified with constraint store

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(west,nono,W) ? owns(nono,W) ? missile(W)
• hostile(N) ? enemy(N,america)
• weapon(W) ? missile(W)
• owns(nono,m1) ? missile(m1) ? american(west) ?
  nation(nono) ? enemy(nono,america) ? true

head sells(west,nono,W) unifies with constraint
sells(P,N,W) but does not match it!
26
CHR Base Example Is West Criminal?CHR are
matched not unified with constraint store

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(west,nono,W) ? owns(nono,W) ? missile(W)
• hostile(N) ? enemy(N,america)
• weapon(W) ? missile(W)
• owns(nono,m1) ? missile(m1) ? american(west) ?
  nation(nono) ? enemy(nono,america) ? true

27
CHR Base Example Is West Criminal?CHR are
matched not unified with constraint store

• criminal(P) ? american(P) ? weapon(W) ? nation(N)
  ? hostile(N) ? sells(P,N,W)
• sells(west,nono,W) ? owns(nono,W) ? missile(W)
• hostile(N) ? enemy(N,america)
• weapon(W) ? missile(W)
• owns(nono,m1) ? missile(m1) ? american(west) ?
  nation(nono) ? enemy(nono,america) ? true

• No new rule applies
• Neither positive nor negative conclusion can be
  reached about the goal to prove
• Like CFOL and unlike Prolog and production
  systems, CHR makes open-world assumption
• Thus, lack of proof for the goal does not result
  in negative conclusion

28
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

29
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

30
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

31
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

32
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

33
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

34
CHR Base Example Is West Criminal?Data-Driven
Version with Production Rules

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(N,W) ? missile(W) ? N nono ? P west
  sells(P,N,W)
• enemy(N,M) ? M america hostile(N)
• missile(W) ? weapon(W)
• start ? N nono ? W m1 ? P west ? M
  america owns(N,W) ? missile(W) ?
  american(P) ? nation(N) ? enemy(N,M)

35
CHR Base Example Is West Criminal?Data Driven
Version with Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

36
CHR Base Example Is West Criminal?Data Driven
Version with Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

37
CHR Base Example Is West Criminal?Data Driven
Version w/ Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

38
CHR Base Example Is West Criminal?Data Driven
Version w/ Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

39
CHR Base Example Is West Criminal?Data Driven
Version w/ Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

40
CHR Base Example Is West Criminal?Data Driven
Version w/ Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

41
CHR Base Example Is West Criminal?Data Driven
Version w/ Partially Instantiated Heads

• american(P) ? weapon(W) ? nation(N) ? hostile(N)
  ? sells(P,N,W) ? criminal(P)
• owns(nono,W) ? missile(W) ? sells(west,nono,W)
• enemy(N,america) ? hostile(N)
• missile(W) ? weapon(W)
• start ? owns(nono,m1) ? missile(m1) ?
  american(west) ? nation(nono) ?
  enemy(nono,america)

42
CHRV Abstract Syntax
2..
body
guard
0..1
CHR Rule
And Formula
Tried Alternative Body

simplified head
CHR Base
0..1

propagated head
0..1
Atomic Formula
Fired Rule
Simpagation Rule
Simplification Rule
Propagation Rule
Constraint

ordered



Built-In Constraint
Rule Defined Constraint
Built-In Constraint Store
Rule Defined Constraint Store
Derivation State
CHR Derivation


true
false
43
CHRV Declarative Semantics inClassical
First-Order Logic (CFOL)

• Simplification rule sh1, ... , shi ltgt g1, ...,
  gj b11, ..., bkp ... b11, ..., blq.
• where X1, ..., Xn vars(sh1 ? ... ? shi ? g1
  ? ... ? gj) and Y1, ... , Ym vars(b1 ? ... ?
  bk) \ X1, ..., Xn?X1, ..., Xn g1 ? ... ? gj
  ? (sh1 ? ... ? shi ?
  ?Y1, ... , Ym ((b11 ? ... ?
  bkp) ? ... ? (b11 ? ... ? bkq))
• Propagation rule ph1, ... , phi gt g1, ..., gj
  b11, ..., bkp ... b11, ..., blq.
• where X1, ..., Xn vars(ph1 ? ... ? phi
  ? g1 ? ... ? gj) and Y1, ... , Ym vars(b1 ?
  ... ? bk) \ X1, ..., Xn?X1, ..., Xn g1 ? ...
  ? gj ? (ph1 ? ... ? phi ?
  ?Y1, ... , Ym ((b11 ? ...
  ? bkp) ? ... ? (b11 ? ... ? bkq))

44
CHRV Operational Semantics

• When rule R with disjunctive body B1 ... Bk
  is fired
• Update both constraint stores using B1
• Start next matching-updating cycle
• When BICS false or when no rule matches the
  RDCS
• Backtrack to last alternative body Bi
• Restore both constraint stores to their states
  prior to their update with Bi
• Update both constraint stores using Bi1
• Start next matching-updating cycle
• Exhaustively try all alternative bodies of all
  fired rules through backtracking

45
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltgd Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

46
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

47
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

48
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

49
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

50
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

51
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

52
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

53
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

54
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

55
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

56
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

57
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

58
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

59
CHRV Base Example Course Enrollment

• enrolled(john,C,T) gt T lt 1994 T gt 1996.
• enrolled(S,C,T), prereq(C,C) gt
  enrolled(S,C,T), T lt T.
• XltY, XltZ ltgt Y ltground Z X lt Y.
• X ltY, X gt Y ltgt false.
• X gt X ltgt true.
• X lt Y ltgt gd(X), gd(Y) X ltgd Y

60
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

61
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

62
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

63
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

64
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

65
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

66
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

67
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

68
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

69
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

70
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

71
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

72
CHRV Base Example AppendCHRV as a Logic
Programming Platform

• a) append(X,Y,Z) ? Y ? ( X ? Y L ?
  Z L ) ? ( X H L1 ? Y L2 ? Z H L3
  ? append(L1,L2,L3))
• b) append(X, ,Z) ? X Z

73
CHRV Practical Applications

• Declarative, easy to extend and compose
  constraint solvers and all their applications
• Scheduling, allocation, planning, optimization,
  recommendation, configuration
• Deductive theorem proving (propositional and
  first-order) and all its applications
• CASE tools, declarative programs analysis,
  formal methods in hardware and software design,
• Hypothetical adbuctive reasoning and all its
  applications
• Diagnosis and repair, observation explanation,
  sensor data integration
• Multi-agent reasoning
• Spatio-temporal reasoning and robotics
• Hybrid reasoning integrating
• Deduction, belief revision, abduction,
  constraint solving and optimization
• with open and closed world assumption
• Heterogeneous knowledge integration
• Semantic web services
• Natural language processing

74
CHRV vs. Production Systems

• Common characteristics
• Forward chains rules
• Requires conflict resolution strategy to choose
• Which of several matching rules to fire
• Non-monotonic reasoning due to
• Constraint retraction in Rule-Defined Constraint
  Store
• Fact retraction in the RHS
• Tricky confluence and termination issues

• CHRV
• Constraint store contains arbitrary atoms
  including functional, non-ground atoms
• Simplification rules allow straightforward
  modeling for goal-driven reasoning, with
  rewriting simulating Prolog-like backward
  chaining
• Disjunctive bodies
• Built-in backtracking search

• Production Systems
• Fact base only contains ground Datalog atoms
• Cumbersome modeling to implement goal-driven
  reasoning
• No disjunctions in RHS
• No built-in search

75
CHRV vs. Rewriting Systems

• Common characteristics
• Forward chains rules
• Requires conflict resolution strategy to choose
• Which of several matching rules to fire
• Non-monotonic reasoning due to
• Constraint retraction in Rule-Defined Constraint
  Store
• Retraction of substituted sub-term
• Tricky confluence and termination issues

• CHRV
• Matching applied to atomic formula conjunctions
• Rule head is matched with constraint store
  sub-set, which requires head to be more general
  than sub-set
• Propagation rules provide further simplification
  opportunities

• Rewriting Systems
• Unification of LHS is applied recursively down to
  sub-terms
• Rule LHS is unified with sub-term which can thus
  be more general than LHS
• All reasoning done through rewriting (no
  propagation rules)

76
CHRV vs. Prolog

• CFOL semantics of CHRV guardless, single head
  simplification rule, equivalent to CFOL semantics
  of pure Prolog clause set sharing same conclusion
  (Clark's completion)
• Simplification rule sh ltgt true b11, ..., bkp
  ... b11, ..., blq.
• where X1, ..., Xn vars(shi), and Y1,
  ... , Ym vars(b1 ? ... ? bk) \ X1, ..., Xn
  ?X1, ..., Xn true ? (sh ? ?Y1, ... , Ym ((b11 ?
  ... ? bkp) ? ... ? (b11 ? ... ? bkq))
• Equivalent Prolog clauses sh - b11, ..., bkp.
  , ... , sh - b11, ..., blq.
• Thus, using Clark's completion, any Prolog
  program can be reformulated into a semantically
  equivalent CHRV program
• CHRV extends Prolog with
• Conjunctions in the heads
• Guards
• Non-ground numerical constraints heads, guards
  and bodies
• Propagation rules