Rule-Based Reasoning

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Rule-Based Reasoning

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Title: Rule-Based 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
Production rules and systems
Abstract syntax
High-level operational semantics
Example rule bases
Special cases
Practical applications
Rewrite rules and systems
Abstract syntax
High-level operational semantics
Comparison with production rules
Example rule bases
Special cases
Practical applications

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 Base
Persistant intentional knowledge,
specific to application domain,
but general to problem class
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 a 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 and
detailed knowledge with the rule engines
algorithm
Especially serious with object-oriented rule
languages that combine rule-based deduction or
abduction with class-based inheritance
But purely relational rule-based language do not
naturally
Embed within mainstream object-oriented modeling
and programming languages
Represent inherently procedural knowledge
Represent taxonomic knowledge

8
Rule Languages Diversity

Production rules
Event-Condition-Action rules (ECA rules)
Rewrite rules
Conditional rewrite rules
Logic program rules
Constraint handling rules (CHR)
Object-oriented rules
Semantic web service rules (SWS rules)
Business rules
Meta-rules

9
Production Rules Abstract Syntax

IF (p(X,a) OR p(X,b)) AND
p(Y,Z) AND q(c) AND X ltgt Y AND X gt 3.5 THEN
Z X 12 AND addr(a,c,Z
AND deletep(Y,Z)
Production Rule Base
Production Rule
Right-Hand Side (RHS)

Action
Left-Hand Side (LHS)
Arithmetic Calculation
Fact Base

Fact
2..
Atom
Predicate Symbol
Functor
args
Arithmetic Predicate Symbol
Non-Ground Atom
Non-Functional Term
Ground Atom
Constant Symbol
Arithmetic Constant Symbol
Variable
10
Production System Architecture
Fact Base Management Component
Fact Base
Rule Firing Policy Component
Action Execution Component
Pattern Matching Component
Rule Base
Host Language API
Candidate Rules
11
Production Rules Operational Semantics
Matching Instantiated Rule Set Empty
Pick one Instantiated Rule to Fire
Match LHS of Rule Base against Fact Base
Matching Instantiated Rule Set
Else
Rule Base
Fact Base
Matching Instantiated Rule Set Singleton
Selected Instantiated Rule
Execute in Sequence Actions in RHS of
Selected Instantiated Rule
12
Conflict Resolution Strategies

Conflict resolution strategy
Heuristic to choose at each cycle which of the
matching rule set to fire
Common strategies
Follow writing order of rules in rule base
Follow absolute priority level annotations
associated with each rule or rule class
Follow relative priority level annotations
associated with pairs of rules or rule classes
Prefer rules whose LHS match the most recently
derived facts in the fact base
Prefer rules that have remained for the longest
time unfired while matching a fact
Apply control meta-rules that declaratively
specify arbitrarily sophisticated strategies

13
Production Rule Base ExampleIs West Criminal?

Production Rule Base
IF american(P) AND weapon(W) AND nation(N) AND
hostile(N) AND sell(P,N,W)
THEN addcriminal(P)
IF owns(nono,W) AND missile(W) THEN
addsells(west,nono,W)
IF missile(W) THEN addweapon(W)
IF enemy(N,america) THEN addhostile(N)
Initial Fact Base
owns(nono,m1)
missile(m1)
american(west)
nation(nono)
enemy(nono,america)
nation(america)

14
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(W)
nation(N)
hostile(N)
sells(P,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
15
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(W)
nation(N)
hostile(N)
sells(P,N,W)
sells(west,nono,W)
missile(m1)
enemy(nono,N)
owns(nono,m1)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
16
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(m1)
nation(N)
hostile(N)
sells(P,N,W)
sells(west,nono,W)
missile(W)
enemy(nono,N)
owns(nono,W)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
17
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(m1)
nation(N)
hostile(N)
sells(P,N,W)
sells(west,nono,W)
missile(m1)
enemy(nono,N)
owns(nono,m1)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
18
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(m1)
nation(N)
hostile(nono)
sells(P,N,W)
sells(west,nono,W)
missile(W)
owns(nono,W)
missile(m1)
american(west)
nation(nono)
enemy(nono,N)
owns(nono,m1)
nation(america)
19
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(m1)
nation(N)
hostile(nono)
sells(P,N,W)
sells(west,nono,W)
missile(m1)
owns(nono,m1)
missile(m1)
american(west)
nation(nono)
enemy(nono,america)
owns(nono,m1)
nation(america)
20
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(m1)
nation(N)
hostile(nono)
sells(P,N,W)
sells(west,nono,m1)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
21
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(P)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
22
Production Rule Inference ExampleIs West
Criminal?
criminal(west)?
criminal(west)
weapon(m1)
hostile(nono)
sells(west,nono,m1)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
23
Production Rule Inference ExampleIs West
Criminal?
criminal(west)? yes
criminal(west)
weapon(m1)
hostile(nono)
sells(west,nono,m1)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
24
Properties of Production Systems

Confluence
From confluent rule base, same final fact base
is derived independently of rule firing order
Makes values of queries made to the system
independent of the conflict resolution strategy
Termination
Terminating system does not enter in an infinite
loop
Example of non-termination production rule base
Initial fact base p(0)
Production rule base IF p(X) THEN Y X 1 AND
addp(Y)
Any production system which conflict resolution
strategy does not prevent the same rule instance
to be fired in two consecutive cycles is
trivially non-terminating.

25
Production System Inference vs.Lifted Forward
Chaining

Common characteristics
Data driven reasoning
Requires conflict resolution strategy to choose
Which of several matching rules to fire
Which of several unifying clauses for next Modus
Ponens inference step

Production System Inference
Sequential conjunction of actions in rule RHS
Non-monotonic reasoning due to delete actions
Matching only Datalog atoms (i.e., atoms which
arguments are all non-functional terms)
Matching ground fact atoms against non-ground
atoms in rule LHS And-Or atom
Neither sound nor complete inference engine

Lifted Forward Chaining
Unique conclusion in Horn Clause
Monotonic reasoning
Unifying arbitrary first-order atoms, possibly
functional
Unifying two arbitrary atoms, possibly both
non-grounds
Sound and refutation-complete

26
Embedded Production System Abstract Syntax
Production Rule Base
Production Rule
Right-Hand Side (RHS)

Action

Fact Base\
Fact
Left-Hand Side (LHS)
HPL Operation
HPL Boolean Operation
HPL Ground Data Structure
HPL Data Structure

2..
args
HPL Operator
HPL Boolean Operator
27
Generalized Production Rules ECA Rules

Event-Condition-Action rule extension of
production rule with triggering event external to
addition of facts in fact base
Only the rule subset which event just occurred is
matched against the fact base
Events thus partition the rule base into subsets,
each one specifying a behavior in response to a
given events

Event Condtion Action Rule
LHS
RHS
Event
Agents Percept
HPL Boolean Operation
28
Production SystemsPractical Applications

Reactive agents
Expert systems in domains without complex entity
taxonomy nor uncertainty
Business rules

29
Semi-Natural Language Syntaxfor Production Rules

Associate key word or key phrase to
Each domain model constant, function and
predicate symbol
Each production rule syntactic construct
Each host programming language construct
Substitute in place of these constructs and
symbols the associated words or phrase
Example Is West Criminal? in semi-natural
language syntax
IF P is American AND P sells a W to N AND
W is a weapon
AND N is a nation AND N is hostile
THEN P is a criminal
IF nono owns a W AND W is a missile THEN west
sells nono to W
IF W is a missile THEN W is a weapon
IF N is an enemy of America THEN N is hostile

30
Rewrite Rules Abstract Syntax
Rewrite Rule Base
Rewrite Rule

plus(X,0) ? X
fib(suc(suc(N)))
? plus(fib(suc(N)),fib(N))

RHS

LHS
Term

args
non-overlapping, complete
non-overlapping, complete
Functional Term
Non-Functional Term
Ground Term
Non-Ground Term
Function Symbol
Constant Symbol
Variable
31
Rewrite Rules Operational Semantics
Rewrite Rule Base
T Term
32
Rewrite Rule Base Example

plus(X,0) ? X
plus(X,suc(Y)) ? suc(plus(X,Y))
fib(0) ? suc(0)
fib(suc(0)) ? suc(0)
fib(suc(suc(N)) ? plus(fib(suc(N)),fib(N))

fib(suc(suc(suc(0)))) w/ rule
e
plus(fib(suc(suc(0))),fib(suc(0))) w/ rule d
plus(fib(suc(suc(0))),suc(0)) w/ rule b
suc(plus(fib(suc(suc(0))),0)) w/ rule a
suc(fib(suc(suc(0)))) w/ rule
e
suc(plus(fib(suc(0)),fib(0))) w/ rule c
suc(plus(fib(suc(0)),suc(0))) w/ rule b
suc(suc(plus(fib(suc(0)),0))) w/ rule a
suc(suc(fib(suc(0)))) w/ rule
c
suc(suc(suc(0)))

33
Conditional Rewrite RulesAbstract Syntax
Rewrite Rule Base
Rewrite Rule

X 0 ? Y 0 X Y ? 0

RHS

LHS
Condition

2
Term
Equation

Rule with matching LHS can only be fired if
condition is also verified
Proving condition can be recursively done by
rewriting it to true

34
Rewrite Rule Base ExampleIs West Criminal?

Rewrite Rule Base
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
A ? B ? B ? A
A ? A ? A
Term
criminal(P)
american(P) ? weapon(W) ? nation(N) ? hostile(N)
? sells(P,N,W)
w/ rule a
american(P) ? weapon(W) ? nation(N) ? hostile(N)
? owns(nono,W) ? missile(W) w/
rule b
american(P) ? weapon(W) ? nation(N) ?
enemy(N,america) ? owns(nono,W) ? missile(W) w/
rule c
american(P) ? missile(W) ? nation(N) ?
enemy(N,america) ? owns(nono,W) ? missile(W)
w/ rule d
american(P) ? missile(W) ? nation(N) ?
enemy(N,america) ? owns(nono,W)
w/ rule g
...

w/ rule f
owns(nono,W) ? missile(W) ? american(P) ?
nation(N) ? enemy(N,america)
w/ rule f
true

w/ rule e

35
Rewriting Systems Practical Application

Theorem proving
CASE
Programming language formal semantics
Program verification
Compiler design and implementation
Model transformation and automatic programming
Data integration
Using XSLT an XML-based language to rewrite
XML-based data and documents
Web server pages and web services (also using
XSLT)

36
Production Rules vs. Rewriting Rules

Common characteristics
Data driven reasoning
Requires conflict resolution strategy to choose
Which of several matching rules to fire
Which of several rules with an LHS unifying with
a sub-term
Non-monotonic reasoning due to
Fact deletion actions in RHS
Retraction of substituted sub-term
Tricky confluence and termination issues

Production System Inference
Fact base implicitly conjunctive
Matching only Datalog atoms (i.e., atoms which
arguments are all non-functional terms)
Matching ground fact atoms against non-ground
atoms in rule LHS And-Or atom

Term Rewriting
Reified logical connectives in term provide full
first-order expressivity
Unifying arbitrary first-order atoms, possibly
functional
Unifying two arbitrary atoms, possibly both
non-grounds

37
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

38
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
39
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
40
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
41
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)

42
CHR Operational Semantics
43
CHR Rule Priority Heuristics

No standard, implementation dependent
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

44
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

Rule Goal UDCS BICS
criminal(P) true true

45
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

Rule Goal UDCS BICS
criminal(P) true true
true criminal(P) true

46
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

Rule Goal UDCS BICS
criminal(P) true true
true criminal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true

47
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

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

Rule Goal UDCS BICS
criminal(P) true true
true criminal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true
b) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? owns(N,W) ? missile(W) N nono ? P west
c) true american(P) ? weapon(W) ? nation(N) ? enemy(N,M) ? owns(N,W) ? missile(W) N nono ? P west ? M america

49
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

Rule Goal UDCS BICS
criminal(P) true true
true criminal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true
b) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? owns(N,W) ? missile(W) N nono ? P west
c) true american(P) ? weapon(W) ? nation(N) ? enemy(N,M) ? owns(N,W) ? missile(W) N nono ? P west ? M america
d) true american(P) ? nation(N) ? enemy(N,M) ? owns(N,W) ? missile(W) N nono ? P west ? M america

50
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

Rule Goal UDCS BICS
criminal(P) true true
true criminal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true
b) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? owns(N,W) ? missile(W) N nono ? P west
c) true american(P) ? weapon(W) ? nation(N) ? enemy(N,M) ? owns(N,W) ? missile(W) N nono ? P west ? M america
d) true american(P) ? nation(N) ? enemy(N,M) ? owns(N,W) ? missile(W) N nono ? P west ? M america
e) true true N nono ? P west ? M america ? W m1
51
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

Rule Goal UDCS BICS
criminal(P) true true

52
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

Rule Goal UDCS BICS
criminal(P) true true
true crimimal(P) true

53
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

Rule Goal UDCS BICS
criminal(P) true true
true crimimal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true

head sells(west,nono,W) unifies with constraint
sells(P,N,W) but does not match it!
54
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

Rule Goal UDCS BICS
criminal(P) true true
true crimimal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true
c) true american(P) ? weapon(W) ? nation(N) ? enemy(N,america) ? sells(P,N,W) true

55
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

Rule Goal UDCS BICS
criminal(P) true true
true crimimal(P) true
a) true american(P) ? weapon(W) ? nation(N) ? hostile(N) ? sells(P,N,W) true
c) true american(P) ? weapon(W) ? nation(N) ? enemy(N,america) ? sells(P,N,W) true
d) true american(P) ? missile(W) ? nation(N) ? enemy(N,america) ? sells(P,N,W) 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

56
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)

Rule Goal UDCS BICS
start true true

57
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)

Rule Goal UDCS BICS
start true true
true start true

58
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)

Rule Goal UDCS BICS
start true true
true start true
e) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) N nono ? W m1 ? P west ? M america

59
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)

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)

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)

Rule Goal UDCS BICS
start true true
true start true
e) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) N nono ? W m1 ? P west ? M america
b) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) N nono ? W m1 ? P west ? M america
c) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) ? hostile(N) N nono ? W m1 ? P west ? M america
d) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) ? hostile(N) ? weapon(W) N nono ? W m1 ? P west ? M america

62
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)

Rule Goal UDCS BICS
start true true
true start true
e) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) N nono ? W m1 ? P west ? M america
b) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) N nono ? W m1 ? P west ? M america
c) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) ? hostile(N) N nono ? W m1 ? P west ? M america
d) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) ? hostile(N) ? weapon(W) N nono ? W m1 ? P west ? M america
a) true start ? owns(N,W) ? missile(W) ? american(P) ? nation(N) ? enemy(N,M) ? sells(P,N,W) ? hostile(N) ? weapon(W) ) ? criminal(P) N nono ? W m1 ? P west ? M america
63
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)

Rule Goal UDCS BICS
start true true

64
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)

Rule Goal UDCS BICS
start true true
true start true

65
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)

Rule Goal UDCS BICS
start true true
true start true
e) true owns(nono,m1) ? missile(m1) ? american(west) ? nation(nono) ? enemy(nono,america) true

66
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)

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)

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)

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)

Rule Goal UDCS BICS
start true true
true start true
e) true owns(nono,m1) ? missile(m1) ? american(west) ? nation(nono) ? enemy(nono,america) true
b) true owns(nono,m1) ? missile(m1) ? american(west) ? nation(nono) ? enemy(nono,america) ? sells(west,nono,m1) true
c) owns(nono,m1) ? missile(m1) ? american(west) ? nation(nono) ? enemy(nono,america) ? sells(west,nono,m1) ? hostile(N) true
d) owns(nono,m1) ? missile(m1) ? american(west) ? nation(nono) ? enemy(nono,america) ? sells(west,nono,m1) ? hostile(N) ? weapon(m1) true
a) owns(nono,m1) ? missile(m1) ? american(west) ? nation(nono) ? enemy(nono,america) ? sells(west,nono,m1) ? hostile(N) ? weapon(m1) ? criminal(west) true
70
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
71
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))

72
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

73
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

Rule UDCS BICS
a1) enrolled(john,cs100,1996), prereq(cs100,cs50) true

74
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

Rule UDCS BICS
a1) enrolled(john,cs100,1996), prereq(cs100,cs50) true
b) enrolled(john,cs100,1996), prereq(cs100,cs50), 1996 lt 1994 true

75
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

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

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

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

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

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

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

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

Rule UDCS BICS
a1) enrolled(john,cs100,1996), prereq(cs100,cs50) true
b) enrolled(john,cs100,1996), prereq(cs100,cs50), 1996 lt 1994 true
f) enrolled(john,cs100,1996), prereq(cs100,cs50), 1996 lt 1994, enrolled(john,cs50,T), T lt 1996 true
solve enrolled(john,cs100,1996), prereq(cs100,cs50), enrolled(john,cs50,T), T lt 1996 gd(1996), gd(1994),1996 ltgd 1994
bt enrolled(john,cs100,1996), prereq(cs100,cs50), enrolled(john,cs50,T), T lt 1996 false
a2) enrolled(john,cs100,1996), prereq(cs100,cs50) true
b) enrolled(john,cs100,1996), prereq(cs100,cs50), 1996 gt 1996 true
e) enrolled(john,cs100,1996), prereq(cs100,cs50), 1996 gt 1996, enrolled(john,cs50,T), T lt 1996 true
a1) enrolled(john,cs100,1996), prereq(cs100,cs50), enrolled(john,cs50,T), T lt 1996 true
c) enrolled(john,cs100,1996), prereq(cs100,cs50), enrolled(john,cs50,T), T lt 1996, T lt 1994 true

83
CHRV Base Example Course Enrollment