1. Introduction to Rule-Based Reasoning
Jacques Robin

2. Outline Rules as a knowledge representation formalism
Common characteristics of rule-based systems
Roadmap of rule-based languages
Common advantages and limitations
Example practical application of rules: declarative business rules
History of rule-based systems
Production Systems
Term Rewriting Systems
Logic Programming and Prolog

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 an agent’s percepts
and/or facts in its Knowledge Base (KB)
that represent 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,
directly on its environment (through actuators) or on the facts in its KB.
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 Fact Base: Environment Rule Engine: Rule Base:
Volatile knowledge
Dependent on problem instance
Data
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 agent’s 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
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 engine’s 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, taxonomic and uncertain knowledge
Current research challenge:
User-friendly reasoning engine for probabilistic object-oriented rules

8. Rule-Based Implementation frequent policy changes
Business Rules
Example of modern commercial application of rule-based knowledge representation
Imperative OO
Program
Language
SQL API
GUI API
Classic
Implementation
GUI Layer
Data Layer
Business Logic
Layer
Classic 3-Tier
Information System
Architecture
Rule-Based Implementation
Imperative OO
Host Language
Embedded
Production
Rule
Engine
Language
SQL API
GUI API
Rule Base
Easier to reflect
frequent policy changes
than imperative code
Generic Component
Reusable in Any
Application Domain

9. Semi-Natural Language Syntax for Business Rules
Associate key word or key phrase to:
Each domain model entity or relation name
Each rule language syntactic construct
Each host programming language construct used in rules
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 W to nono
IF W is a missile THEN W is a weapon
IF N is an enemy of America THEN N is hostile

10. Roadmap of Rule-Based Languages
ELAN
Maude
Otter
EProver
Rewrite
Rules
ISO Prolog
Logic Programming
CLP(X)
Rule-Based
Constraint
Languages
Transaction
Logic
HiLog
Concurrent
Prolog
Courteous
Rules
OPS5
Production
Rules
Frame
Logic
CCLP(X)
Flora
CHRV
CHR
RuleML
OO Rule
Languages
NeOPS
JEOPS
CLIPS
JESS
Java
Smalltalk
C++
Pure OO
Languages
CHORD
OCL
MOF
UML
QVT
SWSL
XML
Web Markup
Languages
XSLT

11. Rewrite Rules: Abstract Syntax
Rule Base
Rewrite
Rule *
RHS
plus(X,0)  X
fib(suc(suc(N)))
 plus(fib(suc(N)),fib(N))
LHS
Term
Functional
Term
Non-Functional
Ground
Non-Ground
Function
Symbol
Constant
Variable *
args
{disjoint, complete}

12. Rewrite Rules: Operational Semantics
Base
T : Term
Unify LHS of Rewrite Rule Base
against sub-terms of Term
: Unifying Set of Pairs:
Instantiated Rule
Instantiated Sub-term
[ Matching Pair Set Empty ]
Pick one
Pair Set
[ Else ]
[ Matching Pair
Set Singleton ]
: Pair
Instantiated Rule R
Instantiated Sub-Term S
Substitute Sub-Term S
in Term T
by RHS of Rule R

13. Rewrite Rule Base Computation 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 d
suc(suc(suc(0)))

14. Conditional Rewrite Rules: Abstract Syntax
Rule Base
Rewrite
Rule *
RHS
X = 0  Y = 0 | X + Y  0
Condition
LHS * 2
Equation
Term
Rule with matching LHS can only be fired if condition is also verified
Proving condition can be recursively done by rewriting it to true

15. Rewrite Rule Base Deduction Example: Is West Criminal?
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

16. 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)

17. Production Rules: Abstract Syntax
Rule Base
Rule * IF
(p(X,a) OR p(X,b))
AND p(Y,Z) AND q(c)
AND X <> Y AND X > 3.5
THEN
Z = X + 12
AND add{r(a,c,Z}
AND delete{p(Y,Z)}
Right-Hand Side
(RHS)
Action *
Arithmetic
Calculation
Fact Base Update
Operator: enum{add,delete}
Atom
And-Or Formula
Connective: enum{and,or}
Left-Hand Side
(LHS)
2..*
Fact
Base *
Predicate
Symbol
Non-Ground
Atom
Constant
Variable
Ground
Non-Functional
Term
args
Functor
Arithmetic
Predicate Symbol
Arithmetic
Constant Symbol

18. Production System Architecture
Fact
Base
Fact Base
Management
Component
Pattern Matching
Component
Action Execution
Component
Rule Firing
Policy
Component
Rule
Base
Host Language
API
Candidate
Rules

19. Production Rules: Operational Semantics
Match LHS
of Rule Base
against
Fact Base
Rule
Base
Fact
Matching
Instantiated
Rule Set
[Matching Instantiated Rule Set Empty]
Pick one
Instantiated
Rule to Fire
[Else]
[Matching Instantiated
Rule Set Singleton]
Selected
Instantiated
Rule
Execute in Sequence
Actions in RHS
of Selected
Instantiated Rule

20. 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

21. Production Rule Base Example: Is West Criminal?
IF american(P) AND weapon(W) AND nation(N) AND hostile(N) AND sell(P,N,W)
THEN add{criminal(P)}
IF owns(nono,W) AND missile(W) THEN add{sells(west,nono,W)
IF missile(W) THEN add{weapon(W)}
IF enemy(N,america) THEN add{hostile(N)}
Initial Fact Base:
owns(nono,m1)
missile(m1)
american(west)
nation(nono)
enemy(nono,america)
nation(america)

22. Production Rule Inference Example: Is 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)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

23. Production Rule Inference Example: Is 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,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

24. Production Rule Inference Example: Is 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,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

25. Production Rule Inference Example: Is 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,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

26. Production Rule Inference Example: Is 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)
american(west)
missile(m1)
nation(nono)
enemy(nono,america)
owns(nono,m1)
nation(america)

27. Production Rule Inference Example: Is 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)
american(west)
missile(m1)
nation(nono)
enemy(nono,america)
owns(nono,m1)
nation(america)

28. Production Rule Inference Example: Is West Criminal?
criminal(west)?
criminal(P)
american(P)
weapon(m1)
nation(N)
hostile(nono)
sells(P,N,W)
sells(west,nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

29. Production Rule Inference Example: Is West Criminal?
criminal(west)?
criminal(P)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

30. Production Rule Inference Example: Is West Criminal?
criminal(west)?
criminal(west)
weapon(m1)
hostile(nono)
sells(west,nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

31. Production Rule Inference Example: Is West Criminal?
criminal(west)? yes
criminal(west)
weapon(m1)
hostile(nono)
sells(west,nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

32. 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 add{p(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.

33. 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

34. Embedded Production System: Abstract Syntax
Right-Hand Side
(RHS)
Action *
Fact Base Update
Operator: enum{add,delete}
Fact
Base
Production
Rule Base
Rule
Left-Hand Side
(LHS)
And-Or Formula
Connective: enum{and,or}
2..*
HPL
Operation
HPL Ground
Data Structure
HPL
args *
HPL Boolean
Operation
HPL Boolean
Operator
HPL

35. 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
Event
RHS
Agent’s
Percept
HPL Boolean
Operation

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. Logic Programming: a Versatile Metaphor
Declarative
Programming
Formal
Software
Specification
Logic
Programming
Formal
Logic
Theory
Automated
Reasoning
Intelligent
Databases

38. Logic Programming: Key Ideas
Logical Theory
Software Specification
Declarative Programming
Automated Reasoning
Intelligent Databases
Logical Formula
Abstract Specification
Declarative Program / Data Structure
Knowledge Base
Database
Theorem Prover
Specification Verifier
Interpreter / Compiler
Inference Engine
Query Processor
Theorem Proof
Specification Verification
Program Execution
Inference
Query Execution
Theorem Proving Strategy
Verification Algorithm
Single, Fixed, Problem-Independent Control Structure
Inference Search Algorithm
Query Processing Algorithm

39. Logic programming vision
Single language with logic-based declarative semantics that is:
A Turing-complete, general purpose programming language
A versatile, expressive knowledge representation language to support deduction and other reasoning services useful for intelligent agents (abduction, induction, constraint solving, belief revision, belief update, inheritance, planning)
Data definition, query and update language for databases with built-in inference capabilities
"One tool solves all" philosophy:
Any computation = resolution + unification + search
Programming = declaring logical axioms (Horn clauses)
Running the program = query about theorems provable from declared axioms
Algorithmic design entirely avoided (single, built-in control structure)
Same language for formal specification (modeling) and implementation
Same language for model checking and code testing

40. Prolog First and still most widely used logic programming language
Falls short to fulfill the vision in many respect:
Many imperative constructs with no logical declarative semantics
No commercial deductive database available
Most other logic programming languages both:
Extend Prolog
Fulfill better one aspect of the logic programming vision
In the 80s, huge Japanese project tried to entirely rebuild all of computing from ground up using logic programming as hardware basis

41. Pure Prolog: abstract syntax
arg *
0..1
Definite Query
+connective = 
arg
Pure Prolog Term *
Pure Prolog Atom * *
Symbol
predicate
body
head
Definite Clause
+connective = 
clauses
Functional Term
Function-Free Term
functor
Numerical Symbol
Variable
c11 (...,Xk1,...) :- p11(...,Xi1,...), ... , pm1(...,Xj1,...).
...
c1n (...,Xkn,...) :- p1n(...,Xin,...), ... , pmn(...,Xjn,...).
Definite Logic Program
+connective = 
parent(al,jim)
 parent(jim,joe)
 anc(A,D)  parent(A,D)
 anc(A,D)  parent(A,P)  anc(P,D)

42. Full Prolog Semantics of full Prolog: Full Prolog Atom
Full Prolog Query
+connective = 
Full Prologl Clause
+connective = 
head
body
0..1
Full Prolog Program *
arg
Prolog Literal
+connective = naf *
Prolog Literal
arg *
Full Prolog Term
predicate
Functional Term
Function-Free Term
functor
Symbol
Built-in
Symbol
User-Defined
Symbol
Variable
Built-in
Imperative Symbol
Built-in
Logical Symbol
Semantics of full Prolog:
Cannot be purely logic-based
Must integrate algorithmic ones
Numerical
Symbol
Meta-Programming
Predicate Symbol
I/O Predicate
Symbol
Search Customization
Predicate Symbol
Program Update
Predicate Symbol

43. Pure Prolog program declarative formal semantics
Declarative, denotational, intentional:
Clark’s transformation to CFOL formula
First-order formula semantically equivalent to program
Declarative, denotational, extentional, model-theoretic:
Least Herbrand Model
Intersection of all Herbrand Models
Conjunction of all ground formulas that are deductive consequences of program

44. CFOL x Prolog Semantic Assumptions
Classical First-Order Predicate Logic:
No Unique Name Assumption (UNA): two distinct symbols can potentially be paraphrases to denote the same domain entity
Open-World Assumption (OWA): If KB |≠ Q but KB |≠ Q , the truth value of Q is considered unknown
Ex: KB: true  core(ai)  true  core(se)  core(C)  offered(C,T)  true  offered(mda,fall)
Q: true  offered(mda,spring)
OWA necessary for monotonic, deductively sound reasoning: If KB |= T then A, KB  A |= T
Logic Programming:
UNA: two distinct symbols necessarily denote two distinct domain entities
Closed-World Assumption (CWA): If KB |≠ Q but KB |≠ Q , the truth value of Q is considered false
Ex: ?- offered(mda,spring) fail
CWA is a logically unsound form of negatively abductive reasoning
CWA makes reasoning inherently non-monotonic as one can have: KB |= T but KB  A |≠ T if the proof KB |= T included steps assuming A false by CWA
Motivation: intuitive, representational economy, consistent w/ databases

45. Clark’s completion semantics
Transform Pure Prolog Program P into semantically equivalent CFOL formula comp(P)
Same answer query set derived from P (abductively) than from FP (deductively)
Example P: core(se). core(ai). offered(mda,fall). offered(C,T) :- core(C).
?- offered(mda,spring) no ?-
Naive semantics naive(P): C,T true  core(ai)  true  core(se)  true  offered(mda,fall)  core(C)  offered(C,T) naive(P) |≠ offered(mda,spring)
Clark’s completion semantics comp(P):
C,T,C1,T1
(core(C1)  (C1=ai  C1=se))  (offered(C1,T1)  (C1=mda  T1=fall)  (C,T (C1=C  T1=T  core(C))))  (ai=se)  (ai=mda)  (ai=fall)  (se=fall)  (se=mda)  (mda=fall) Fp |= offered(mda,spring)

46. Clark’s transformation
Axiomatizes closed-world and unique name assumptions in CFOL
Starts from naive Horn formula semantics of pure Prolog program
Partitions program in clause sets, each one defining one predicate (i.e., group together clauses with same predicate c(t1, ..., tn) as conclusion)
Replaces each such set by a logical equivalence
One side of this equivalence contains c(X1, ..., Xn) where X1, ..., Xn are fresh universally quantified variables
The other side contains a disjunction of conjunctions, one for each original clause
Each conjunction is either of the form:
Xi = ci  ...  Xj = ci, if the original clause is a ground fact
Yi ... Yj Xi = Yi  ...  Xj = Yj  p1( ...)  ...  pn( ... ) if the original clause is a rule with body p1( ...)  ...  pn( ... ) containing variables Yi ... Yj
Joins all resulting equivalences in a conjunction
Adds conjunction of the form (ci = cj) for all possible pairs (ci,cj) of constant symbols in pure Prolog program

47. SLD Resolution
SLD resolution (Linear resolution with Selection function for Definite logic programs):
Joint use of goal-driven set of support and input resolution heuristics
Always pick last proven theorem clause with next untried axiom clause
Always questions last pick even if unrelated to failure that triggered backtracking
A form of goal-driven backward chaining of Horn clauses seen as deductive rules
Prolog uses special case of SLD resolution where:
Axiom clauses tried in top to bottom program writing order
Atoms to unify in the two picked clauses:
Conclusion of selected axiom clause
Next untried premise of last proven theorem clause in left to right program writing order (goal)
Unification without occur-check
Generates proof tree in top-down, depth-first manner
Failure triggers systematic, chronological backtracking:
Try next alternative for last selection even if clearly unrelated to failure
Reprocess from scratch new occurrences of sub-goals previously proven true or false
Simple, intuitive, space-efficient, time-inefficient, potentially non-terminating (incomplete)

48. Refutation Resolution Proof Example
Refutation proof principle:
To prove KB |= F
Prove logically equivalent: (KB  F) |= True
In turn logically equivalent to: (KB   F) |= False
(american(P)  weapon(W) 
nation(N)  hostile(N) 
sells(P,N,W)  criminal(P)) //1
 (T  owns(nono,m1)) //2a
 (T  missile(m1)) //2b
 (owns(nono,W)  missile(W)  sells(west,nono,W)) //3
 (T  american(west)) //4
 (T  nation(nono)) //5
 (T  enemy(nono,america)) //6
 (missile(W)  weapon(W)) //7
 (enemy(N,america)  hostile(N)) //8
 (T  nation(america)) //9
 (criminal(west)  F) //0
1. Solve 0 w/ 1 unifying P/west: american(west)  weapon(W)  nation(N)  hostile(N)  sells(west,N,W)  F //10
2. Solve 10 w/ 4: weapon(W)  nation(N)  hostile(N)  sells(west,N,W)  F //11
3. Solve 11 w/ 7: missile(W)  nation(N)  hostile(N)  sells(west,N,W)  F //12
4. Solve 12 w/ 2b unifying W/m1: nation(N)  hostile(N)  sells(west,N,m1)  F //13
5. Solve 13 w/ 5 unifying N/nono: hostile(nono)  sells(west,nono,m1)  F //14
6. Solve 14 w/ 8 unifying N/nono: enemy(nono,america)  sells(west,nono,m1)  F //15
7. Solve 15 w/ 6: sells(west,nono,m1)  F //16
8. Solve 16 w/ 3 unifying W/m1: owns(nono,m1)  missile(m1)  F //17
9. Solve 17 with 2a: missile(m1)  F //18
10. Solve 18 with 2b: F

49. SLD resolution example
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)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

50. SLD resolution example
criminal(west)?
criminal(west)
american(P)
weapon(W)
nation(N)
hostile(N)
sells(P,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

51. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(W)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

52. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(W)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

53. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(W)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

54. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(W)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

55. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(W)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(W)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

56. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(W)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

57. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,W)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

58. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

59. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

60. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

61. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

62. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

63. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

64. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(nono,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

65. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(nono,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

66. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(nono,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

67. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(nono,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

68. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,W)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

69. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

70. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

71. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

72. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

73. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

74. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

75. SLD resolution example
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

76. SLD resolution example
criminal(west)? yes
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,m1)
missile(m1)
enemy(nono,america)
owns(nono,m1)
american(west)
missile(m1)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)

77. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

78. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

79. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(america)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

80. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(america)
hostile(america)
sells(west,america,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

81. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(america)
hostile(america)
sells(west,america,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

82. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(america)
hostile(america)
sells(west,america,m1)
sells(west,nono,W)
missile(m1)
owns(nono,W)
enemy(america,america)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

83. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(america)
hostile(america)
sells(west,america,m1)
sells(west,nono,W)
enemy(america,america)
missile(m1)
fail
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

84. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(america)
hostile(america)
sells(west,america,m1)
sells(west,nono,W)
backtrack
missile(m1)
enemy(N,america)
owns(nono,W)
fail
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

85. SLD resolution example with backtracking
criminal(west)?
criminal(west)
backtrack
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

86. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(N)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

87. SLD resolution example with backtracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(N)
sells(west,N,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

88. SLD resolution example with bactracking
criminal(west)?
criminal(west)
american(west)
weapon(m1)
nation(nono)
hostile(nono)
sells(west,nono,m1)
sells(west,nono,W)
missile(m1)
enemy(N,america)
owns(nono,W)
american(west)
missile(m1)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)

89. Limitations of Pure Prolog and ISO Prolog
For knowledge representation:
search customization:
No declarative constructs
Limited support procedimental constructs (cuts)
no support for uncertain reasoning
forces unintuitive rule-based encoding of inherently taxonomic and procedural knowledge
knowledge base updates non-backtrackable and without logical semantics
ab :- assert(a), b. if b fails, a remains as true
For programming:
no fine-grained encapsulation
no code factoring (inheritance)
poor data structures (function symbols as only construct)
mismatch with dominant object-oriented paradigm
not integrated to comprehensive software engineering methodology
IDE not friendly enough
scarce middleware
very scarce reusable libraries or components (ex, web, graphics)
mono-thread
For declarative logic programming:
imperative numerical computation, I/O and meta-programming without logical semantics

90. Limitation of SLD resolution engines
Unsound: unification without occur-check
Incomplete: left-recursion
Correct ancestor Prolog program:
anc(A,D) :- parent(A,D).
anc(A,D) :- parent(P,D), anc(A,P).
Logically equivalent program (since conjunction is a commutative connective) that infinitely loops: anc(A,D) :- anc(A,P), parent(P,D). anc(A,D) :- parent(A,D).
Inefficient:
repeated proofs of same sub-goals
irrelevant chronological backtracking

91. Prolog’s imperative arithmetics
fac(0,1) :- !.
fac(I,O) :- I1 is I - 1,
fac(I1,O1), O is I * O1.
?- fac(1,X).
X = 1
?- fac(3,X).
X = 6
?- I1 is I -1
error
?- I1 = I -1
I1 = I -1
?- I = 2, I1 = I -1
I1 = 2 -1
?- I = 2, I1 is I -1
I1 = 1
Arithmetic functions and predicates generate exception if queried with non-ground terms as arguments
is: requires a uninstantiated variable on the left and an arithmetic expression on the right
Relational programming property lost
Syntax more like assembly than functional !
is: Prolog’s sole explicit variable assignment predicate (only for arithmetics)
=: is a bi-directional, general-purpose unification query predicate

92. Prolog’s redundant sub-goal proofs

93. Extensions of pure Prolog
Inductive LP
Aleph
Abductive LP
Preference LP
Functional LP
Pure Prolog
Constraint LP
ISO Prolog
XSB
High-Order LP
Tabled LP
Transaction LP
Flora
Frame (OO) LP

94. Constraint Logic Programming (CLP)
CSP libraries’ strengths:
Specification level, declarative programming for predefined sets of constraints over given domains
Efficient
Extensive libraries cover both finite domain combinatorial problems and infinite numerical domain optimization problems
CSP libraries’ weaknesses:
Constraint set extension can only be done externally through API using general-purpose programming language
Same problem for mixed constraint problems (ex, mixing symbolic finite domain with real variables)
Prolog’s strengths:
Allows specification level, declarative programming of arbitrary general purpose problems
A constraint is just a predicate
New constraints easily declaratively specified through user predicate definitions
Mixed-domain constraints straightforwardly defined as predicate with arguments from different domains
Prolog’s weakness:
Numerical programming is imperative, not declarative
Very inefficient at solving finite domain combinatorial problem

95. CLP Application Rule Base Solver Programming Language L
Integrate Prolog with constraint satisfaction and solving libraries in a single inference engine
Get the best of both worlds:
Declarative user definition of arbitrary constraints
Declarative definition of arbitrarily mixed constraints
Declarative numerical and symbolic reasoning, seamlessly integrated
Efficient combinatorial and optimization problem solving
Single language to program constraint satisfaction or solving problems and the rest of the application
CLP Application Rule Base
CLP Engine
Prolog Engine
...
Procedural Solver
for Domain D1
Procedural Solver
for Domain Dk
Prolog/L Bridge
Solver Programming Language L