Ontologies Reasoning Components Agents Simulations Introduction to Rule-Based Reasoning Jacques...
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OntologiesReasoningComponentsAgentsSimulations
Introduction to Rule-Based Introduction to Rule-Based ReasoningReasoning
Jacques Robin
OutlineOutline
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
Rules as a Rules as a Knowledge Representation Knowledge Representation
FormalismFormalism 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.
Rule-Based AgentRule-Based Agent
Enviro
nm
ent
Sensors
Effectors
Rule Base:Rule Base:• Persistant intentional knowledge• Dependent on problem class, not instance• Declarative code
Ask
Fact Base:Fact Base:• Volatile knowledge• Dependent on problem instance• Data
Rule Engine:Rule Engine:• Problem class independent• Only dependent on rule language• Declarative code interpreter or compiler
Tell Retract
Ask
Rule Languages: Common Rule Languages: Common CharacteristicsCharacteristics
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.
Rule Languages: General AdvantagesRule 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
Rule Languages: General LimitationsRule 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
Business Rules Business Rules
Example of modern commercial application of rule-based knowledge representation
GUI Layer
Data Layer
Business LogicLayer
Classic 3-TierInformation System
Architecture
Imperative OOProgram
Imperative OOLanguageSQL API
Imperative OOLanguageGUI API
ClassicImperative OOImplementation
Rule-Based Implementation
Imperative OOHost Language
EmbeddedProduction
RuleEngine
Imperative OOLanguageSQL API
Imperative OOLanguageGUI API
ProductionRule Base
Generic ComponentReusable in Any
Application Domain
Easier to reflect frequent policy changes
than imperative code
Semi-Natural Language SyntaxSemi-Natural Language Syntaxfor Business Rulesfor 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
OO RuleLanguages
NeOPS
JEOPS
CLIPS
JESS
XML
Web MarkupLanguages
CLP(X)
Rule-BasedConstraintLanguages
Roadmap of Rule-Based LanguagesRoadmap of Rule-Based Languages
XSLT
OPS5
ProductionRules
ISO Prolog
Logic Programming
TransactionLogic
HiLogConcurrentProlog
CourteousRules
CCLP(X)
FrameLogic
Flora
OCLMOF
UML
CHORD
RuleML
ELAN
Maude
Otter
EProver
RewriteRules
SWSL
CHRV
CHR
QVT
Java
Smalltalk
C++
Pure OOLanguages
Rewrite Rules: Abstract SyntaxRewrite Rules: Abstract Syntax
RewriteRule Base
RewriteRule
* RHS plus(X,0) X
fib(suc(suc(N))) plus(fib(suc(N)),fib(N))
FunctionalTerm
Non-FunctionalTerm
GroundTerm
Non-GroundTerm
FunctionSymbol
ConstantSymbol
Variable
*args
{disjoint, complete} {disjoint, complete}
LHS
Term
Rewrite Rules: Operational Rewrite Rules: Operational SemanticsSemantics
: RewriteRuleBase
T : Term
Unify LHS of Rewrite Rule Baseagainst sub-terms of Term
: Unifying Set of Pairs:- Instantiated Rule- Instantiated Sub-term
[ Matching Pair Set Empty ]
[ Matching Pair Set Singleton ] : Pair
- Instantiated Rule R- Instantiated Sub-Term S
Pick onePair Set[ Else ]
Substitute Sub-Term S in Term T
by RHS of Rule R
Rewrite Rule Base Computation Rewrite Rule Base Computation ExampleExample
a) plus(X,0) X
b) plus(X,suc(Y)) suc(plus(X,Y))
c) fib(0) suc(0)
d) fib(suc(0)) suc(0)e) fib(suc(suc(N))
plus(fib(suc(N)),fib(N))
1. fib(suc(suc(suc(0)))) w/ rule e2. plus(fib(suc(suc(0))),fib(suc(0))) w/ rule d3. plus(fib(suc(suc(0))),suc(0)) w/ rule
b4. suc(plus(fib(suc(suc(0))),0)) w/ rule
a5. suc(fib(suc(suc(0)))) w/ rule e6. suc(plus(fib(suc(0)),fib(0))) w/ rule c7. suc(plus(fib(suc(0)),suc(0))) w/ rule
b8. suc(suc(plus(fib(suc(0)),0))) w/ rule
a9. suc(suc(fib(suc(0)))) w/ rule d10.suc(suc(suc(0)))
Conditional Rewrite Rules:Conditional Rewrite Rules:Abstract SyntaxAbstract Syntax
RewriteRule Base
RewriteRule
*
LHS
Term
RHS
Condition
Equation
*
2
X = 0 Y = 0 | X + Y 0
• Rule with matching LHS can only be fired if condition is also verified• Proving condition can be recursively done by rewriting it to true
Rewrite Rule Base Deduction Rewrite Rule Base Deduction Example:Example:
Is West Criminal?Is West Criminal?Rewrite Rule Base:a) criminal(P) american(P) weapon(W) nation(N) hostile(N) sells(P,N,W)b) sells(west,nono,W) owns(nono,W) missile(W)c) hostile(N) enemy(N,america)d) weapon(W) missile(W)e) owns(nono,m1) missile(m1) american(west) nation(nono) enemy(nono,america) truef) A B B Ag) A A A
Term:criminal(P) american(P) weapon(W) nation(N) hostile(N) sells(P,N,W) w/ rule aamerican(P) weapon(W) nation(N) hostile(N) owns(nono,W) missile(W) w/ rule bamerican(P) weapon(W) nation(N) enemy(N,america) owns(nono,W) missile(W) w/ rule camerican(P) missile(W) nation(N) enemy(N,america) owns(nono,W) missile(W) w/ rule damerican(P) missile(W) nation(N) enemy(N,america) owns(nono,W) w/ rule g ... w/ rule fowns(nono,W) missile(W) american(P) nation(N) enemy(N,america) w/ rule ftrue w/ rule e
Rewriting Systems: Practical Rewriting Systems: Practical ApplicationApplication
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)
Production Rules: Abstract SyntaxProduction Rules: Abstract Syntax
ProductionRule Base
ProductionRule
*
Arithmetic Predicate Symbol
Right-Hand Side(RHS)
Action*
Arithmetic Constant Symbol
IF (p(X,a) OR p(X,b)) AND p(Y,Z) AND q(c) AND X <> Y AND X > 3.5THEN Z = X + 12 AND add{r(a,c,Z} AND delete{p(Y,Z)}
FactFactBase
*
Atom
And-Or Formula
Connective: enum{and,or}
Left-Hand Side(LHS)
2..*
ArithmeticCalculation
Fact Base Update
Operator: enum{add,delete}
PredicateSymbol
Non-GroundAtom
ConstantSymbol
VariableGroundAtom
Non-FunctionalTerm
argsFunctor
Production System ArchitectureProduction System Architecture
Fact Base ManagementComponent
Rule FiringPolicy
Component
Host LanguageAPI
Action ExecutionComponent
FactBase
RuleBase
Pattern MatchingComponent
CandidateRules
Production Rules: Operational Production Rules: Operational SemanticsSemantics
[Matching Instantiated Rule Set Empty]
Pick one Instantiated Rule to Fire [Else]
[Matching InstantiatedRule Set Singleton]
SelectedInstantiated
Rule
Execute in SequenceActions in RHS
of SelectedInstantiated Rule
Match LHSof Rule BaseagainstFact Base
RuleBase
FactBase
MatchingInstantiated
Rule Set
Conflict Resolution StrategiesConflict 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
Production Rule Base Example:Production Rule Base Example:Is West Criminal?Is West Criminal?
Production Rule Base: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)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(W) nation(N) hostile(N) sells(P,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(W) nation(N) hostile(N) sells(P,N,W)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,W)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(m1) nation(N) hostile(N) sells(P,N,W)
criminal(west)?
missile(W) enemy(nono,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,W)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(m1) nation(N) hostile(N) sells(P,N,W)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,W)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(m1) nation(N) hostile(nono) sells(P,N,W)
criminal(west)?
missile(W) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enemy(nono,america) owns(nono,m1)
sells(west,nono,W)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(m1) nation(N) hostile(nono) sells(P,N,W)
criminal(west)?
missile(m1) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enemy(nono,america) owns(nono,m1)
sells(west,nono,W)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(P) weapon(m1) nation(N) hostile(nono) sells(P,N,W)
criminal(west)?
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(P)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(west)
weapon(m1) hostile(nono)
criminal(west)?
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
Production Rule Inference Example:Production Rule Inference Example:Is West Criminal?Is West Criminal?
criminal(west)
weapon(m1) hostile(nono)
criminal(west)? yes
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
Properties of Production SystemsProperties 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.
Production System Inference vs.Production System Inference vs.Lifted Forward ChainingLifted Forward Chaining
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
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
Embedded Production System: Embedded Production System: Abstract SyntaxAbstract Syntax
HPL Operation
HPL GroundData Structure
HPL Data Structureargs
*HPL BooleanOperation
Right-Hand Side(RHS)
Action*Fact Base Update
Operator: enum{add,delete}
FactFactBase
*
ProductionRule Base
ProductionRule
*
Left-Hand Side(LHS)
And-Or Formula
Connective: enum{and,or}2..*
HPL BooleanOperator
HPLOperator
Generalized Production Rules: ECA Generalized Production Rules: ECA RulesRules
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
EventCondtion
ActionRule
EventLHS RHS
Agent’sPercept
HPL BooleanOperation
Production Rules vs. Rewriting RulesProduction Rules vs. Rewriting Rules
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
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
FormalLogic
Theory
IntelligentDatabases
FormalSoftware
Specification
AutomatedReasoning
DeclarativeProgramming
Logic Programming: Logic Programming: a Versatile Metaphora Versatile Metaphor
Logic Programming
Logic Programming: Key IdeasLogic 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
Logic programming visionLogic programming vision
Single language with logic-based declarative semanticslogic-based declarative semantics that is: A Turing-complete, general purpose programmingprogramming language A versatile, expressive knowledge representationknowledge 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 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
PrologProlog
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
Definite Logic Program+connective =
Definite Query+connective =
Definite Clause+connective =
clauses
Pure Prolog: abstract syntaxPure Prolog: abstract syntax
Pure Prolog Atom
arg
headbody
0..1 *
*arg Pure Prolog Term*
c11 (...,Xk
1,...) :- p11(...,Xi
1,...), ... , pm1(...,Xj
1,...)....c1
n (...,Xkn,...) :- p1
n(...,Xin,...), ... , pm
n(...,Xjn,...).
parent(al,jim) parent(jim,joe) anc(A,D) parent(A,D) anc(A,D) parent(A,P) anc(P,D)
Numerical Symbol
Function-Free TermFunctional Term
Variable
functor
*
Symbol
predicate
Full PrologFull Prolog
Full Prolog Atom
Full Prolog Query+connective =
Full Prologl Clause+connective =
head
body
0..1
Full Prolog Program+connective =
*
Semantics of full Prolog:• Cannot be purely logic-based• Must integrate algorithmic ones
User-Defined Symbol
Built-in Symbol
Built-inLogical Symbol
Built-in Imperative Symbol
Meta-ProgrammingPredicate Symbol
NumericalSymbol
I/O Predicate Symbol
Program UpdatePredicate Symbol
Search CustomizationPredicate Symbol
Full Prolog Term
Function-Free TermFunctional Term
Variable
Symbol
functor
arg
predicate
* Prolog LiteralProlog Literal
+connective = naf
arg
*
Pure Prolog program Pure Prolog program declarative formal semantics 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
CFOL x Prolog Semantic AssumptionsCFOL 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 |≠ Tif the proof KB |= T included steps assuming A false by CWA
Motivation: intuitive, representational economy, consistent w/ databases
Clark’s completion semanticsClark’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)
Clark’s transformationClark’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
SLD ResolutionSLD 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)
Refutation Resolution Proof ExampleRefutation 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
SLD resolution example SLD resolution example
criminal(P)
american(P) weapon(W) nation(N) hostile(N) sells(P,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(P) weapon(W) nation(N) hostile(N) sells(P,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(W) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(W) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(W) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(W) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(W) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(W) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(W) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,W)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,W)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america)owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
SLD resolution example SLD resolution example
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)? yes
missile(m1) enemy(nono,america) owns(nono,m1)
missile(m1)american(west) nation(nono)
nation(america)
enermy(nono,america) owns(nono,m1)
sells(west,nono,m1)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(america) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(america) hostile(america) sells(west,america,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(america) hostile(america) sells(west,america,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(america) hostile(america) sells(west,america,m1)
criminal(west)?
missile(m1)
enemy(america,america)
owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(america) hostile(america) sells(west,america,m1)
criminal(west)?
missile(m1)
enemy(america,america)
owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
fail
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(america) hostile(america) sells(west,america,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
fail
backtrack
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
backtrack
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(N) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with backtrackingbacktracking
criminal(west)
american(west) weapon(m1) nation(nono) hostile(N) sells(west,N,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
SLD resolution example with SLD resolution example with bactrackingbactracking
criminal(west)
american(west) weapon(m1) nation(nono) hostile(nono) sells(west,nono,m1)
criminal(west)?
missile(m1) enemy(N,america) owns(nono,W)
missile(m1)american(west)
nation(nono)
nation(america) enermy(nono,america)owns(nono,m1)
sells(west,nono,W)
Limitations of Pure Prolog Limitations of Pure Prolog and ISO Prologand ISO Prolog
For knowledge representationknowledge 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 programmingprogramming: 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 logicdeclarative logic programming:
imperative numerical computation, I/O and meta-programming without logical semantics
Limitation of SLD resolution enginesLimitation 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
Prolog’s imperative arithmeticsProlog’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 -1error?- I1 = I -1I1 = I -1?- I = 2, I1 = I -1I1 = 2 -1?- I = 2, I1 is I -1I1 = 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 variable assignmentassignment predicate (only for arithmetics) =: is a bi-directional, general-purpose unification queryquery predicate
Prolog’s redundant sub-goal proofsProlog’s redundant sub-goal proofs
Extensions of pure PrologExtensions of pure Prolog
Pure Prolog
ISO Prolog
High-Order LP
Functional LP
Constraint LP
Abductive LP
Transaction LP
Frame (OO) LP
Preference LP
Tabled LP
Inductive LP
XSB
Aleph
Flora
Constraint Logic Programming (CLP)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 predicateNew 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 declarativeVery inefficient at solving finite
domain combinatorial problem
CLPCLP
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 Engine CLP Application Rule Base
Prolog EngineProcedural Solver
for Domain D1
Procedural Solverfor Domain Dk
Solver Programming Language L
...Prolog/L Bridge