Logic Programming

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Logic Programming

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Title: Logic Programming

1
Logic Programming

Jacques Robin

2
Roteiro

A unifying metaphor for computer science
Syntax
Pure Prolog abstract syntax
Full Prolog abstract syntax
Concrete syntax
Declarative semantics
Open x closed world hypothesis
Unique name hypothesis
Clarks completion
Least Herbrand model
Operational semantics
Lifted SLD resolution and backward chaining
Limitations
Of ISO Prolog
Of SLD resolution and pure backward chaining

Tabled logic programming
SLG resolution for pure Prolog
High order logic programming
HiLog abstract syntax
HiLog first-order semantics
Constraint logic programming
Limitations of CSP
CLP engine architeture
CLP abstract syntax
Operational semantics
Other extensions

3
Logic Programming a Versatile Metaphor
4
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
5
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)
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

6
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
Fulfills better one aspect of the logic
programming vision
Basis for aborted attempt to Japanese to fully
revolutionize all computing from ground up in the
80s

7
Pure Prolog abstract syntax
arg

0..1
Pure Prolog Term

Pure Prolog Atom
arg

predicate
head
body
Function-Free Term
Functional Term
functor
Symbol
clauses
Numerical Symbol
Variable
c11 (...,Xk1,...) - p11(...,Xi1,...), ... ,
pm1(...,Xj1,...). ... c1n (...,Xkn,...) -
p1n(...,Xin,...), ... , pmn(...,Xjn,...).

parent(al,jim)
? parent(jim,joe
? anc(A,D) ? parent(A,D)
? anc(A,D) ? parent(A,P) ? anc(P,D)

8
Full Prolog
arg
0..1

arg
Prolog Literal
Full Prolog Term

body
head
Functional Term
predicate
Function-Free Term
functor

Symbol
User-Defined Symbol
Variable
Built-in Symbol
Built-in Logical Symbol
Built-in Imperative Symbol

Semantics of full Prolog
Cannot be purely logic-based
Must integrate algorithmic ones

Meta-Programming Predicate Symbol
Numerical Symbol
I/O Predicate Symbol
Search Customization Predicate Symbol
Program Update Predicate Symbol
9
Pure Prolog program declarative formal semantics

Declarative, denotational, intentional
Clarks 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

10
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 haveKB T but KB ? A ? Tif the
proof KB T included steps assuming A false by
CWA
Motivation intuitive, representational economy,
consistent w/ databases

11
Clarks 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 Pcore(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)
Clarks completion semantics comp(P)
?C,T,C1,T1
(core(C1) ? (C1ai ? C1se)) ?(offered(C1,T1) ?
(C1mda ? T1fall) ? (?C,T (C1C
? T1T ? core(C)))) ??(aise) ? ?(aimda) ?
?(aifall) ? ?(sefall) ? ?(semda) ?
?(mdafall)
Fp ?offered(mda,spring)

12
Clarks transformation

Axiomatizes closed-world and unique name
assumptions in CFOL
Start from naive Horn formula semantics of pure
Prolog program
Partition program in clause sets, each one
defining one predicate (i.e., group together
clauses with same predicate c(t1, ..., tn) as
conclusion)
Replace 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
Join all resulting equivalences in a conjunction
Add conjunction of the form ?(ci cj) for all
possible pairs (ci,cj) of constant symbols in
pure Prolog program

13
SLD Resolution

SLD resolution (Linear resolution with Selection
function for Definite logic programs)
Joint use of input and goal-driven set of support
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)

14
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/westamerican(west) ?
weapon(W) ? nation(N) ?hostile(N) ?
sells(west,N,W) ? F //10 2. Solve
10 w/ 4weapon(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/m1nation(N) ? hostile(N) ? sells(west,N,m1) ?
F //13 5. Solve 13 w/ 5 unifying
N/nonohostile(nono) ? sells(west,nono,m1) ? F
//14 6. Solve 14 w/ 8 unifying
N/nonoenemy(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
15
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
16
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
17
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
18
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
19
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
20
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
21
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
22
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
23
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
24
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
25
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
26
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
27
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
28
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
29
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
30
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
31
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
32
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
33
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
34
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
35
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
36
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
37
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
38
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
39
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
40
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
41
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
42
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)
missile(m1)
american(west)
nation(nono)
enermy(nono,america)
owns(nono,m1)
nation(america)
43
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
44
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
45
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
46
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
47
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
48
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
49
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)
fail
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
50
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
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
51
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
52
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
53
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
54
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)
missile(m1)
american(west)
nation(america)
enermy(nono,america)
owns(nono,m1)
nation(nono)
55
Dual use of Prolog for AI

As knowledge representation language

As Turing-complete programming language

Inference Engine Implementing in
Prolog Reasoning R in formalism F
Knowledge Base Prolog Facts and Rules
Inference Engine Prolog Compiler/ Interpreter
Knowledge Base Sentences in formalism F
Ask
Ask
Tell
Tell
Retract
Retract
Inference Engine Prolog Compiler/ Interpreter
56
Logic vs. imperative and functional programming

Single relational definition
append(,L,L).
append(HT1,L,HT2) - append(T1,L,T2).
Usable as verification function
?- append(a,b,c,a,b,c). -gt yes.
?- append(a,b,c,a). -gt no.
Usable as function and its various input/output
converse functions
?- append(a,b,c,R). -gt R a,b,c.
?- append(H,c,a,b,c). -gt H a,b.
Usable as constraint solver
?- append(X,Y,a,b,c). -gt X , Y a,b,c
-gt X a, Y b,c ...
Usable for enumeration
?- append(X,Y,Z). -gt X , Y , Z -gt X
_, Y , Z _ ...
Implements alone 8 different functions or methods!

57
Limitations of Pure Prolog and ISO Prolog

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

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

58
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

59
Prologs 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 Prologs sole explicit variable assignment
predicate (only for arithmetics)
is a bi-directional, general-purpose
unification query predicate

60
Prologs redundant sub-goal proofs
61
Extensions of pure Prolog
Aleph
Inductive LP
Abductive LP
Preference LP
Functional LP
Pure Prolog
Constraint LP
ISO Prolog
XSB
High-Order LP
Tabled LP
Transaction LP
Flora
Frame (OO) LP
62
Table logic programming SLG resolution

Substitute goal stack by goal and answer table
caching containing for each goal
Calling goal (parent in top-down proof tree)
Pointer to variant goal modulo
variable renaming, ex, p(f(X),Y) ? p(f(A),B)
or term subsumption, ex, p(X,g(d)) ? p(A,B)
Instantiated program clause which conclusion it
unified with
Position of that clause in the program
All its answers
To prove goal G
Look for goal G variant of G in the table
If there is one, then use answers of G to
answer G
Else backchain on G to get first answer to G
After finding such answer (or failing)
cache it in the table
backtrack to find all the other answers and cache
them as well
Resume proof to next premise of calling goal

63
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1
64
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
65
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
66
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
67
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a, 2b
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
68
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a, 2b
1b anc(A,dan) - parent(P,dan), anc(A,P) C2
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
3a parent(P,dan) 2 1b
4a anc(A,fay) 1b anc(A,fay) - parent(A,fay) C1 5a
4b 1b anc(A,fay) - parent(P,fay), anc(A,P) C2
5a parent(A,fay) 4a fail
6a parent(P,fay) 5 4b
75
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a, 2b
1b anc(A,dan) - parent(P,dan), anc(A,P) C2
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
3a parent(P,dan) 2 1b
4a anc(A,fay) 1b anc(A,fay) - parent(A,fay) C1 5a
4b 1b anc(A,fay) - parent(P,fay), anc(A,P) C2 6a
5a parent(A,fay) 4a fail
6a parent(P,fay) 5 4b
76
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a, 2b
1b anc(A,dan) - parent(P,dan), anc(A,P) C2
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
3a parent(P,dan) 2 1b
4a anc(A,fay) 1b anc(A,fay) - parent(A,fay) C1 5a
4b 1b anc(A,fay) - parent(P,fay), anc(A,P) C2 6a
5a parent(A,fay) 4a fail
6a parent(P,fay) 5 4b
7a anc(A,jim) 1b anc(A,jim) - parent(A,jim) C1
77
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A dan
A fay
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a, 2b
1b anc(A,dan) - parent(P,dan), anc(A,P) C2
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
3a parent(P,dan) 2 1b
4a anc(A,fay) 1b anc(A,fay) - parent(A,fay) C1 5a
4b 1b anc(A,fay) - parent(P,fay), anc(A,P) C2 6a
5a parent(A,fay) 4a fail
6a parent(P,fay) 5 4b
7a anc(A,jim) 1b anc(A,jim) - parent(A,jim) C1 8a
7b 1b anc(A,jim) - parent(P,jim), anc(A,P) C2 9a
8a parent(A,jim) 4a fail
9a parent(P,jim) 5 4b
78
SLG resolution for definite programs

anc(A,D) - parent(A,D). C1
anc(A,D) - parent(P,D), anc(A,P). C2
parent(fay,dan). C3
parent(jim,dan). C4
?- anc(A,dan).
A fay
A jim
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - parent(A,dan) C1 2a, 2b
1b anc(A,dan) - parent(P,dan), anc(A,P) C2 fail
2a parent(A,dan) 1a parent(fay,dan) C3 A fay
2b 1a parent(jim,dan) C4 A jim
3a parent(P,dan) 2 1b
4a anc(A,fay) 1b anc(A,fay) - parent(A,fay) C1 5a
4b 1b anc(A,fay) - parent(P,fay), anc(A,P) C2 6a
5a parent(A,fay) 4a fail
6a parent(P,fay) 5 4b
7a anc(A,jim) 1b anc(A,jim) - parent(A,jim) C1 8a
7b 1b anc(A,jim) - parent(P,jim), anc(A,P) C2 9a
8a parent(A,jim) 4a fail
9a parent(P,jim) 5 4b
79
SLG resolution for definite programs

Avoids redundant repeated proofs of same subgoals
ex, calculates each fibonacci number exactly
once
Also avoids left-recursive loops
The table allows detection of goal G that is a
variant of the goal G that called it
Looping is avoided by suspending G and moving to
the following goals of the clause body and then
backpropagating to G the variable bindings
necessary to prove these following goals
When G becomes sufficiently instantiated to no
longer be a variant of G, its proof resumes

80
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
81
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
82
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
83
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
4a parent(P,P) 1a parent(fay,dan) C3 P fay
84
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), anc(P,fay) C1
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
86
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
87
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
88
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
89
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
90
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
91
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
3a anc(A,P) 2a 2a suspend fail
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
92
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
2b 1a anc(A,P) - parent(A,P) C2
3a anc(A,P) 2a 2a suspend fail
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay
4b 1a parent(jim,dan) C4 P jim
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
93
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
2b 1a anc(A,P) - parent(A,P) C2 4a, 4b
3a anc(A,P) 2a 2a suspend fail
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay, P dan
4b 1a parent(jim,dan) C4 P jim, P dan
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
8a parent(A,P) 4a 4a,4b
95
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
2b 1a anc(A,P) - parent(A,P) C2 4a, 4b
3a anc(A,P) 2a 2a suspend fail
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay, P dan
4b 1a parent(jim,dan) C4 P jim, P dan
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
8a parent(A,P) 4a 4a,4b
9a parent(fay,dan) parent(fay,dan) yes
96
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1 4a
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
2b 1a anc(A,P) - parent(A,P) C2 4a, 4b
3a anc(A,P) 2a 2a suspend fail
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay, P dan
4b 1a parent(jim,dan) C4 P jim, P dan
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
8a parent(A,P) 4a 4a,4b
9a parent(fay,dan) 1a parent(fay,dan) yes
97
SLG resolution for definite programs

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

anc(A,D) - anc(A,P), parent(P,D). C1
anc(A,D) - parent(A,D). C2
parent(fay,dan).
C3
parent(jim,dan).
C4
?- anc(A,dan).
A fay
A dan
no
?-

gid goal vg cg instantiated rule rn answers
1a anc(A,dan) anc(A,dan) - anc(A.P), parent(P,dan) C1 4a, 4b
2a anc(A,P) 1a anc(A.P) - anc(A,P), parent(P,P) C1
2b 1a anc(A,P) - parent(A,P) C2 4a, 4b
3a anc(A,P) 2a 2a suspend fail
3b anc(A,fay) 3a anc(A,fay) - anc(A,P), parent(P,fay) C1 fail
3b 3a anc(A,fay) - parent(A,fay) C2 fail
4a parent(P,P) 1a parent(fay,dan) C3 P fay, P dan
4b 1a parent(jim,dan) C4 P jim, P dan
5a anc(A,P) 2a 3a suspend
6a parent(P,fay) 3a fail
7a parent(A,fay) 6a
8a parent(A,P) 4a 4a,4b
9a parent(fay,dan) 1a parent(fay,dan) yes
10a parent(jim,dan) 1a parent(jim,dan) yes
99
HiLog extending Prolog with high-order syntax

Meta-programming facilities of functional
languages
Single language for reasoning and meta-reasoning
Schema browsing in databases

Examples of HiLog atomsthat are not Prolog
atoms
P(a)
p(F(a)
f(a)(X,Y)
p(f(a)(X,Y))
P(F(X))(G(Y))

Prolog atoms
HiLog atoms
arg
arg

Prolog Atom
Prolog Term
predicate
HiLog Atom
HiLog Term

predicate
arg
arg

Function-Free Term
Function-Free Term
Functional Term
Functional Term
functor
functor
Variable
Variable
Symbol
Symbol
100
HiLog example

Prolog program
parent(bob, bill).parent(bill,
mary).supervisor(john, mary).supervisor(mary,
kathy).
anc(A,D) - parent(A,D).anc(A,D) - parent(A,C),
anc(C,D).
boss(B,E) - supervisor(B,E).
boss(B,E) - supervisor(B,D), boss(D,E).
obeys(P,A) - anc(A,P).
obeys(P,A) - boss(A,P).
?- obeys(mary,X).
Xbill Xbob X mary Xjohn no

More concise and extensible HiLog program
parent(bob, bill).parent(bill,
mary).supervisor(john, mary).supervisor(mary,
kathy).
relation(parent).relation(supervisor).
trans(R)(X, Y) - R(X, Y)trans(R)(X, Y) - R(X,
Z), trans(R)(Z,Y).obeys(P)(A) - relation(R),
trans(R)(A,P).
?- obeys(mary)(X).Xbill Xbob X mary
Xjohn no

101
HiLog

High-order syntax ...
... but first-order semantics (and thus
semi-decidable)
Why?
HiLog is merely a very convenient syntactic sugar
since
All HiLog program can be translated into a barely
legible, verbose but semantically equivalent
Prolog program
How?
Axiomatize predicates and functions with single
additional functor apply
Complex terms or variables as functors become
first argument of apply
Arguments of these functors become second to last
arguments of apply

102
HiLog to Prolog translation example

parent(bob, bill).
parent(bill, kathy).
supervisor(john, mary).
supervisor(mary, kathy).
relation(parent).
relation(supervisor).
trans(R)(X, Y) - R(X, Y)
trans(R)(X, Y) - R(X, Z), trans(R)(Z,Y).
obeys(P)(A) - relation(R), trans(R)(A,P)
?- obeys(kathy)(X).Xbill Xbob X mary
Xjohn no

parent(bob, bill).
parent(bill, kathy).
supervisor(john, mary).
supervisor(mary, kathy).
relation(parent).
relation(supervisor).
apply(trans(R),X,Y) - apply(R,X,Y).
apply(trans(R),X,Y) - apply(R,X,Y),
apply(trans(R),Z,Y).
apply(obeys(P,A)) - relation(R),
apply(trans(R),A,P)).
?- apply(obeys(john),X).Xbill Xbob X mary
Xjohn no

103
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)

Prologs 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
Prologs weakness
Numerical programming is imperative, not
declarative
Very inefficient at solving finite domain
combinatorial problem

Logic Programming - PowerPoint PPT Presentation

Logic Programming

Logic Programming Jacques Robin – PowerPoint PPT presentation