Optimisation of Declarative Programs Lecture 1: Introduction

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Optimisation of Declarative Programs Lecture 1: Introduction

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Title: Optimisation of Declarative Programs Lecture 1: Introduction

1
Optimisation of Declarative Programs Lecture 1
Introduction

Michael Leuschel
Declarative Systems Software Engineering
Dept. Electronics Computer Science
University of Southampton
http//

www.ecs.soton.ac.uk/mal
Morten Heine Sørensen DIKU, Copenhagen
2
Lecture 1
1. Overview 2. PE 1st steps 3. Why PE 4. Issues
in PE

Explain what these lectures are about
Program Optimisation in general
Partial Evaluation in particular
Program Transformation, Analysis, and
Specialisation
Explain why you should attend the lectures
Why are the topics interesting ?
Why are they useful ?
Overview of the whole course

3
Part 1Program Optimisation and Partial
Evaluation A first overview
1. Overview 2. PE 1st steps 3. Why PE 4. Issues
in PE
4
(Automatic) Program Optimisation

When
Source-to-source
during compilation
at link-time
at run-time
Why
Make existing programs faster (10 ? 500 ? ?
...)
Enable safer, high-level programming style (??)
(Program verification,)

5
Program Optimisation II

What
constant propagation, inlining , dead-code
elimination, eliminate redundant computations,
change of algorithm or data-structures,
Similar to highly optimising compilers
much more aggressive,
(much) greater speedups and
much more difficult to control
How ?

6
Program Transformation Unfold/fold
Final Program Pn
Initial Program P0
Program P1
...
Under some conditions Same semantics Same
termination properties Better performance
7
Program Specialisation
Drawing Program P

What
Specialise a program for aparticular application
domain
How
Partial evaluation
Program transformation
Type specialisation
Slicing

P
8
Overview
Partial Evaluation
Program Transformation
Program Specialisation
Program Optimisation
9
Program Analysis

What
Find out interesting properties of your programs
Values produced, dependencies, usage,
termination, ...
Why
Verification, debugging
Type checking
Check assertions
Optimisation
Guide compiler (compile time gc, )
Guide source-to-source optimiser

10
Program Analysis II

How
Rices theorem all non-trivial properties are
undecidable !
? approximations or sufficient conditions
Techniques
Ad hoc
Abstract Interpretation Cousot77
Type Inference

11
Abstract Interpretation
Concrete domain
-2 -1 0 1 2 3 ...

Principle
Abstract domain
Abstract operations
safe approximation of concrete operations
Result
Correct
Termination can be guaranteed
? Decidable approximation of undecidable
properties

N N N N 0 N ...
12
Part 2A closer look at partial evaluation
First Steps
1. Overview 2. PE 1st steps 3. Why PE 4. Issues
in PE
13
A closer look at PE

Partial Evaluation versus Full Evaluation
Why PE
Issues in PE Correctness, Precision, Termination
Self-Application and Compilation

14
Full Evaluation

Full input
Computes full output

function power(b,e) is if e 0 then 1
else bpower(b,e-1)
15
Partial Evaluation

Only part of the input
? produce part of the output ?
power(?,2)

? evaluate as much as you can
? produce a specialized program
power_2(?)

16
PE A first attempt
function power(b,e) is if e 0 then 1
else bpower(b,e-1)

Small (functional) language
constants (N), variables (a,b,c,)
arithmetic expressions ,/,,-,
if-then-else, function definitions
Basic Operations
Evaluate arithmetic operation if all arguments
are known
23 ? 5 54 ? false x(23) ? x5
Replace if-then-else by then-part (resp.
else-part) if test-part is known to be true
(resp. false)

17
Example power

power(?,2)

function power(b,2) is if e 0 then 1
else bpower(b,e-1)
function power(b,2) is if 2 0 then 1
else bpower(b,e-1)
function power(b,2) is if false then 1
else bpower(b,2-1)
Residual code
function power_2(b) is bpower_1(b)

power(?,1)

function power(b,1) is if e 0 then 1
else bpower(b,e-1)
function power(b,1) is if false then 1
else bpower(b,1-1)
function power_1(b) is bpower_0(b)
18
Example power (contd)

power(?,0)

function power(b,0) is if e 0 then 1
else bpower(b,e-1)
function power(b,0) is if 0 0 then 1
else bpower(b,e-1)
function power(b,0) is if 0 0 then 1
else bpower(b,e-1)
Residual code
function power_0(b) is 1
19
Example power (contd)

Residual code

function power_1(b) is bpower_0(b)
function power_2(b) is bpower_1(b)
function power_0(b) is 1

What we really, really want

function power_2(b) is bb
20
Extra Operation Unfolding
evaluating the function call operation

Replace call by definition

function power(b,2) is if 2 0 then 1
else bpower(b,1)
function power(b,2) is if 2 0 then 1
else b (if 1 0 then 1 else
bpower(b,0))
function power(b,2) is if 2 0 then 1
else b (if 1 0 then 1 else b (if 0
0 then 1 else bpower(b,-1)))
Residual code
function power_2(b) is bb1
SQR function
21
PE Not always beneficial

power(3,?)
Residual code

function power(3,e) is if e 0 then 1
else 3power(3,e-1)
function power(3,e) is if e 0 then 1
else bpower(b,e-1)
function power_3(e) is if e 0 then 1
else 3power_3(e-1)
22
Part 3Why Partial Evaluation (and Program
Optimisation)?
1. Overview 2. PE 1st steps 3. Why PE 4. Issues
in PE
23
Constant Propagation

Static values in programs

function my_program(x) is y 2
power(x,3)
captures inlining and more
function my_program(x) is y 2 x x x

24
Abstract Datatypes / Modules

Get rid of inherent overhead by PE

function my_program(x) is if not
empty_tree(x) then y get_value(x)
get rid of redundant run-time tests
function my_program(x) is if x ! null then
if x ! null then y x-gtval
25
Higher-Order/Reusing Generic Code

Get rid of overhead ? incite usage

function my_program(x,y,z) is r
reduce(,1,map(inc,x,y,z))
function my_program(x,y,z) is r
(x1)(y1)(z1)
26
Higher-Order II

Can generate new functions/procedures

function my_program(x) is r
reduce(,1,map(inc,x))
function my_program(x) is r
red_map_1(x) function red_map_1(x) is if
xnil then return 1 else return x-gthead
red_map_1(x-gttail)
27
Staged Input

Input does not arrive all at once

5
2
2.1
my_program (x , y , z)
42
28
Examples of staged input

Ray tracing calculate_view(Scene,Lights,Viewpoint
)Interpretation interpreter(ObjectProgram,Call)
prove_theorem(FOL-Theory,Theorem) check_integrit
y(Db_rules,Update) schedule_crews(Rules,Facts)
Speedups
2 you get your money back
10 quite typical for interpretation overhead
100-500 (and even 8) possible

29
Ray Tracing
Static
30
Why go beyond partial evaluation ?

Improve algorithms (not necessarily PS)
Tupling
Deforestation
Superlinear speedups
Exponential ? linear algorithm
Non-executable ? executable programs
Software Verification, Inversion, Model Checking

31
Tupling
Corresponds to loop-fusion
32
Deforestation
33
Part 4Issues in Partial Evaluation and Program
Optimisation
1. Overview 2. PE 1st steps 3. Why PE 4. Issues
in PE
Efficiency/Precision
Correctness
Termination
Self-application
34
Correctness

Language
Power/Elegance unfolding LP easy, FP ok, C
aargh
Modularity global variables, pointers
Semantics
Purely logical
Somewhat operational (termination,)
Purely operational
Informal/compiler

admissive
restrictive
35
Programming Language for Course

Lectures use mainly LP and FP
Dont distract from essential issues
Nice programming paradigm
A lot of room for speedups
Techniques can be useful for other languages, but
Correctness will be more difficult to establish
Extra analyses might be required (aliasing, flow
analysis, )

36
Efficiency/Precision
x 2yp(x) y5 ? p(7)

Unfold enough but not too much
Enough to propagate partial information
But still ensure termination
Binding-time Analysis (for offline systems)
Which expressions will be definitely known
Which statements can be executed
Allow enough polyvariance but not too much
Code explosion problem
Characteristic trees

p(7) p(9) p(11) p(13) ...
37
Termination

Who cares
PE should terminate when the program does
PE should always terminate
" within reasonable time bounds

State of the art
38
Self-Application

Principle
Write a partial evaluator (program specialiser,)
which can specialise itself
Why
Ensures non-trivial partial evaluator
Ensures non-trivial language
Optimise specialisation process
Automatic compiler generation !

39
Compiling by PE
Call2
Call1
Call3
Result1
Interpreter
Object Program P
Result2
Result3
Call2
Call1
Call3
PE
Result1
P-Interpreter
Result2
Result3
40
Compiler Generation by PE
Object Program R
static
Object Program Q
Object Program R
Object Program P
Object Program Q
Interpreter I
Object Program P
PE
PE
I-PE
Compiler !
Useful for Lge Extensions, Debugging, DSL,...
P-Interpreter
Q-Interpreter
R-Interpreter
41
Cogen Generation by PE
Interpreter R
static
Interpreter Q
Interpreter R
Interpreter P
Interpreter Q
PE
Interpreter P
PE
PE
PE-PE
Compiler Generator !
3rd Futamura Projection
P-Compiler
Q-Compiler
R-Compiler
42
Part 5Overview of the course (remainder)
43
Overview of the Course Lecture 2

Partial Evaluation of Logic ProgramsFirst Steps
Topics
A short (re-)introduction to Logic Programming
How to do partial evaluation of logic
programs(called partial deduction) first steps
Outcome
Enough knowledge of LP to follow the rest
Concrete idea of PE for one particular language

44
Overview of the Course Lecture 3

(Online) Partial Deduction Foundations,
Algorithms and Experiments
Topics
Correctness criterion and results
Control Issues Local vs. Global Online vs.
Offline
Local control solutions Termination
Global control Ch. trees, WQO's at the global
level
Relevance for other languages and applications
Full demo of Ecce system
Outcome
How to control program specialisers (analysers,
...)
How to prove them correct

45
Overview of the Course Lecture 4(?)

Extending Partial Deduction
Topics
Unfold/Fold vs PD
Going from PD to Conjunctive PD
Control issues, Demo of Ecce
Abstract Interpretation vs Conj. PD (and
Unfold/Fold)
Integrating Abstract Interpretation
Applications ( Demo) Infinite Model Checking
Outcome
How to do very advanced/fancy optimisations

46
Overview of the CourseOther Lectures

By Morten Heine Sørensen
Context
Functional Programming Languages
Topics
Supercompilation
Unfold/fold
...

47
Summary of Lecture 1What to know for the exam

Terminology
Program Optimisation, Transformation, Analysis,
Specialisation, PE
Basics of PE and Optimisation in general
Uses of PE and Program Optimisation
Common compiler optimisations
Enabling high-level programming
Staged input, optimising existing code
Self-application and compiler generation

48
Optimisation of Declarative Programs Lecture 2

Michael Leuschel
Declarative Systems Software Engineering
Dept. Electronics Computer Science
University of Southampton
http//

www.ecs.soton.ac.uk/mal
49
Overview of Lecture 2
1. LP Refresher 2. PE of LP 1st steps

Recap of LP
First steps in PE of LP
Why is PE non-trivial ?
Generating Code Resultants
Correctness
Independence, Non-triviality, Closedness
Results
Filtering
Small demo of the ecce system

50
Part 1Brief Refresher onLogic Programming and
Prolog
1. LP Refresher 2. PE of LP 1st steps
51
Brief History of Logic Programming

Invented in 1972
Colmerauer (implementation, Prolog
PROgrammation en LOGique)
Kowalski (theory)
Insight subset of First-Order Logic
efficient, procedural interpretation based on
resolution (Robinson 1965 Herbrand 1931)
Program theory
Computation logical inference

52
Features of Logic Programming

Declarative Language
State what is to be computed, not how
Uniform Language to express and reason about
Programs, specifications
Databases, queries,
Simple, clear semantics
Reasoning about correctness is possible
Enable powerful optimisations

53
Logical Foundation SLD-Resolution

Selection-rule driven Linear Resolution for
Definite Clauses
Selection-rule
In a denial selects a literal to be resolved
Linear Resolution
Derive new denial, forget old denial
Definite Clauses
Limited to (Definite) Program Clauses
(Normal clause negation allowed in body)

54
2. Higher-Order Logic
? R. R(tom) ? human(tom)
variables over objects and relations
1. First-Order Logic - constants, functions,
variables (tom mother(tom) X) -
relations ( human(.) married(. , .) ) -
quantifiers (? ? )
human(sokrates)
? X. human(X) ? mortal(X)
variables over objects
0. Propositional Logic - propositions (basic
units either true or false) - logical
connectives (? ? ? ?)
rains
rains ? carry_umbrella
p ?? p
rains ?? rains
no variables
55
FO Resolution Principle Example
1. select literals
1.) ?X umbrella(X) ? ?rains(X) 2.)
rains(london) 1.) umbrella(london) ?
?rains(london) 2.) rains(london) 12)
umbrella(london)
2. no renaming required
3. mgu ? X / london
4. apply ?
5. resolution
56
Terminology

SLD-Derivation for P ? ? G
Sequence of goals and mgus, at every step
Apply selection rule on current goal (very first
goal G)
Unify selected literal with the head of a program
clause
Derive the next goal by resolution
SLD-Refutation for P ? ? G
SLD-Derivation ending in ?
Computed answer for P ? ? G
Composition of mgus of a SLD-refutation,
restricted to variables in G

57
SLDDerivation
G

? knows_logic(Z)
? good_student(X) ? teacher(Y,X) ?
logician(Y)
? teacher(Y,tom) ? logician(Y)
? logician(peter)
?

1) knows_logic(X) ? good_student(X) ?
teacher(Y,X) ? logician(Y)
2) good_student(tom)
3) logician(peter)
4) teacher(peter,tom)

Z/X
X/tom
Y/peter
Computed answer Z/tom ? P J knows_logic(tom)
??

58
Backtracking

? knows_logic(Z)
? good_student(X) ? teacher(Y,X) ?
logician(Y)
? teacher(Y,jane) ? logician(Y)
? teacher(Y,tom) ? logician(Y)
? logician(peter)
?

1) knows_logic(X) ? good_student(X) ?
teacher(Y,X) ? logician(Y)
2) good_student(jane)
3) good_student(tom)
4) logician(peter)
5) teacher(peter,tom)

?
Backtracking required !
Try to resolve with another clause
59
SLD-Trees

? knows_logic(Z)
? good_student(X) ? teacher(Y,X) ?
logician(Y)
? teacher(Y,tom) ? logician(Y)
? logician(peter)
?

? teacher(Y,jane) ? logician(Y)

?
Prolog explores this tree Depth-First, always
selects leftmost literal
60
SLD-Trees II

Branches of SLD-tree SLD-derivations
Types of branches
refutation (successful)
finitely failed
infinite

61
Some info about Prolog

Depth-first traversal of SLD-tree
LD-resolution
Always select leftmost literal in a denial

62
Prologs Syntax

Facts
human(socrates).
Rules (clauses)
carry_umbrella - rains,no_hat.
Variables
mortal(X) - human(X).
exists_human - human(Any).
Queries
?- mortal(Z).

? (implication)
? (conjunction)
existential variable (?)
universal variable (?)
63
Prologs Datastructures
lower-case

Constants
human(socrates).
Integers, Reals
fib(0,1).
pi(3.141).
(Compound) Terms
head_of_list(cons(X,T),X).
tail_of_list(cons(X,T),T).
?- head_of_list(cons(1,cons(2,nil)),R).

functor
Prolog actually uses . (dot) for lists
64
Lists in Prolog

Empty list
(syntactic sugar for nil )
Non-Empty list with first element H and tail T
HT (syntactic sugar for .(H,T) )
Fixed-length list
a (syntactic sugar for a
.(a,nil) )
a,b,c (syntactic sugar for a b c
)

65
Part 2Partial Evaluation of Logic
ProgramsFirst Steps
1. LP Refresher 2. PE of LP 1st steps
66
What is Full Evaluation in LP

In our context full evaluation constructing
complete SLD-tree for a goal
? every branch either
successful, failed or infinite
SLD can handle variables !!
? can handle partial data-structures
(not the case for imperative or functional
programming)
? Unfolding ordinary SLD-resolution !
? Partial evaluation full evaluation ??
? Partial evaluation trivial ??

67
Example
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
app (a,B,C)
C/aC
app (,B,C)
C/B
?
app(a,B,aB). app_a(B,aB).
68
ExampleRuntimeQuery
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
app (a,b,C)
C/aC
app (,b,C)
app (a,b,C)
app_a (b,C)
C/b
C/a,b
C/a,b
?
?
?
app(a,B,aB). app_a(B,aB).
c.a.s. C/a,b
69
Example II
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
app (A,B, a)
A/aA
A/,B/a
app (A,B,)
?
A/
?
app(,a,a). app(a,,a). app_a(,a).
app_a(a,).
70
Example III
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).

BUT complete SLD-tree usually infinite !!

app (A,a,B)
A/HA,B/HB
A/,B/a
app (A,a,B)
?
A/HA,B/HB
A/,B/a
app (A,a,B)
?
...
71
Partial Deduction

Basic Principle
Instead of building one complete SLD-treeBuild
a finite number of finite SLD- trees !
SLD-trees can be incomplete
4 types of derivations in SLD-trees
Successful, failed, infinite
Incomplete no literal selected

72
Example (revisited)
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
app_a(,a). app_a(HA,HB) -
app_a(A,B).
73
Example (revisited)
app (A,a,B)
app (A,a,B)
?
Infinite complete tree is covered
app (A,a,B)
?
app (A,a,B)
?
...
74
Main issues in PD ( PE in general)

How to construct the specialised code ?
When to stop unfolding (building SLD-tree) ?
Construct SLD-trees for which goals ?
When is the result correct ?
? Will be addressed in this Lecture !

75
Generating CodeResultants
G0

? A1,,Ai,,An
? G1
? G2
...
? Gk

Resultant ofSLD-derivation
is the formula
G0?1...?k ? Gk
if n1 Horn clause !

?0
?1
?k
76
Resultants Example
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
app(,a,a). app(HA,a,HB) -
app(A,a,B).
77
Formalisation of Partial Deduction

Given a set S A1,,An of atoms
Build finite, incomplete SLD-trees for each ? Ai
For every non-failing branch
generate 1 specialised clause bycomputing the
resultants
When is this correct ?

78
Correctness 1 Non-trivial trees

Trivial tree
Resultant app (A,a,B) ? app (A,a,B)
loops!
Correctness conditionPerform at least one
derivation step

? app (A,a,B)
79
Correctness 2 Closedness
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
? app (aA,B,C)
C/aC
? app (A,B,C)
app(aA,B,aC) - app(A,B,C).
Not an instance of app (aA,B,C)
Not C/a,b !!
80
Closedness Condition

To avoid uncovered calls at runtime
All predicate calls
in the bodies of specialised clauses( atoms in
leaves of SLD-trees !)
in the calls to the specialised program
must be instances of at least one of the atoms in
S A1,,An

81
Correctness 3 Independence
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
? app (aA,B,C)
C/aC
? app (A,B,C)
app(aA,B,aC) - app(A,B,C). app(,C,C).
app(HA,B,HC) - app(A,B,C).
Closedness Ok
82
Extra answers
app(aA,B,aC) - app(A,B,C). app(,C,C).
app(HA,B,HC) - app(A,B,C).
? app (X,Y,Z)
X/a,Z/aZ
Z/XZ
? app (,Y,Z)
? app (,Y,Z)
Extra answer ! more instantiated !!
?
?
X a, ZaY
ZXY
83
Independence Condition

To avoid extra and/or more instantiated answers
require that no two atoms in S A1,,An have a
common instance
S app (aA,B,C), app(A,B,C)
common instances app(a,X,Y), app(a,b,,X),
How to ensure independence ?
Just Renaming !

84
Renaming No Extra answers
app_a(aA,B,aC) - app(A,B,C). app(,C,C)
. app(HA,B,HC) - app(A,B,C).
app(aA,B,aC) - app(A,B,C). app(,C,C).
app(HA,B,HC) - app(A,B,C).
? app (X,Y,Z)
Z/XZ
? app (,Y,Z)
No extra answer !
?
ZXY
85
Soundness Completeness

P obtained by partial deduction from P
If non-triviality, closedness, independence
conditions are satisfied
P ? ?G has an SLD-refutation with c.a.s. ? iff
P ? ?Ghas
P ? ?G has a finitely failed SLD-tree iffP ?
?Ghas
Lloyd, Shepherdson J. Logic Progr. 1991

86
A more detailed example
? map (inv,L,R)
map(P,,). map(P,HT,PHPT) -
C..P,H,PH, call(C),map(P,T,PT). inv(0,1).
inv(1,0).
map(P,,). map(P,HT,PHPT) -
C..P,H,PH, call(C),map(P,T,PT). inv(0,1).
inv(1,0).
Overhead removed 2? faster
map(inv,,). map(inv,0L,1R) -
map(inv,L,R). map(inv,1L,0R) -
map(inv,L,R).
87
Filtering
? map (inv,L,R)
map(P,,). map(P,HT,PHPT) -
C..P,H,PH, call(C),map(P,T,PT). inv(0,1).
inv(1,0).
map(inv,,). map(inv,0L,1R) -
map(inv,L,R). map(inv,1L,0R) -
map(inv,L,R).
map_1(,). map_1(0L,1R) -
map_1(L,R). map_1(1L,0R) -
map_1(L,R).
88
RenamingFiltering Correctness

Correctness results carry over
Benkerimi,Hill J. Logic Comp 3(5), 1993
No worry about Independence
Non-triviality also easy to ensure
? only worry that remains (for correctness)
Closedness

89
Small Demo (ecce system)

Append
Higher-Order
Match
Interpreter

90
Summary of part 2

Partial evaluation in LP
Generating Code
Resultants, Renaming, Filtering
Correctness conditions
non-triviality
closedness
independence
Soundness Completeness

91
Optimisation of Declarative Programs Lecture 3

Michael Leuschel
Declarative Systems Software Engineering
Dept. Electronics Computer Science
University of Southampton
http//

www.ecs.soton.ac.uk/mal
92
Part 1Controlling Partial DeductionLocal
Control Termination
1. Local Control Termination 2. Global
Control Abstraction
93
Issues in Control
A
A1 A2 A3 A4 ...

Correctness
ensure closedness? add uncovered leaves to set
A
Termination
build finite SLD-trees
build only a finite number of them !
Precision
unfold sufficiently to propagate information
have a precise enough set A

global
local
CONFLICTS!
94
ControlLocal vs Global
A
A1 A2 A3 A4 ...

Local Control
Decide upon the individual SLD-trees
Influences the code generated for each Ai
? unfolding rule
given a program P and goal G returns finite,
possibly incomplete SLD-tree
Global control
decide which atoms are in A
? abstraction operator

95
Generic Algorithm
Input program P and goal G Output specialised
program P Initialise i0, A0 atoms in G
repeat Ai1 Ai for each a ? Ai do
Ai1 Ai1 ? leaves(unfold(a)) end for
Ai1 abstract(Ai1) until Ai1
Ai compute P via resultantsrenaming
96
Overview Lecture 3

Local Control
Determinacy
Ensuring Termination
Wfos, Wqos
Global Control
abstraction
most specific generalisation
characteristic trees

97
(Local) Termination

Who cares
PE should terminate when the program does
PE should always terminate
" within reasonable time bounds

State of the art
Well-founded orders Well-quasi orders
98
Determinate Unfolding

Unfold atoms which match only 1 clause(only
exception first unfolding step)
avoids code explosion
no duplication of work
good heuristic for pre-computing
too conservative on its own
termination
always ? ? No
when program terminates ??

99
Duplication
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
?app(A,B,C),app (_I,C,a,a)
a2(A,B,a,a) - app(A,B,a,a). a2(A,B,a) -
app(A,B,a). a2(A,B,) - app(A,B,).
100
Duplication 2
a2(A,B,a,a) - app(A,B,a,a). a2(A,B,a) -
app(A,B,a). a2(A,B,) - app(A,B,).
?a2(a,,C)

Solutions
Determinacy!
Follow runtime selection rule

101
Order of Solutions
t(A,B) - p(A,C),p(C,B). p(a,b). p(b,a).
?t(A,C)

Same solutions
Determinacy!
Follow runtime selection rule

?p(A,B),p(B,C)
t(b,b). t(a,a).
?
?
102
Lookahead

Also unfold if
atom matches more than 1 clause
only 1 clause remains after further unfolding
less conservative
still ensures no duplication, explosion

fail
103
Lookahead example
? s4(5,R)
? sel (4,5,R)
sel(P,,). sel(P,HT,HPT) - PltH,
sel(P,T,PT). sel(P,HT,PT) - PgtH,
sel(P,T,PT). s4(L,PL) - sel(4,L,PL).

R/5R
? 4gt5, sel(4,,R)
? 4lt5, sel(4,,R)
s4(5,5).
104
Example
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
?app (A,a,A)
At runtime app(,a,) terminates !!!
?
105
Well-founded orders

lt (strict) partial order (transitive,
anti-reflexive, anti-symmetric) with no ?
descending chainss1 gt s2 gt s3 gt
To ensure termination
define lt on expressions/goals
unfold only if sk1 lt sk
Example termsize norm (number of function and
constant symbols)

106
Example
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
?app (a,bC,A,B)
. 5
Ok
. 3
Ok
. 1
Stop
. 1
. 0
?
107
Other wfos

linear norms, semi-linear norms
f(t1,,tn) cf cf,1 t1 cf,n tn
X cV
lexicographic ordering
.1, , .k first use .1, if equal use
.2...
multiset ordering
t1,,tn lt s1,,sk, t2,,tn if all si lt t1
recursive path ordering
...

108
More general-than/Instance-of

AgtB if A strictly more general than B
p(X)
p(s(X))
p(s(s(X)))
not a wfo

AgtB if A strict instance of B
p(s(s(X)))
p(s(X))
p(X)
X
is a wfo (Huet80)

109
Refining wfos
rev(,L,L). rev(HX,A,R) - rev(X,HA,R).
. 4
?rev (a,bC,,R)
. 6
? measure just arg 1
. 2
. 6
Ok
. 0
Stop
. 0
?
110
Well-quasi orders

? quasi order (reflexive and transitive)
withevery infinite sequence s1,s2,s3, we can
findiltj such that si ? sj
To ensure termination
define ? on expressions/goals
unfold only if for no iltk1 si ? sk1

?
111
Alternative Definitions of wqos

All quasi orders ? which contain ? (a?b ? a?b)
are wfos
can be used to generate wfos
Every set has a finite generating set (a?b ? a
generates b)
For every infinite sequence s1,s2, we can find
an infinite subsequence r1,r2, such that ri ?
ri1
Wfo with finitely many incomparable elements
...

112
Examples of wqos

A?B if A and B have same top-level functor
p(s(X))
q(s(s(X)))
r(s(s(s(X))))
p(s(s(s(s(X)))))
...
Is a wqo for a finite alphabet

A?B if not A lt B, where gt is a wfo
p(s(s(X)))
r(s((X))
q(X))
p(s(s(s(s(X)))))
Is a wqo (always) with same power as gt

?
Termsize not decreasing
113
Instance-of Relation

AgtB if A strict instance of B is a wfo
A?B if A instance of B wqo ??
p(s(X))
p(s(s(X)))
...
p(s(X))
p(s(X))

?
p(0) p(s(0)) p(s(s(0))) p(s(s(s(0)))) p(s(s(s(s(0)
)))) ...
?
? Every wqo is a wfo but not the other way around
114
Comparison WFOs and WQOs

No ? descending chains s1 gt s2 gt s3 gt
Ensure that sk1 lt sk
More efficient(only previous element)
Can be used statically
(admissible sequences can be composed)

Every ? sequencesi ? sj with iltj
Ensure that not si ? sk1
uncomparable elements
More powerful (SAS98)
? more powerful than all monotonic and
simplification orderings

115
Homeomorphic Embedding ?

Diving s ? f(t1,,tn) if ?i s ? ti
Coupling f(s1,,sn) ? f(t1,,tn) if ?i si ?
ti

n0
f(a)
g(f(f(a)),a)
g
f
a
f
a
f
a
116
Admissible transitions

rev(a,bT,,R)
solve(rev(a,bT,,R),0)
t(solve(rev(a,bT,,R),0),)
path(a,b,)
path(b,a,)
t(solve(path(a,b,),0),)

rev(aT,a,R)
solve(rev(aT,a,R),s(0))
t(solve(rev(aT,a,R),s(0)),rev))
path(b,a,a)
path(a,b,b)
t(solve(path(b,a,a),s(0)),path))

117
Higman-Kruskal Theorem (1952/60)

? is a WQO (over a finite alphabet)
Infinite alphabets associative operators
f(s1,,sn) ? g(t1,,tm) if f ? g and ?i
si ? tji with 1?j1ltj2ltltjn?m and(q,p(b)) ?
and(r,q,p(f(b)),s)
Variables X ? Y more refined solutions
possible (DSSE-TR-98-11)

118
An Example for ?

rev(a,bX,)
rev(bX,b)
rev(X,b,a)
rev(Y,H,b,a)

eval(rev(a,bX,),0)
eval(rev(bX,b),s(0))
eval(rev(X,b,a),s(s(0)))
eval(rev(Y,H,b,a),s(s(s(0))))

? Data consumption ? Termination ? Stops at the
right time ? Deals with encodings
119
Summary of part 1

Local vs Global Control
Generic Algorithm
Local control
Determinacy
lookahead
code duplication
Wfos
Wqos
-homeomorphic embedding

120
Part 2 Controlling Partial DeductionGlobal
control and abstraction
1. Local Control Termination 2. Global
Control Abstraction
121
Most Specific Generalisation (msg)

A is more general than B iff ?? BA?
for every B,C there exists a most specific
generalisation (msg) M
M more general than B and C
if A more general then B and C then A is also
more general than M
there exists an algorithm for computing the msg
(anti-unification, least general generalisation)

122
MSG examples

a a
a b
X Y
p(a,b) p(a,c)
p(a,a) p(b,b)
q(X,a,a) q(a,a,X)

a
X
X
p(a,X)
p(X,X)
q(X,a,Y)

123
First abstraction operator

Benkerimi,Lloyd90 abstract two atoms in Ai1
by their msg if they have a common instance
will also ensureindependence(no renaming
required)
Termination ?

Input program P and goal G Output specialised
program P Initialise i0, A0 atoms in G
repeat Ai1 Ai for each a ? Ai do
Ai1 Ai1 ? leaves(unfold(a)) end for
Ai1 abstract(Ai1) until Ai1
Ai compute P via resultantsrenaming
124
Rev-acc
rev(,L,L). rev(HX,A,R) - rev(X,HA,R).
?rev (L,,R)
unfold
unfold
?
rev(L,,R)
unfold
rev(L,H,R)
rev(L,H,H,R)
rev(L,H,H,H,R)
125
First terminating abstraction

If more than one atom with same predicate in
Ai1 replace them by their msg
rev (L,,R), rev (L,H,R) ? rev(L,A,R)
Ensures termination
strict instance-of relation wfo
either fixpoint or strictly smaller atom
But loss of precision !
rev (a,bL,,R), rev (L,b,a,R) ?
rev(L,A,R) ? No specialisation !!

126
Global trees

Also use wfos MartensGallagher95 or wqos for
the global control SørensenGlück95,
LeuschelMartens96
Arrange atoms in Ai1 as a tree
register which atoms descend from which(more
precise spotting of non-termination)
Only if termination is endangered apply msg on
the offending atoms
use wfo or wqo on each branch

127
Rev-acc
rev(,L,L). rev(HX,A,R) - rev(X,HA,R).
?rev (L,,R)
unfold
unfold
rev(L,A,R)
rev(L,,R)
DANGER
rev(L,H,R)
128
More precise spotting
app(,L,L). app(HX,Y,HZ) -
app(X,Y,Z). t(X,L) - app(X,2,L). t(X,L) -
app(X,2,3,5,L).
Global Tree
Set
DANGER ABSTRACT
t(X,L)
t(X,L)
No Danger
app(X,2,L)
app(X,2,3,5,L)
129
Examining Syntactic Structure

p(a,B,C) ? p(A,a,C)
abstract yes or no ???

130
pappend
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).

Do not abstract

app_a(,a). app_a(HA,HC) -
app_a(A,C).
app_a(B,aB).
131
pcompos
compos(HTX,HTY,H). compos(XTX,YTY,E)
- compos(TX,TY,E).

Do abstract

compos_a(a,a).
compos_a(a,a).
132
Characteristic Trees

Register structure of the SLD-trees
position of selected literals
clauses that have been resolved with
GallagherBruynooghe91 Only abstract atoms if
same characteristic tree

133
Characteristic Trees II

Abstract

Do not abstract

134
Rev-acc revisited (Blackboard)
rev(,L,L). rev(HX,A,R) - rev(X,HA,R).
?rev (a,bC,,R)
135
Path Example (Blackboard)
path(A,B,L) - arc(A,B). path(A,B,L)
- not(member(A,L)), arc(A,C), path(C,B,aL).
member(X,X_). member(X,_L) -
member(X,L). arc(a,a).
?path (a,b,)
?path (a,b,a)
136
Characteristic Trees Problems

Leuschel95
characteristic trees not preserved upon
generalisation
Solution ecological PD or constrained PD
LeuschelMartens96
non-termination can occur for natural examples
Solution apply homeomorphic embedding o n
characteristic trees
? ECCE partial deduction system (Full DEMO)

137
Summary of part 2

Most Specific Generalisation
Global trees
reusing wfos, wqos at the global level
Characteristic trees
advantage over purely syntactic structure

138
Summary of Lectures 2-3

Foundations of partial deduction
resultants, correctness criteria results
Controlling partial deduction
generic algorithm
local control
determinacy, wfos, wqos
global control
msg, global trees, characteristic trees

139
Optimisation of Declarative ProgramsLecture 4

Michael Leuschel
Declarative Systems Software Engineering
Dept. Electronics Computer Science
University of Southampton
http//

www.ecs.soton.ac.uk/mal
140
Part 1A Limitation of Partial Deduction
141
SLD-Trees

? knows_logic(Z)
? good_student(X) ? teacher(Y,X) ?
logician(Y)
? teacher(Y,tom) ? logician(Y)
? logician(peter)
?

? teacher(Y,jane) ? logician(Y)

?
Prolog explores this tree Depth-First, always
selects leftmost literal
142
Partial Deduction

Basic Principle
Instead of building one complete SLD-treeBuild
a finite number of finite SLD- trees !
SLD-trees can be incomplete
4 types of derivations in SLD-trees
Successful, failed, infinite
Incomplete no literal selected

143
Formalisation of Partial Deduction

Given a set S A1,,An of atoms
Build finite, incomplete SLD-trees for each ? Ai
For every non-failing branch
generate 1 specialised clause bycomputing the
resultants
Why only atoms ?
Resultants Horn clauses
Is this a limitation ??

144
ControlLocal vs Global
A
A1 A2 A3 A4 ...

Local Control
Decide upon the individual SLD-trees
Influences the code generated for each Ai
? unfolding rule
given a program P and goal G returns finite,
possibly incomplete SLD-tree
Global control
decide which atoms are in A
? abstraction operator

145
Generic Algorithm
Input program P and goal G Output specialised
program P Initialise i0, A0 atoms in G
repeat Ai1 Ai for each a ? Ai do
Ai1 Ai1 ? leaves(unfold(a)) end for
Ai1 abstract(Ai1) until Ai1
Ai compute P via resultantsrenaming
146
Limitation 1 Side-ways Information

t(X) - p(X), q(X).
p(a).
q(b).

Original program !
147
Limitation 1 Solution ?

t(X) - p(X), q(X).
p(a).
q(b).
Just unfold deeper

t(X) - fail.
? q(a)
fail
148
Limitation 1 Solution ?

t(X) - p(X), q(X).
p(a).
p(X) - p(X).
q(b).
We cannot fullyunfold !!

? q(a)
? p(Y),q(Y)
fail
149
Limitation 1

Side-ways information passing
solve(Call,Res),manipulate(Res)
Full unfolding not possible
termination (e.g. ? of Res)
code explosion
efficiency (work duplication)
Ramifications
difficult to automatically specialise
interpreters(aggressive unfolding required)
no tupling, no deforestation

150
Limitation 1 Solutions

Pass least upper bound around
For a query ?A,B
Take msg of all (partial) solutions of A
Apply msg to B
Better
Specialise conjunctions instead of atoms
? Conjunctive Partial Deduction

151
Part 2Conjunctive Partial Deduction
152
Conjunctive Partial Deduction

Given a set S C1,,Cn of atoms
Build finite, incomplete SLD-trees for each ? Ci
For every non-failing branch
generate 1 specialised formula Ci ?L
bycomputing the resultants
To get Horn clauses
Rename conjunctions into atoms !
? Assign every Ci an atom with the same
variables and each with a different predicate name

153
Renaming Function
p(Y),q(Y)
pq(Y)

p(f(X)),q(b) ? p(X),q(X)
p(f(X)),q(b) ? p(X),r(X),q(X)

pq(f(X),b) ? pq(X) pq(f(X),b) ? pq(X), r(X)
or pq(f(X),b) ? r(X), pq(X)
? Resolve ambiguities contiguous if no
reordering allowed
154
Soundness Completeness

P obtained by conjunctive partial deduction from
P
If non-triviality, closedness wrt renaming
function
P ? ?G has an SLD-refutation with c.a.s. ? iff
P ? ?Ghas
If in addition fairness (or weak fairness)
P ? ?G has a finitely failed SLD-tree iffP ?
?Ghas
? see Leuschel et al JICSLP96

155
Maxlength Example (Blackboard)
max_length(X,M,L) - max(X,M),len(X,L). max(X,M)
- max(X,0,M). max(,M,M). max(HT,N,M) -
HltN,max(T,N,M). max(HT,N,M) -
HgtN,max(T,H,M). len(,0). len(HT,L) -
len(T,TL),L is TL1.
?maxlen (X,M,L)
?max(X,N,M),len(X,L)
ml(X,N,M,L)
156
Double-Append Example (blackboard)
app(,L,L). app(HX,Y,HZ) - app(X,Y,Z).
? app (A,B,I), app(I,C,Res)
da(,B,B,C,Res) - app(B,C,Res). da(HA,B,HI
,C,HR) - da(A,B,I,C,R).
da(,B,C,Res) - app(B,C,Res). da(HA,B,C,HR
) - da(A,B,C,R).
157
Controlling CPD

Local Control
becomes easier
we have to be less aggressive
we can concentrate on efficiency of the code
Global Control
gets more complicated
handle conjunctions
structure abstraction (msg) no longer sufficient !

158
Global Control of CPD

p(0,0).
p(f(X,Y),R) - p(X,R),p(Y,R).
Specialise p(Z,R)
We need splitting
How to detect infinite branches
homeomorphic embedding on conjunctions
How to split
most specific embedded conjunction

159
Homeomorphic Embedding on Conj.

Define and/m ? and/n
f(s1,,sn) ? g(t1,,tm) if f ? g
and ?i si ? tji with 1?j1ltj2ltltjn?m
and(q,p(b)) ? and(r,q,p(f(b)),s)

160
Most Specific Embedded Subconj.
p(X) ? q(f(X))
t(Z) ? p(s(Z)) ? q(a) ? q(f(f(Z)))
t(Z), q(a), p(s(Z)) ?
q(f(f(Z)))
161
Generic Algorithm

Plilp96, Lopstr96, JLP99

Input program P and goal G Output specialised
program P Initialise i0, A0 G repeat
Ai1 Ai for each c ? Ai do Ai1
Ai1 ? leaves(unfold(c)) end for Ai1
abstract(Ai1) until Ai1 Ai compute P
via resultantsrenaming
Changesover PD
162
Ecce demo of conjunctive partial deduction

Max-length
Double append
depth, det. Unf. with PD and CPD
rotate-prune

163
Comparing with FP

Conjunctions
can represent nested function calls
(supercompilation Turchin, deforestation
Wadler)g(X,RG),f(RG,Res) f(g(X))
can represent tuples (tupling Chin)g(X,ResG),f(
X,ResF) ?f(X),g(X)?
We can do both deforestation and tupling !
No such method in FP
Semantics for infinite derivations (WFS) in LP
safe removal of infinite loops, etc
applications interpreters,model checking,...

164
Comparing with FP II

Functionality has to be inferred in LP
fib(N,R1),fib(N,R2) vs fib(N)fib(N)
Dataflow has to be analysed
rot(Tree,IntTree),prune(IntTree,Result)
rot(L,Tl),rot(R,Tr),prune(Tl,Rl),prune(Tr,Rr)
vs
prune(rot(Tree)) tree( prune(rot(L)), I,
prune(rot(R)) )
Failure possible
reordering problematic ! fail,loop ? loop,fail

165
Part 3Another Limitation of Partial Deduction
(and CPD)
166
Limitation 2 Global Success Information
? p(Y)

p(a).
p(X) - p(X).
We get sameprogram back
The fact that in all successful derivations Xa
is not explicit !

?
? p(Y)
167
Limitation 2 Failure example

t(X) - p(X), q(X).
p(a).
p(X) - p(X).
q(b).
Specialised program
t(X) - pq(X).
pq(X) - pq(X).
But even better
t(X) -fail.

? q(a)
? p(Y),q(Y)
fail
168
Append-last Example (blackboard)
app(,L,L). app(HX,Y,HZ) -
app(X,Y,Z). last(X,X). last(HT,X) -
last(T,X).
? app (A,a,I), last(I,X)
al(,a,a). al(HA,HI,X) - al(A,I,X).
al(,a,a). al(HA,HI,a) - al(A,I,a).
169
Limitation 2 Ramifications
Fully known value
?
Known binding
Binding Lost !
Partially knownDatastructure
Environment for interpreter
170
Part 4Combining Abstract Interpretation with
(Conjunctive) Partial Deduction
171
More Specific Version (Msv)