The implicit complexity content of Quasi-interpretations G. - - PowerPoint PPT Presentation

The implicit complexity content of Quasi-interpretations

• G. Bonfante, J-Y Marion and J-Y Moyen

Ecole des Mines - INPL - Loria

Appsem – 04/04 – p.1/21

Motivations

Control program resources: time, memory, stack size. Static analysis provides a “resource certificate” A resource certificate can be attached to a program (see PCC) A resource certificate can give hints to run a program satisfying some computational property. Resource controls at runtime slowdown a program and is problematic in case of attack

Appsem – 04/04 – p.2/21

Proof theoretical analysis of computation and complexity

Specification Proof of termination Program synthetesis

Appsem – 04/04 – p.3/21

Proof theoretical analysis of computation and complexity

Specification 1st order functional programs Proof of termination Program synthetesis

Appsem – 04/04 – p.3/21

Proof theoretical analysis of computation and complexity

Specification 1st order functional programs Proof of termination Complexity analysis Program synthetesis

Appsem – 04/04 – p.3/21

Proof theoretical analysis of computation and complexity

Specification 1st order functional programs Proof of termination Complexity analysis Program synthetesis with transformation and guaranted complexity

Appsem – 04/04 – p.3/21

1st order fct prog

Constructors :

• , functions :

, variables :

.

(Terms)

• ☎

f

(Patterns)

(Rules)

f

Appsem – 04/04 – p.4/21

1st order fct prog

Constructors :

• , functions :

, variables :

.

(Terms)

• ☎

f

(Patterns)

(Rules)

f

concat

Nil

• ✆

• concat

Cons

• ✆

Cons

concat

• ✆

Appsem – 04/04 – p.4/21

Termination

Termination orderings : Polynomial interpretation (Lankford)

• Multiset path ordering (MPO) (Dershowitz)

Lexicographic path ordering (LPO)

Appsem – 04/04 – p.5/21

MPO (1/2)

• ✄

is an ordering on

• .

g

• ✌

g

f

g

Appsem – 04/04 – p.6/21

MPO (2/2)

• ✡

iff

• and there is a permutation

s.t.

s.t.

Appsem – 04/04 – p.7/21

MPO (2/2)

• ✡

iff

• and there is a permutation

s.t.

s.t.

A program terminates by MPO if for each rule

• ✙

, we have

• .

Appsem – 04/04 – p.7/21

Extensional characterization of MPO

A MPO function is a function computed by a program which terminates by MPO. The set of MPO functions is exactly the set of primitive recursive functions. (Cichon (90), Hofbauer (92))

Appsem – 04/04 – p.8/21

Intentionnality

The “good” algorithms which computes

• ✁✄✂

in

steps is not definable by primitive recursive schema (Colson (89)). However, it terminates by MPO.

Appsem – 04/04 – p.9/21

Intentionnality

The “good” algorithms which computes

• ✁✄✂

in

steps is not definable by primitive recursive schema (Colson (89)). However, it terminates by MPO. On the other hand, quicksort does not terminate by MPO but the sort function is a PR function. The goal is to capture “good” programs within a resource bound.

Appsem – 04/04 – p.9/21

Quasi-interpretations

(Marion et Moyen, Bonfante,

• ✁

) A quasi-interprétation of

is a function

satisfying:

is bounded by a polynomial.

for all

.

is increasing (not-strictly).

Program with quasi-interpretation if for each rule

• ✙

, we have

• ✄

.

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Characterization of PTIME

The set of functions which are computed by a program

• 1. which terminates by MPO
• 2. which admits a quasi-interpretation

is exactly the set PTIME of functions computable in polynomial time. (Marion et Moyen,

• ✁

) ICAR system implements this resource analysis method. (Moyen,

• ✁

)

Appsem – 04/04 – p.11/21

Characterization of PSPACE

The set of functions which are computed by a program

• 1. which terminates by LPO
• 2. which admits a quasi-interpretation

is exactly the set PSPACE of functions computable in polynomial space. (Bonfante, Marion et Moyen,

• ✁

• )

A upper bound on memory can be produced from termina- tion proof (Amadio et al.

• ✁

• ).

Appsem – 04/04 – p.12/21

Length of the longest common subsequence

Appsem – 04/04 – p.13/21

Length of the longest common subsequence

A B B A B A

Appsem – 04/04 – p.13/21

Length of the longest common subsequence

A B B A B A B A B A

Appsem – 04/04 – p.13/21

Length of the longest common subsequence

A B B A B A B A B A

Appsem – 04/04 – p.13/21

Length of the longest common subsequence

A B B A B A B A B A

Appsem – 04/04 – p.13/21

Length of the longest common subsequence

A B B A B A B A B A

lcs

• ✁✄✂

lcs

• ☎

lcs

• i
• ✁

i

• ✟

lcs

• ✁

lcs

• i
• ✁

j

• ✟

max

• lcs
• ✁

j

• ✟

lcs

• i
• ✁

Appsem – 04/04 – p.13/21

Explicit vs Implicit

lcs

• ✁✄✂

lcs

• ☎

lcs

• i
• ✁

i

• ✟

lcs

• ✁

lcs

• i
• ✁

j

• ✟

max

• lcs
• ✁

j

• ✟

lcs

• i
• ✁

Explicit complexity : runtime is

• ✄
• ✄

Appsem – 04/04 – p.14/21

Explicit vs Implicit

lcs

• ✁✄✂

lcs

• ☎

lcs

• i
• ✁

i

• ✟

lcs

• ✁

lcs

• i
• ✁

j

• ✟

max

• lcs
• ✁

j

• ✟

lcs

• i
• ✁

Explicit complexity : runtime is

• ✄
• ✄

Appsem – 04/04 – p.14/21

Explicit vs Implicit

lcs

• ✁✄✂

lcs

• ☎

lcs

• i
• ✁

i

• ✟

lcs

• ✁

lcs

• i
• ✁

j

• ✟

max

• lcs
• ✁

j

• ✟

lcs

• i
• ✁

Explicit complexity : runtime is

• ✄
• ✄

lcs terminates by MPO and admits a QI :

lcs

• ✁

. Implicit complexity is polynomial.

Appsem – 04/04 – p.14/21

Memoisation

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

Appsem – 04/04 – p.15/21

Memoisation

• ✁

• ✁✂

• ✁✂

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

• ✁✂

• ✁

• ✁✂

• ✁✂

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

• ✁✂

• ✁✂

• ✁

• ✟
• ✁

• ✁✂

• ✁✂

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

• ✁✂

• ✟
• ✁✂

• ✁✂

• ✟
• ✁✂

• ✁✂

• ✁✂

• ✁

• ✟
• ✁

• ✁✂

• ✁✂

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

• ✁✂

• ✟
• ✁✂

• ✁✂

• ✁✂

• ✁

• ✟
• ✁

• ✁✂

• ✁✂

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

• ✁✂

• ✟
• ✁✂

• ✁✂

• ✁✂

• ✁

• ✟
• ✁

• ✁✂

• ✁✂

k-2PDA Cook, Jones (00)

Appsem – 04/04 – p.15/21

Memoisation

• ✁✂

• ✁✂

• ✟
• ✁✂

• ✁✂

• ✁✂

• ✁

• ✟
• ✁

• ✁✂

• ✁✂

k-2PDA Cook, Jones (00) Cache is mintained minimal because of MPO (Marion,

• ✁

).

Appsem – 04/04 – p.15/21

CBV with cache

• ✡

• ✄

• ✂

• ✄

• ✄

f

• ✄

f

• ✄

f

f

• ✄

f

• ✎

• ✎

• ✄

f

f

Appsem – 04/04 – p.16/21

Finding QI

Finding quasi-interpretations is undecidable. NP-hard when considering max-plus quasi-interpretations (Amadio 2003) Open : Finding heuristics to synthesis QI

Appsem – 04/04 – p.17/21

In-place computation

A function

• computes in place if

f

. Compilation without malloc (M. Hofmann,

• ).

A program which admits a linear quasi-interpretation

• ✁

computes in place.

Appsem – 04/04 – p.18/21

Conclusions

Appsem – 04/04 – p.19/21

Fight : Functions vs Algorithms

About functions Calculability Logical characterization of complexity classes Extensional characterization of complexity classes About Algorithms Studying algorithms : Colson, Gurevich, Moschovakis Implicit computational complexity classes Intentional characterization of complexity classes including “good” algorithms (Amadio & al, Jones, Hofmann, Bonfante, Marion and Moyen)

Appsem – 04/04 – p.20/21

On-going research

Source 1st order functional programs Compilation Resource analysis Prf of termination & Quasi-int. Virtual Machine Byte code + Resource Certificate

Appsem – 04/04 – p.21/21