Polynomials over the reals in proofs of termination: from theory to practice Salvador Lucas Dep. de Sistemas Informáticos y Computación Universidad Politécnica

Polynomials over the reals in proofs of termination: from theory to practice

Salvador LucasDep. de Sistemas Informáticos y Computación

Universidad Politécnica de [email protected]

Introduction: termination of TRSs Orderings and termination:

Theorem: A TRS is terminating iff there is a well-founded ordering > such that

t > s

whenever

t s

for all terms t,s.

Problem: infinite number of checkings; automatization?

Introduction: termination of TRSs Algebras and orderings:

Theorem: Algebraic interpretations (A, FA, >A) can be used to induce term orderings:

t > s if and only if [t] >A [s]

Well-foundedness of > is ensured if (A,>A) is a well-founded ordering

Introduction: termination of TRSs Reduction orderings and termination:

Theorem: A TRS is terminating iff there is a well-founded, stable and monotonic ordering > such that

l > r

for each rule

l r

of the TRS.

Finite number of checkings: automatization!

Introduction: termination of TRSs Algebras and reduction orderings:


Stability of > is automatically achieved by definition Monotonicity of > is guaranteed if

x >A y

implies

fA(x1,...,xi-1,x,...,xk) >A fA (x1,..., xi-1, y,...,xk)

for all x1,...,xk,x,y in A and all i between 1 and k

Introduction: termination of TRSs Example: [Der95, Example 1] (TPDB version)

\(x,x) -> e .(x,\(x,y)) -> y/(x,x) -> e .(/(y,x),x) -> y.(e,x) -> x \(x,.(x,y)) -> y.(x,e) -> x /(.(y,x),x) -> y\(e,x) -> x /(x,\(y,x)) -> y/(x,e) -> x \(/(x,y),x) -> y

[\](X1,X2) = X1 + X2 + 2[e] = 1[/](X1,X2) = X1 + X2 + 2[.](X1,X2) = X1 + X2

Introduction: termination of TRSs Example: [Der95, Example 1] (TPDB version)

\(x,x) -> e .(x,\(x,y)) -> y/(x,x) -> e .(/(y,x),x) -> y.(e,x) -> x \(x,.(x,y)) -> y.(x,e) -> x /(.(y,x),x) -> y\(e,x) -> x /(x,\(y,x)) -> y/(x,e) -> x \(/(x,y),x) -> y

[\](X1,X2) = X1 + X2 + 2[e] = 1[/](X1,X2) = X1 + X2 + 2[.](X1,X2) = X1 + X2

[.(e,x)] = 1+x > x = [x]

Introduction: termination of TRSs Dependency pairs and termination (I):

A reduction pair (≥ , ) consists of a weakly monotonic and stable quasi-ordering ≥ and a stable and well-founded ordering such that

≥ ° is a subset of or ° ≥ is a subset of

Given a rule f(l1,…,lm) r, a dependency pair is a pair F(l1,…,lm),G(s1,…,sn )), where g(s1,…,sn ) is an operation rooted subterm of r, and F, G are new symbols (called ‘tuple symbols’) associated to f and g

Introduction: termination of TRSs Dependency pairs and termination (II):

Theorem: A TRS is terminating iff there is a reduction pair (≥ , ) such that

l ≥ r for each rule l rt s for each dependency pair

(t,s)of the TRS.

Less restrictive checkings: better automatization!

Example: [AG01, Example 1]

minus(x,0) -> xminus(s(x),s(y)) -> minus(x,y)quot(0,s(y)) -> 0quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))

[minus](X1,X2) = X1 [quot](X1,X2) = X1[0] = 0 [MINUS](X1,X2) = X1[QUOT](X1,X2) = X1 [s](X) = X + 1

MINUS(s(x),s(y)) -> MINUS(x,y)QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))QUOT(s(x),s(y)) -> MINUS(x,y)

Introduction: termination of TRSs

Example: [AG01, Example 1]

minus(x,0) -> xminus(s(x),s(y)) -> minus(x,y)quot(0,s(y)) -> 0quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))

[minus](X1,X2) = X1 [quot](X1,X2) = X1[0] = 0 [MINUS](X1,X2) = X1[QUOT](X1,X2) = X1 [s](X) = X + 1

MINUS(s(x),s(y)) -> MINUS(x,y)QUOT(s(x),s(y)) -> QUOT(minus(x,y),s(y))QUOT(s(x),s(y)) -> MINUS(x,y)

Introduction: termination of TRSs

[minus(x,0)] = x x = [x]

[MINUS(s(x),s(y))] = x + 1 > x = [MINUS(x,y)]

Introduction: termination of CSR -reduction orderings and termination:

Theorem: A TRS is -terminating iff there is a well-founded, stable and -monotonic ordering > such that

l > r

for each rule

l r

of the TRS.

Introduction: termination of CSR Algebras and -reduction orderings:


Stability of > is automatically achieved by definition -monotonicity of > is guaranteed if

x >A y

implies

fA(x1,...,xi-1,x,...,xk) >A fA (x1,..., xi-1, y,...,xk)

for all x1,...,xk,x,y in A and i in (f)

Example: [Zan97, Example 1]

g(x) -> h(x)c -> dh(d) -> g(c)

[g](x) = x2 - 3x + 4 [c] = 1[h](x) = x2 - 3x + 4 [d] = 0

Introduction: termination of CSR

(g)={}(h)={}

Example: [Zan97, Example 1]

g(x) -> h(x)c -> dh(d) -> g(c)

[g](x) = x2 - 3x + 4 [c] = 1[h](x) = x2 - 3x + 4 [d] = 0


(g)={}(h)={}

[h(d)] = 0-0+4 = 4 > 3 = 1-3+4 = [g(c)]

Termination and monotonicity Termination of rewriting: ™f, (f)={1,…,ar(f)} Termination of rewriting with DPs: ™f, (f) =

Termination of CSR: f, (f) {1,…, ar(f)} Top-termination: ™ f, can(f) (f) Termination of innermost rewriting: inn

Example: [GM04, Introduction]

nats -> adx(zeros)zeros -> cons(0,zeros)incr(cons(X,Y)) -> cons(s(X),incr(Y))adx(cons(X,Y)) -> incr(cons(X,adx(Y)))hd(cons(X,Y)) -> Xtl(cons(X,Y)) -> Y

[nats] = 5 [incr](x) = x + 1 [cons](x,y) = x + 1/2 y [adx](x) = x + 3 [s](x) = 0 [tl](x) = 2x + 1 [zeros] = 1 [hd](x) = x + 1 [0] = 0


(0)=ˆ(s)=ˆ(nil)=ˆ(cons)=ˆ(adx)={1}(incr)={1}(hd)={1}(tl)={1}(nats)={1}

Introduction: termination of CSR Algebras and -reduction orderings:

Well-foundedness of > gets lost: e.g., (0, >) is not well-founded:

1 > 1/2 > 1/4 > ··· > 1/n > ··· > ··· ∞ ·· > 0

On the other hand, it is possible to prove that the termination of this TRS cannot be proved by a polynomial interpretation over the naturals.

Summary Specifying non-monotonic orderings Term (quasi-) orderings induced from

algebras over the reals Polynomial algebras over the reals Automatic generation of polynomial

algebras over the reals Related work Conclusions and future work

Specifying non-monotonic orderings

Specifying non-monotonic orderings Replacement map [Luc98]:

A mapping which associates a subset (f) of {1,…,ar(f))} to each function symbol f

Used to specify -monotonic orderings >: for all arguments i in (f),

t > s implies f(t1,...,ti-1,t,...,tk) > f(t1,..., ti-1, s,...,tk)

Well-founded orderings from algebras

Algebras over the reals Algebraic interpretations (A, FA, >A) can be

used to induce (-)reduction orderings:

t > s if and only if [t] >A [s]

We consider algebras whose carrier A is a subset of and whose functions are real functions

Algebras over the reals We guarantee well-foundedness as follows:

let (A, FA) be an algebra over the real numbers and be a number. Define:

t > s if and only if [t] - [s] ≥

This idea has already been used by Geser (1991) and Hofbauer (2001) among others

Algebras over the reals

Monotonicity (or -monotonicity) has tobe proved apart…

Theorem: Let A m for some real number m and >0. Then > is a well-founded and stable ordering on terms.

Example: The following TRS

can be proved terminating by using the 0-bounded algebra (0,{[f],[g]}), where

[f](x) = x + 1/2

[g](x) = x


f(f(x)) f(g(f(x)))

Proposition: Let T be a finite set of pairs of terms. If for each (t,s) T there is t,s > 0 such that [t]-[s] ≥ t,s, then

min({t,s | (t,s) T})

issuch that t > s for all (t,s) T.

Avoiding the choice of

Avoiding the choice of Example: Since

[f(f(x))] = x + 3/2

[f(g(f(x)))] = x + 1/2

we have [f(f(x))] - [f(g(f(x)))] = 1

Then, f(f(x)) >1 f(g(f(x))) and, since >1 is monotonic (ad-hoc proof!), termination of R is proved.

[f](x)

x


Monotone!

[s] [t]

[f(t)]

[f(s)]

<

<

Theorem: If ∂f/∂xi ≥ 1 for all symbols f and replacing arguments i, then > is -monotone for all >0.

Criterion for monotonicity

Example: The following TRS

can be proved -terminating by using the well-founded, stable and monotonic ordering >1 induced by the 0-bounded algebra (1,{[f],[g]}), where

[f](x,y,z) = x + xy-1 + zy-1 + z[a] = 2[b] = 1[c] = 3


c a f(a,b,x) f(x,x,x)

c b

(a)=ˆ(b)=ˆ(c)=ˆ(f) ={1,3}

Example: Since

[f(a,b,x)] = 2x + 4

[f(x,x,x)] = 2x + 2

we have [f(a,b,x)] - [f(x,x,x)] = 2and [c] - [a] = 1; [c] - [b] = 2

Thus, the rules are compatible with >1

Furthermore, ∂[f]/∂x = 1 + y-1 ≥ 1 for all y ≥ 1

∂[f]/∂z = 1 + y-1 ≥ 1 for all y ≥ 1

Then, >1 is monotonic and the -termination of R is proved.


Polynomial algebras

Proposition: Let ≥ 0 be such that A and P be a polynomial without negative coefficients.

Then, = P(,…, ) is the minimum of P in An


Example: The polynomial with negative coefficients

P(x,y) = (xy - 1)2 + x2

is positive in 2 but it does not reach any minimum value!


Corollary: Let T be a finite set of pairs of terms. Let ≥ 0 be such that A and (A, FA) be a polynomial algebra.

If for all (t,s) T, Pt,s = [t] - [s] has no negative coefficient and t,s = P(,…, ) > 0,then

min({t,s | (t,s) T}) satisfies t > s for all (t,s) T.



When comparing terms t and s (from rules or dependency pairs) we will require that Pt,s be

a positive polynomial without negative coefficients

Proposition: There is a TRS which can be proved -terminating by using a polynomial interpretation over the reals, whereas it cannot be proved -terminating by using a polynomial interpretation over the naturals

Generating non-monotonic orderings

zeros -> 0:zerostl(x:y) -> y

(zeros)=ˆ(0)=ˆ(:) ={1}(tl)={1}

[zeros] = 0 x[:]y = x + 1/2 y [0] = 0 [tl](x) = 2x + 1

Automatic proofs of termination

Generic framework: -terminationWe let A =

1. Restrictions due to the algebraicity of the interpretation:

[f](x1,…,xk) ≥ 0 for all symbols f and x1,…,xk ≥ 0 2. Restrictions due to the monotonicity specification :

[f] / xi ≥ 1 for all symbols f and i (f)

3. Compatibility of > with the rules:

Pl,r = [l] - [r] > 0

and Pl,r has no negative coefficient

Implicit !

Generic framework: DP terminationWe let A =


[f](x1,…,xk) ≥ 0 for all symbols f and x1,…,xk ≥ 0 2. Restrictions due to the weak monotonicity of :

[f] / xi ≥ 0 for all symbols f and i {1,..,ar(f)}

3. Compatibility of with the rules:

Pl,r = [l] - [r] 0

4. Compatibility of > with the dependency pairs:

Pt,s = [t] - [s] > 0

and Pt,s has no negative coefficient

Implicit !

Example: [GM04, Introduction]

nats -> adx(zeros)zeros -> cons(0,zeros)incr(cons(X,Y)) -> cons(s(X),incr(Y))adx(cons(X,Y)) -> incr(cons(X,adx(Y)))hd(cons(X,Y)) -> Xtl(cons(X,Y)) -> Y

Proving termination of CSR

(0)=ˆ(s)=ˆ(nil)=ˆ(cons)=ˆ(adx)={1}(incr)={1}(hd)={1}(tl)={1}(nats)={1}

Proving termination of CSR We start with undeterminate polynomial interpretations Example: (linear interpretations)

[nats] = a0

[adx](x) = b1x + b0

[zeros] = c0

[cons](x,y) = d10x + d01y + d00

[0] = e0

[incr](x) = f1x + f0

[s](x) = g1x + g0

[hd](x) = h1x + h0

[tl](x) = i1x + i0

Coefficients can bearbitrary real numbers

Proving termination of CSR The previous results are used to put restrictions over the

undeterminate coefficients: let A =


af,i ≥ 0 for all symbols and coefficients

3. Restrictions due to the monotonicity specification given by :

f / xi = af,i ≥ 1 if i is in (f)

2. Compatibility with the rules:

Pl,r = [l] - [r] > 0


Proving termination of CSR Example (cont’d): Algebraicity: only non-negative coefficients are allowed Monotonicity restrictions in the example:

b1 ≥ 1 f1 ≥ 1 h1 ≥ 1 i1 ≥ 1

Compatibility with the rule l rzeros cons(0,zeros)

[zeros ] - [cons(0,zeros)] > 0 iff

c0 - (d10e0 + d01c0 + d00) > 0 iff

c0 - d10e0 - d01c0 - d00 > 0

since Pl,r only has a constant coefficient, this restriction implies the absence of negative coefficients, as required.

Proving termination of CSR Consider the rule l r

incr(x:y) s(x):incr(y)

[incr(x:y)] - [s(x):incr(y)] > 0 iff

(f1d10 - d10g1)x + (f1d01 - d01f1)y + (f1 d00 + f0 - d10 g0 - d01 f0 - d00) > 0

in order to guarantee that Pl,r only has non-negative coefficients, we further impose:

f1d10 - d10g1 ≥ 0

f1d01 - d01f1 ≥ 0

f1 d00 + f0 - d10 g0 - d01 f0 - d00 > 0

Proving termination of CSR Example: (cont’d) The collected set of constraints

can be sent to a constraint solving system (over the reals) like CON’FLEX to obtain, e.g.:

[nats] = 1.740 [incr](x) = 1.000x + 0.251 [cons](x,y) = 0.431x + 0.431y + 0.288[adx](x) = 1.100x + 0.658 [s](x) = 0.300x + 0.224[tl](x) = 2.322x + 0.300 [zeros] = 0.932 [hd](x) = 2.322x + 0.300 [0] = 0.300

Implicit : 0.046275

More… Dependency pairs: similar scheme; no significative

differences Non-linear interpretations: simple and simple mixed

interpretations discussed Polynomials with negative coefficients: new family of

intrpretations: 2-simple mixed Implemented: the techniques are the basis of MU-TERM:

http://www.dsic.upv.es/~slucas/csr/termination/muterm

Related work

Polynomials over the naturals

Introduced by Lankford (1979) All coefficients are natural numbers The interpretation domain N is obviously

well-founded

Polynomials over the naturals Recently, Contejean, Marché, Tomás and Urbain have described

how to automatically compute such polynomials [CMTU04]: let A = N

1. Restrictions due to the algebraicity of the interpretation: Only natural coefficients are allowed

3. Restrictions due to (full) monotonicity:For each symbol f and argument i,

there is a linear monomial af,i xi where af,i ≥ 12. Compatibility with the rules:

Pl,r = [l] - [r] > 0


Polynomials over the naturals

Our framework subsumes this!

They (CiME, AProVE, TTT designers) just need to solve the obtained

constraints over the reals!!

Polynomials over the reals

Proposed by Dershowitz (1979) All coefficients are real numbers The interpretation domain is not well-founded,

so a subterm property is additionally required:

[f](x1,…, xi,…,xk) > xi

for all symbols f, arguments i, and x1,…,xk

Polynomials over the reals Giesl has investigated how to automatically compute

such polynomials [Gie95]: let A = Compatibility with the rules:

[l] - [r] > 0 Weak subterm

[f](x1,…, xi,…,xk) ≥ xi Weak monotonicity

[f] / xi ≥ 0 for all symbols f and i {1,..,ar(f)} Non-negative constants:

[c] ≥ 0 for all constant symbol c

Polynomials over the reals Then, Giesl applies two differentiation rules to

obtain a final set of constraints over the underterminate coefficients

According to Hong and Jakuš, all polynomials involved in the initial constraints above must have non-negative coefficients

Again, our framework subsumes this withoutrequiring any subterm property!

There are TRSs whose termination can be directly proved by means of a polynomial interpretation over the reals (or rationals) but it cannot be proved by using polynomials over the naturals!

RECENTLY SOLVED OLD OPEN PROBLEM !!!

Polynomials over the reals

Conclusions / future work

Conclusions

New technique for defining reduction pairs by using algebras over the reals

Useful for proving termination of rewriting by using non-monotonic polynomial interpretations together with the Dependency Pairs approach

Implemented in MU-TERM

Future work

Investigate other families of interpretations over the reals which can be well-suited for automatization (e.g., polynomial fractions)

Implement in MU-TERM: More efficient use of rational coefficients

OPEN PROBLEMS

Efficient representation of real coefficients Analysis techniques to detect when to use them Use of bounded intervals Use of transcendental real numbers Connections/use with/of Algebraic Geometry …

Salvador LucasDep. de Sistemas Informáticos y Computación

Universidad Politécnica de [email protected]

Polynomials over the reals in proofs of termination: from theory to practice

Documents

Polynomials over the reals in proofs of termination: from theory to practice Salvador Lucas Dep. de Sistemas Informáticos y Computación Universidad Politécnica