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Forward Closure and the Finite Variant Property Christopher Bouchard 1 Kimberly A. Gero 1 Christopher Lynch 2 Paliath Narendran 1 1 University at AlbanySUNY, Albany, NY, USA 2 Clarkson University, Potsdam, NY, USA Frontiers of Combining Systems

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Forward Closure and the Finite Variant Property

Christopher Bouchard1 Kimberly A. Gero1 Christopher Lynch2 Paliath Narendran1

1University at Albany—SUNY, Albany, NY, USA 2Clarkson University, Potsdam, NY, USA

Frontiers of Combining Systems 2013

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Motivation

Consider the following two theories: E1: f (g(u, x), g(v, y)) ≈ g(f (u, v), f (x, y)) E2: f (f (x, y), f (x, y)) ≈ f (x, y)

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Motivation

Consider the following two theories: E1: f (g(u, x), g(v, y)) ≈ g(f (u, v), f (x, y))

Undecidable unification problem∗

E2: f (f (x, y), f (x, y)) ≈ f (x, y)

Decidable unification problem Can be shown by forward closure

∗ S. Anantharaman, et al.

“Unification modulo Synchronous Distributivity.” 2012.

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Section 1 Introduction

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Terms

Purely syntactic Composed of. . .

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Terms

Purely syntactic Composed of. . . Constants: t1 = a

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Terms

Purely syntactic Composed of. . . Constants: t1 = a Variables: t2 = x

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Terms

Purely syntactic Composed of. . . Constants: t1 = a Variables: t2 = x Function Symbols: t3 = g(a, f (x))

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Terms

Purely syntactic Composed of. . . Constants: t1 = a Variables: t2 = x Function Symbols: t3 = g(a, f (x)) t3 = g a

f

x

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Terms

Purely syntactic Composed of. . . Constants: t1 = a Variables: t2 = x Function Symbols: t3 = g(a, f (x)) t3 = g a

f

x

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Terms

Purely syntactic Composed of. . . Constants: t1 = a Variables: t2 = x Function Symbols: t3 = g(a, f (x)) t3 = t3

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Rewriting

Rewrite Rule: t1 → t2 Rewrite System: Set of rewrite rules Example (Associativity) f (x, f (y, z)) − → f (f (x, y), z)

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Rewriting

Rewrite Rule: t1 → t2 Rewrite System: Set of rewrite rules Example (Associativity) f (x, f (y, z)) − → f (f (x, y), z)

f

x

f

y z − →

f f

x y z

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Convergence

Confluence: s t1 t2

∗ ∗ 6

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Convergence

Confluence: s t1 t2 u

∗ ∗ ∗ ∗ 6

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Convergence

Confluence: s t1 t2 u

∗ ∗ ∗ ∗

Termination: No infinite descending chain t0 → t1 → t2 → · · ·

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Convergence

Confluence: s t1 t2 u

∗ ∗ ∗ ∗

Termination: No infinite descending chain t0 → t1 → t2 → · · · Confluence + Termination = Convergence

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Equational Unification

Unification modulo a set of axioms E Given a set of equations EQ = { s1

?

= t1, . . . , sn

?

= tn } Substitution σ is an E-unifier of EQ iff: σ(s1) ≈E σ(t1) ∧ · · · ∧ σ(sn) ≈E σ(tn)

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Section 2 Motivation

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Motivation

Equational unification has lots of applications:

Automated reasoning Programming languages (e.g., Maude) Protocol analysis (e.g., Maude-NPA)

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Motivation

Equational unification has lots of applications:

Automated reasoning Programming languages (e.g., Maude) Protocol analysis (e.g., Maude-NPA)

But equational unification is undecidable in general

Even when restricted to “nice” theories

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Motivation

How to identify decidable cases?

∗ C. Lynch and B. Morawska. “Basic Syntactic Mutation.” 2002. † H. Comon-Lundh and S. Delaune. “The finite variant property: How to get

rid of some algebraic properties.” 2005.

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Motivation

How to identify decidable cases? Two important syntactic approaches:

Basic Syntactic Mutation∗ The Finite Variant Property†

∗ C. Lynch and B. Morawska. “Basic Syntactic Mutation.” 2002. † H. Comon-Lundh and S. Delaune. “The finite variant property: How to get

rid of some algebraic properties.” 2005.

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Motivation

How to identify decidable cases? Two important syntactic approaches:

Basic Syntactic Mutation∗ The Finite Variant Property†

Forward closure unifies these approaches.

∗ C. Lynch and B. Morawska. “Basic Syntactic Mutation.” 2002. † H. Comon-Lundh and S. Delaune. “The finite variant property: How to get

rid of some algebraic properties.” 2005.

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Section 3 Forward Closure

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Forward Closure

Extension of work by Hermann on chain properties∗ Compress rewriting steps New rules capture chains of original rules

∗ M. Hermann. “Chain Properties of Rule Closures.” 1990. 12

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Overlap

Overlap two rules and get a new rule General idea: If t1

ǫ

− − − →

ρ1

t2

p

− − − →

ρ2

t3 then t1

ǫ

− − − − − − − →

ρ1 p ρ2

t3

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Overlap

To compute ρ1 p ρ2 for p ∈ FPos(r1): ρ1 : l1 − → r1 ρ2 : l2 − → r2

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Overlap

To compute ρ1 p ρ2 for p ∈ FPos(r1): ρ1 : l1 − →

p

r1|p

ρ2 : l2 − → r2

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Overlap

θ = mgu(

r1|p

?

= l2 )

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Overlap

ρ1 p ρ2 : θ(l1) − → θ(r1[r2]p)

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Overlap

ρ1 p ρ2 : θ(l1) − →

p

θ(r2)

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Overlap

Example ρ1 : f (g(u1, x1), g(v1, y1)) → g(f (u1, v1), f (x1, y1)) ρ2 : f (g(u2, x2), g(v2, y2)) → g(f (u2, v2), f (x2, y2))

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Overlap

Example ρ1 : f (g(u1, x1), g(v1, y1)) → g(f (u1, v1), f (x1, y1)) ρ2 : f (g(u2, x2), g(v2, y2)) → g(f (u2, v2), f (x2, y2)) ρ1 1 ρ2 :

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Overlap

Example ρ1 : f (g(u1, x1), g(v1, y1)) → g(f (u1, v1), f (x1, y1)) ρ2 : f (g(u2, x2), g(v2, y2)) → g(f (u2, v2), f (x2, y2)) θ: {u1 → g(u2, x2), v1 → g(v2, y2)} ρ1 1 ρ2 :

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Overlap

Example ρ1 : f (g(u1, x1), g(v1, y1)) → g(f (u1, v1), f (x1, y1)) ρ2 : f (g(u2, x2), g(v2, y2)) → g(f (u2, v2), f (x2, y2)) θ: {u1 → g(u2, x2), v1 → g(v2, y2)} ρ1 1 ρ2 : f (g(g(u2, x2), x1), g(g(v2, y2), y1)) → g(g(f (u2, v2), f (x2, y2)), f (x1, y1))

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Overlap

Example ρ1 : f (g(u1, x1), g(v1, y1)) → g(f (u1, v1), f (x1, y1)) ρ2 : f (g(u2, x2), g(v2, y2)) → g(f (u2, v2), f (x2, y2)) θ: {u1 → g(u2, x2), v1 → g(v2, y2)} ρ1 1 ρ2 : f (g(g(u2, x2), x1), g(g(v2, y2), y1)) → g(g(f (u2, v2), f (x2, y2)), f (x1, y1))

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Overlap

Example ρ1 : f (g(u1, x1), g(v1, y1)) → g(f (u1, v1), f (x1, y1)) ρ2 : f (g(u2, x2), g(v2, y2)) → g(f (u2, v2), f (x2, y2)) θ: {u1 → g(u2, x2), v1 → g(v2, y2)} ρ1 1 ρ2 : f (g(g(u2, x2), x1), g(g(v2, y2), y1)) → g(g(f (u2, v2), f (x2, y2)), f (x1, y1))

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Redundancy

Not all overlaps are added to the forward closure. A rule l → r is redundant in a rewrite system R iff:

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Redundancy

Not all overlaps are added to the forward closure. A rule l → r is redundant in a rewrite system R iff:

l → r is an instance of a more general rule in R, or

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Redundancy

Not all overlaps are added to the forward closure. A rule l → r is redundant in a rewrite system R iff:

l → r is an instance of a more general rule in R, or Every ground instance of l ≈ r can be proven by smaller ground instances of rules in R

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Redundancy

Not all overlaps are added to the forward closure. A rule l → r is redundant in a rewrite system R iff:

l → r is an instance of a more general rule in R, or Every ground instance of l ≈ r can be proven by smaller ground instances of rules in R

l r t

R ∗ R ∗ 18

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Redundancy

Not all overlaps are added to the forward closure. A rule l → r is redundant in a rewrite system R iff:

l → r is an instance of a more general rule in R, or Every ground instance of l ≈ r can be proven by smaller ground instances of rules in R

l r t

R ∗ R ∗ 18

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Computing Forward Closure

Start with a rewrite system R

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Computing Forward Closure

Start with a rewrite system R Overlap each rule in R with each rule in R

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Computing Forward Closure

Start with a rewrite system R Overlap each rule in R with each rule in R Throw out redundant rules

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Computing Forward Closure

Start with a rewrite system R Overlap each rule in R with each rule in R Throw out redundant rules Call this FC 1(R)

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Computing Forward Closure

Start with a rewrite system FC k(R) Overlap each rule in FC k(R) with each rule in R Throw out redundant rules Call this FC k+1(R)

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Computing Forward Closure

Finally, FC(R) =

k≥0

FC k(R)

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Computing Forward Closure

Finally, FC(R) =

k≥0

FC k(R) If FC k(R) = FC k+1(R) for some k, FC(R) is finite Otherwise, FC(R) is infinite

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Forward Closure

A term t is an innermost redex of R if it can only be rewritten by R at the root. Key Idea: In FC(R), every innermost redex of R can be rewritten to its normal form in one step.

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Section 4 Equivalence of FC and the FVP

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Forward Closure and the Finite Variant Property

We show that a system has a finite forward closure if and only if it has the finite variant property.

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Finite Variant Property

The variants of t are all pairs (t′, θ) such that:

θ is a normalized substitution θ(t) →! t′

Variants capture the idea of rewriting to normal form Finite Variant Property: Every term has a finite set of most general variants

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Equivalence of FC and the FVP

FVP FC BP IR-BP

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Boundedness Property

Bound on the lengths of rewrite chains For each term t there is a bound #(t) such that (θ↓)(t)

≤ #(t)

− − − − − − → θ(t)↓

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Equivalence of FC and the FVP

FVP FC BP IR-BP

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IR-Boundedness Property

Bound on the lengths of rewrite chains from the root There is a global bound n such that, if a term t is an innermost redex, then t

≤n

− − − → t↓.

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Equivalence of FC and the FVP

FVP FC BP IR-BP

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Boundedness ⇒ IR-Boundedness

t − → t↓ Suppose t is an innermost redex

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Boundedness ⇒ IR-Boundedness

f (t1, . . . , tn) − → t↓ Suppose t is an innermost redex Then t = f (t1, . . . , tn), where t1, . . . , tn are normalized

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Boundedness ⇒ IR-Boundedness

θ(f (x1, . . . , xn)) − → t↓ Suppose t is an innermost redex Then t = θ(f (x1, . . . , xn)) Where θ = {x1 → t1, . . . , xn → tn} is normalized

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Boundedness ⇒ IR-Boundedness

θ(tf ) − → t↓ Suppose t is an innermost redex Then t = θ(tf ) Where θ = {x1 → t1, . . . , xn → tn} is normalized And tf = f (x1, . . . , xn)

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Boundedness ⇒ IR-Boundedness

θ(tf )

≤ #(tf )

− − − − − − → t↓ Suppose t is an innermost redex Then t = θ(tf ) Where θ = {x1 → t1, . . . , xn → tn} is normalized And tf = f (x1, . . . , xn)

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Boundedness ⇒ IR-Boundedness

θ(tf )

≤n

− − − → t↓ Suppose t is an innermost redex Then t = θ(tf ) Where θ = {x1 → t1, . . . , xn → tn} is normalized And tf = f (x1, . . . , xn) Let n = max{#(tf ) | f ∈ Σ}

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Boundedness ⇒ IR-Boundedness

t

≤n

− − − → t↓ Suppose t is an innermost redex Then t = θ(tf ) Where θ = {x1 → t1, . . . , xn → tn} is normalized And tf = f (x1, . . . , xn) Let n = max{#(tf ) | f ∈ Σ}

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) =

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n + n

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n + n + n

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n + n + n + n

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n + n + n + n + · · ·

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n + n + n + n + · · · + n

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IR-Boundedness ⇒ Boundedness

t

≤ #(t)

− − − − − − → t↓ #(t) = n · |FPos(t)|

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IR-Boundedness ⇒ Boundedness

(θ↓)(t)

≤ #(t)

− − − − − − → θ(t)↓ #(t) = n · |FPos(t)|

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IR-Boundedness ⇒ Boundedness

(θ↓)(t)

≤ #(t)

− − − − − − → θ(t)↓ #(t) = n · |FPos(t)|

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Equivalence of FC and the FVP

FVP FC BP IR-BP

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Equivalence of FC and the FVP

FVP FC BP IR-BP

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Section 5 Undecidability of FC

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Undecidability of Finiteness of Forward Closure

Reduction from the Uniform Mortality Problem for deterministic Turing machines

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Uniform Mortality Problem

· · · 1 0 1 1 0 1 0 · · · q Given a deterministic Turing machine Does every configuration halt in k or fewer steps (for some k)? Undecidable∗

∗ G.G. Hillebrand, et al. “Undecidable Boundedness Problems for Datalog Pro-

grams.” 1995.

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Undecidability of Finiteness of Forward Closure

Reduction from the Uniform Mortality Problem for deterministic Turing machines Start with a Turing machine M Create a rewrite system RM FC(RM) is finite iff M is uniformly mortal

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Section 6 Modularity of FC

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Modularity

Given rewrite systems R1 and R2, when does the following condition hold? |FC(R1)| + |FC(R2)| < ∞ = ⇒ |FC(R1 ∪ R2)| < ∞ We consider conditions on the signatures of R1 and R2.

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Modularity for Disjoint Systems

If R1 and R2 are rewrite systems with disjoint signatures Then FC(R1) ∪ FC(R2) = FC(R1 ∪ R2).

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Shared Constants

If R1 and R2 share constants Then FC(R1 ∪ R2) may be infinite even if FC(R1) and FC(R2) are finite. Example R1 := {f (a, h(x)) → h(f (b, x))} R2 := {b → a} f (a, h(x)) → h(f (a, x)) ∈ FC(R1 ∪ R2)

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Shared Constants

If R1 and R2 share constants Then FC(R1 ∪ R2) may be infinite even if FC(R1) and FC(R2) are finite. Example R1 := {f (a, h(x)) → h(f (b, x))} R2 := {b → a} f (a, h(h(x))) → h(h(f (b, x))) ∈ FC(R1 ∪ R2)

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Shared Constants

If R1 and R2 share constants Then FC(R1 ∪ R2) may be infinite even if FC(R1) and FC(R2) are finite. Example R1 := {f (a, h(x)) → h(f (b, x))} R2 := {b → a} f (a, h(h(x))) → h(h(f (a, x))) ∈ FC(R1 ∪ R2)

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Summary of Results

Finiteness of forward closure is equivalent to the finite variant property Finiteness of forward closure is undecidable Having the finite variant property is undecidable Finiteness of forward closure is preserved by union if the signatures are disjoint, but not if they share constants.

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Future Work

Forward closure modulo theory More detailed modularity results

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Thank You

cbou@cs.albany.edu Research supported in part by NSF grant CNS-0905286

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