Constant Runtime Complexity of Term Rewriting is Semi-Decidable

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Constant Runtime Complexity of Term Rewriting is Semi-Decidable

J¨urgen Giesl

LuFG Informatik 2, RWTH Aachen University, Germany

joint work with Florian Frohn

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Runtime Complexity

Example TRS

R

f (d)

→

c(g(d)) g(c(x))

→

c(g(f(x)))

Rewrite Sequence

g(c(d))

→_R

c(g(f(d)))

→_R

c(g(c(g(d))))

→_R

c(c(g(f(g(d)))))

Defined Symbols:

roots of lhs f, g

Constructors:

other function symbols c, d

Basic Term:

defined symbol only at the root

f(x), g(c(x)), g(c(d))

Runtime Complexity rc

R(n)

length of longest

−→_R

-sequence starting with basic term t where

|t| ≤

n

Example: rc

R(3)≥3

since

|

g(c(d))

|= 3

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Runtime Complexity

Contribution:

Constant runtime complexity is semi-decidable

R

has constant runtime complexity

⇔

rc

R

(n)

∈ O(1)

⇔ ∃m∈N

. all

→_R

-evaluations of basic terms take at most m steps

Motivation

Complexity Analysis: bounds on program’s resource usage Constant Bounds: detect bugs

Runtime Complexity rc

R(n)

length of longest

−→_R

-sequence starting with basic term t where

|t| ≤

n

Example: rc

R(3)≥3

since

|

g(c(d))

|= 3

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Constructor-Based Narrowing

Example TRS

R

f (d)

→

c(g(d)) g(c(x))

→

c(g(f(x)))

Constructor-Based Narrowing Sequence

g(x)

(c⁰)

{x/c(x0)}

ε c(g(f(x⁰))) ^{x_1.1⁰^/d} c(g(c(g(d)))) ^∅₁ c(c(g(f(g(d)))))

)

Narrowing sequencet₀ ^σ¹ · · · ^σn t_n isconstructor basedift₀σ₁· · ·σ_n is basic.

Constructor-basedR-narrowing terminates:

g(x), g(c(x)), g(c(d)) 3 steps g(c(t))for other termst 1 step g(t)for other termst 0 steps

f(x), f(d) 1 step

f(t)for other termst 0 steps

Goal: Show termination of cb narrowing by inspecting finitely many start terms

Main Theorem

Rhas constant runtime complexity iff constructor-basedR-narrowing terminates

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Constructor-Based Narrowing

Example TRS

R

f (d)

→

c(g(d)) g(c(x))

→

c(g(f(x)))

Constructor-Based Narrowing Sequence

g(x)

(c⁰)

{x/c(x0)}

)

Narrowing sequencet₀ ^σ¹ · · · ^σn t_n isconstructor basedift₀σ₁· · ·σ_n is basic.

Forg(c(x))→c(g(x)), cb narrowing would not terminate:

g(x) ^{x/c(x⁰^)} c(g(x⁰)) ^{x⁰^/c(x⁰⁰^)} c(c(g(x⁰⁰))) · · ·

)

Main Theorem

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Constructor-Based Narrowing

Example TRS

R

f (d)

→

c(g(d)) g(c(x))

→

c(g(f(x)))

Constructor-Based Narrowing Sequence

g(x)

(c⁰)

{x/c(x0)}

)

g(c(x⁰)) ^∅_ε c(g(f(x⁰))) ^{x_1.1⁰^/d} c(g(c(g(d)))) ^∅₁ c(c(g(f(g(d))))) g(c(d)) ^∅_ε c(g(f(d))) _1.1^∅ c(g(c(g(d)))) ^∅₁ c(c(g(f(g(d)))))

Def: s0

σ1

π1 · · · ^σn_π_n sn is more generalthan t0

θ1

π1 · · · _π^θn_n tn if

Def:

there is a substitution η with s₀ σ₁σ₂· · ·σ_n η = t₀ θ₁θ₂· · ·θ_n

s₁ σ₂· · ·σ_n η = t₁ θ₂· · ·θ_n

. . .

s_n η = t_n

Narrowing Lemma

For everyf(. . .) ⁿtthere is a more general sequence f(x₁, . . . , x_k) ⁿs.

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Main Theorem

Proof of “⇐”

Assume thatRdoes not have constant runtime complexity

⇒ f(q~₁) −→ⁿ¹ t₁, f(q~₂) −→ⁿ² t₂, . . . withn₁< n₂<· · ·

⇒ f(q~1) ^∅ⁿ¹ t1, f(q~2) ^∅ⁿ² t2, . . .

⇒ f(x1, . . . , xk) ^σ¹ⁿ¹ s1, f(x1, . . . , xk) ^σ²ⁿ² s2, . . . by Narrowing Lemma

⇒ cb narrowing tree with rootf(x1, . . . , xk) has infinitely many nodes

is finitely branching (as Ris finite)

has infinite path (by K¨onig’s Lemma), i.e., infinite cb narrowing sequence

Narrowing Lemma

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Main Theorem

Proof of “⇒”

Assume that there is an infinite sequencet0

σ1

t1

σ2

· · ·

⇒ t0σ1· · ·σm+1 −→^m+1 tm+1 for allm∈N

⇒ ∀m. there is an→R-evaluation of a basic term with more thanmsteps

⇔ Rdoes not have constant runtime complexity

Narrowing Lemma

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Semi-Decision Procedure for Constant Runtime Complexity

Example TRS

R

f (d)

→

c(g(d)) g(c(x))

→

c(g(f(x)))

CB Narrowing Trees

f(x) ^{x/d} c(g(d))

f(x)

g(x) ^{x/c(x⁰^)} c(g(f(x⁰))) ^{x⁰^/d} c(g(c(g(d))))

(

∅ c(c(g(f(g(d)))))

Semi-Decision Procedure

For all defined symbols f, build cb narrowing tree for f(x

1

, . . . , x

k

).

If constructing the trees terminates, then return “constant runtime”.

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Undecidability of Constant Runtime Complexity

Theorem

Constant runtime complexity of TRSs is semi-decidable, but not decidable.

Proof

Turing machine

M

is immortal

⇔

rewriting

infinite

basic terms with

R_M

does not terminate

⇔

narrowing basic terms f(x

1

, . . . , x

_k

) with

R_M

does not terminate

⇔ R_M

does not have constant runtime complexity

(Im)mortality undecidable

⇒

constant runtime complexity undecidable

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Experiments

Theorem

Constant runtime complexity of TRSs is semi-decidable, but not decidable.

Implementation and integration of semi-decision procedure inAProVE

Full rewriting (959examples from the TPDB, 60 s per example) 57TRSs with constant runtime

57TRSs detected by semi-decision procedure, 1.8 s avg. on successes 51TRSs detected byTcTorAProVEwithout semi-decision procedure

Innermost rewriting (1022examples from the TPDB, 60 s per example) 59TRSs with constant runtime

58TRSs detected by semi-decision procedure, 1.4 s avg. on successes 1TRS withrelativerules not detected by semi-decision procedure 55TRSs detected byTcTorAProVEwithout semi-decision procedure