A Dependency Pair Framework for Innermost Complexity Analysis of Term Rewrite Systems

A Dependency Pair Framework for Innermost Complexity Analysis of

Term Rewrite Systems

J¨urgen Giesl

LuFG Informatik 2, RWTH Aachen University, Germany

joint work withLars NoschinskiandFabian Emmes

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . )

Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity

adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting)

new approach: directadaption of DP framework (CADE ’11) modular combination of different techniques

automated and more powerful than previous approaches

Termination Analysis of TRSs

useful for termination ofprograms (Java, Haskell, Prolog, . . . ) Dependency Pair Framework

modular combination of different techniques automatable

Complexity Analysis of TRSs

should be useful for ofprograms⇒ Innermost Runtime Complexity adaptDependency Pair Framework

Hirokawa & Moser (IJCAR ’08, LPAR ’08) first adaption of DPs for complexity not modular

Zankl & Korp (RTA ’10)

modular approach based on relative rewriting forDerivational Complexity

(cannot exploit strength of DPs for innermost rewriting) new approach: directadaption of DP framework (CADE ’11)

modular combination of different techniques

automated and more powerful than previous approaches

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht

dh(double(s^k(0)))=k+ 1 dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0) Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0) Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0) Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0) Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0)

Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0) Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0)

Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0)

Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n ιR=Pol₀ iff length ∈ O(1)

ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0)

Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n)

ιR=Pol₂ iff length ∈ O(n²) . . . Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0)

Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Innermost Runtime Complexity

R: double(0) → 0

double(s(x)) → s(s(double(x)))

Derivation Height dh(t): length of longest →ⁱ _R-sequence witht dh(double(s^k(0)))=k+ 1

dh(double^k(s(0)))≈2^k

Basic Terms f(t₁, . . . ,t_n)

f defined symbol (double), t₁, . . . ,t_n no defined symbols (s,0)

Complexity ιR of TRS R:

length of longest →ⁱ _R-sequence with basic termt where|t| ≤n

ιR=Pol₀ iff length ∈ O(1) ιR =Pol₁ iff length∈ O(n) ιR=Pol₂ iff length ∈ O(n²) . . .

Example: ιR=Pol₁

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m

]

(x,y)→if

]

(gt(x,y),x,y) if

]

(true,x,y)→m

]

(p(x),y) m

]

(x,y)→gt

]

(x,y) if

]

(true,x,y)→p

]

(x) gt

]

(s(n),s(k))→gt

]

(n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m

]

(x,y)→if

]

(gt(x,y),x,y) if

]

(true,x,y)→m

]

(p(x),y) m

]

(x,y)→gt

]

(x,y) if

]

(true,x,y)→p

]

(x) gt

]

(s(n),s(k))→gt

]

(n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol

m

]

(x,y)→if

]

(gt(x,y),x,y) if

]

(true,x,y)→m

]

(p(x),y) m

]

(x,y)→gt

]

(x,y) if

]

(true,x,y)→p

]

(x) gt

]

(s(n),s(k))→gt

]

(n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol

m

]

(x,y)→if

]

(gt(x,y),x,y) if

]

(true,x,y)→m

]

(p(x),y) m

]

(x,y)→gt

]

(x,y) if

]

(true,x,y)→p

]

(x) gt

]

(s(n),s(k))→gt

]

(n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m

]

(x,y)→if

]

(gt(x,y),x,y) if

]

(true,x,y)→m

]

(p(x),y) m

]

(x,y)→gt

]

(x,y) if

]

(true,x,y)→p

]

(x) gt

]

(s(n),s(k))→gt

]

(n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m^](x,y)→if^](gt(x,y),x,y) if^](true,x,y)→m^](p(x),y) m^](x,y)→gt^](x,y) if^](true,x,y)→p^](x)

gt^](s(n),s(k))→gt^](n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m^](x,y)→if^](gt(x,y),x,y) if^](true,x,y)→m^](p(x),y) m^](x,y)→gt^](x,y) if^](true,x,y)→p^](x)

gt^](s(n),s(k))→gt^](n,k)

Complexity Analysis: Dependency Tuples

compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m^](x,y)→if^](gt(x,y),x,y) if^](true,x,y)→m^](p(x),y) m^](x,y)→gt^](x,y) if^](true,x,y)→p^](x)

gt^](s(n),s(k))→gt^](n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Dependency Tuples

m(x,y)→if(gt(x,y),x,y) gt(0,k)→false p(0)→0 if(true,x,y)→s(m(p(x),y)) gt(s(n),0)→true p(s(n))→n if(false,x,y)→0 gt(s(n),s(k))→gt(n,k)

Termination Analysis: Dependency Pairs

compare lhs with subterms of rhs that start with definedsymbol m^](x,y)→if^](gt(x,y),x,y) if^](true,x,y)→m^](p(x),y) m^](x,y)→gt^](x,y) if^](true,x,y)→p^](x)

gt^](s(n),s(k))→gt^](n,k)

Complexity Analysis: Dependency Tuples compare lhs with alldefined subterms of rhsat once

m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com2(m^](p(x),y),p^](x)) p^](s(n))→Com0

if^](false,x,y)→Com⁰ gt^](0,k)→Com⁰ gt^](s(n),0)→Com⁰

gt^](s(n),s(k))→Com¹(gt^](n,k))

Chain Trees

DT(R) : m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com²(m^](p(x),y),p^](x)) p^](s(n))→Com⁰ if^](false,x,y)→Com0 gt^](0,k)→Com0

gt^](s(n),0)→Com0

gt^](s(n),s(k))→Com1(gt^](n,k)) (D,R)-Chain Tree:

Edgeσ₁(

u^]→Comn(v₁^], . . . ,v_n^])

) toσ₂(w^]→Comm(. . .)) if v_i^]σ₁→^{i ∗}_R w^]σ₂

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0))

if^](true,s(0),0)→Com2(m^](p(s(0)),0),p^](s(0))) gt^](s(0),0)→Com0

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0)) p^](s(0))→Com0

if^](false,s(0),0)→Com0 gt^](s(0),0)→Com0

Chain Trees

DT(R) : m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com²(m^](p(x),y),p^](x)) p^](s(n))→Com⁰ if^](false,x,y)→Com0 gt^](0,k)→Com0

gt^](s(n),0)→Com0

gt^](s(n),s(k))→Com1(gt^](n,k)) (D,R)-Chain Tree:

Edgeσ₁(

u^]→Comn(v₁^], . . . ,v_n^])

) toσ₂(w^]→Comm(. . .)) if v_i^]σ₁→^{i ∗}_R w^]σ₂

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0))

if^](true,s(0),0)→Com2(m^](p(s(0)),0),p^](s(0))) gt^](s(0),0)→Com0

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0)) p^](s(0))→Com0

if^](false,s(0),0)→Com0 gt^](s(0),0)→Com0

Chain Trees

DT(R) : m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com²(m^](p(x),y),p^](x)) p^](s(n))→Com⁰ if^](false,x,y)→Com0 gt^](0,k)→Com0

gt^](s(n),0)→Com0

gt^](s(n),s(k))→Com1(gt^](n,k)) (D,R)-Chain Tree:

Edge

σ₁(u^]→Comn(v₁^], . . . ,v_n^]))

toσ₂(w^]→Comm(. . .)) if v_i^]σ₁→^{i ∗}_R w^]σ₂

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0))

if^](true,s(0),0)→Com2(m^](p(s(0)),0),p^](s(0))) gt^](s(0),0)→Com0

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0)) p^](s(0))→Com0

if^](false,s(0),0)→Com0 gt^](s(0),0)→Com0

Chain Trees

DT(R) : m^](x,y)→Com2(if^](gt(x,y),x,y),gt^](x,y)) p^](0)→Com0

if^](true,x,y)→Com²(m^](p(x),y),p^](x)) p^](s(n))→Com⁰ if^](false,x,y)→Com0 gt^](0,k)→Com0

gt^](s(n),0)→Com0

gt^](s(n),s(k))→Com1(gt^](n,k)) (D,R)-Chain Tree:

Edgeσ₁(u^]→Comn(v₁^], . . . ,v_n^])) to σ₂(w^]→Comm(. . .)) if v_i^]σ₁→^{i ∗}_Rw^]σ₂

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0))

if^](true,s(0),0)→Com2(m^](p(s(0)),0),p^](s(0))) gt^](s(0),0)→Com0

m^](s(0),0)→Com2(if^](gt(s(0),0),s(0),0),gt^](s(0),0)) p^](s(0))→Com0

if^](false,s(0),0)→Com0 gt^](s(0),0)→Com0