ttt2 with termination templates for teachingwst2018.webs.upv.es/papers/p07_slides.pdf · 2018. 7....

TTT2 with Termination Templates for Teaching

Jonas Schopf and Christian Sternagel

Department of Computer ScienceUniversity of Innsbruck

16th International Workshop on TerminationOxford

July 19, 2018

http://cl-informatik.uibk.ac.at

Motivation

f(x, g(x))→ g(f(x, x))f(x, x)→ h(g(x))

g(f(x, x))→ f(h(x), x)

Correct Termination Proofs?

student 1:

w0 = 1w(f) = 4, w(g) = 2, w(h) = 1h > f > g

student 2:

[h](x) = x

[f](x, y) = 8x + 24y + 16[g](x) = x + 1

student 3:

[h](x) =(

1 40 0

)x, [f](x, y) =

(1 150 0

)x +(

16 10 0

)y +(

80

),

[g](x) =(

1 00 4

)x +(

10

)

J. Schopf, C. Sternagel (DCS @ UIBK) WST 2018 2/27

http://informatik.uibk.ac.at/

http://www.uibk.ac.at/

Motivation

f(x, g(x))→ g(f(x, x))f(x, x)→ h(g(x))

g(f(x, x))→ f(h(x), x)

student 1:

w0 = 1w(f) = 4, w(g) = 2, w(h) = 1h > f > g

student 2:

[h](x) = x

[f](x, y) = 8x + 24y + 16[g](x) = x + 1

student 3:

[h](x) =(

1 40 0

)x, [f](x, y) =

(1 150 0

)x +(

16 10 0

)y +(

80

),

[g](x) =(

1 00 4

)x +(

10

)

Motivation

f(x, g(x))→ g(f(x, x))f(x, x)→ h(g(x))

g(f(x, x))→ f(h(x), x)

student 1:

w0 = 1w(f) = 4, w(g) = 2, w(h) = 1h > f > g

student 2:

[h](x) = x

[f](x, y) = 8x + 24y + 16[g](x) = x + 1

student 3:

[h](x) =(

1 40 0

)x, [f](x, y) =

(1 150 0

)x +(

16 10 0

)y +(

80

),

[g](x) =(

1 00 4

)x +(

10

)

Motivation

f(x, g(x))→ g(f(x, x))f(x, x)→ h(g(x))

g(f(x, x))→ f(h(x), x)

student 1:

w0 = 1w(f) = 4, w(g) = 2, w(h) = 1h > f > g

student 2:

[h](x) = x

[f](x, y) = 8x + 24y + 16[g](x) = x + 1

student 3:

[h](x) =(

1 40 0

)x, [f](x, y) =

(1 150 0

)x +(

16 10 0

)y +(

80

),

[g](x) =(

1 00 4

)x +(

10

)

Motivation

f(x, g(x))→ g(f(x, x))f(x, x)→ h(g(x))

g(f(x, x))→ f(h(x), x)

student 1:

w0 = 1w(f) = 4, w(g) = 2, w(h) = 1h > f > g

student 2:

[h](x) = x

[f](x, y) = 8x + 24y + 16[g](x) = x + 1

student 3:

[h](x) =(

1 40 0

)x, [f](x, y) =

(1 150 0

)x +(

16 10 0

)y +(

80

),

[g](x) =(

1 00 4

)x +(

10

)J. Schopf, C. Sternagel (DCS @ UIBK) WST 2018 2/27

Overview

• main idea• TTT2

• template mechanism• URL encoding via the web interface

Overview

• main idea• TTT2• template mechanism• URL encoding via the web interface

Main Idea

• check specific proofs with TTT2

• modify proof construction• provide a mechanism to show termination examples• in general: make TTT2 more useful for teaching

Solve following Questions

• is the following termination proof correct?• exists another termination proof with specific parameters?• ...

Main Idea

• check specific proofs with TTT2• modify proof construction

• provide a mechanism to show termination examples• in general: make TTT2 more useful for teaching

Main Idea

• check specific proofs with TTT2• modify proof construction• provide a mechanism to show termination examples

• in general: make TTT2 more useful for teaching

Main Idea

• check specific proofs with TTT2• modify proof construction• provide a mechanism to show termination examples• in general: make TTT2 more useful for teaching

Main Idea

• check specific proofs with TTT2• modify proof construction• provide a mechanism to show termination examples• in general: make TTT2 more useful for teaching

Main Idea: Templates

• based on SAT/SMT encodings

• concrete instances seem entirely random• provide method-specific template mechanism to modify proof

construction

Main Idea: URL encoded examples

• setting up examples during lecture is error-prone and tedious• encode examples into URL• automatically restore examples• derive a convenient way to present examples

• based on SAT/SMT encodings• concrete instances seem entirely random

• provide method-specific template mechanism to modify proofconstruction

• based on SAT/SMT encodings• concrete instances seem entirely random• provide method-specific template mechanism to modify proof

construction

• setting up examples during lecture is error-prone and tedious

• encode examples into URL• automatically restore examples• derive a convenient way to present examples

construction

• setting up examples during lecture is error-prone and tedious• encode examples into URL

• automatically restore examples• derive a convenient way to present examples

construction

• setting up examples during lecture is error-prone and tedious• encode examples into URL• automatically restore examples

• derive a convenient way to present examples

construction

Template Mechanism for LPO, KBO, PIs and MIs

Addition on Natural Numbers

0 + y → y

s(x) + y → s(x + y)

Termination Proof without restrictions by TTT2LPO:

+ > s ∼ 0

Also correct?

Is there a precedence with

0 > +, s > + or 0 = +

0 + y → y

s(x) + y → s(x + y)

+ > s ∼ 0

Also correct?

0 > +, s > + or 0 = +

0 + y → y

s(x) + y → s(x + y)

+ > s ∼ 0

Also correct?

0 > +, s > + or 0 = +

How to call TTT2./ttt2 [options] <file> [timeout]

Call with specific Strategy

./ttt2 -s ’matrix’ <file> [timeout]

Call with a Template

./ttt2 -s ’poly [processor flags]’ <file> [timeout]

Example: Full Call

./ttt2 -s ’poly -inters "+ = x0 + x1"’ add.trs

Example: Full Call

Encoding in TTT2

TRS Strategy

SAT/SMT Formula

Solver Backend

UNSAT

SAT

Proof

TemplateTemplate Constraint∧

Encoding in TTT2

TRS StrategySAT/SMT Formula

Solver Backend

UNSAT

SAT

Proof

Encoding in TTT2

Solver Backend

UNSAT

SAT

Proof

Encoding in TTT2

Solver Backend

UNSAT

SAT

Proof

Encoding in TTT2

Solver Backend

UNSAT

SAT

Proof

Encoding in TTT2

Solver Backend

UNSAT

SAT

Proof

Template

Template Constraint∧

Encoding in TTT2

Solver Backend

UNSAT

SAT

Proof

Template

Template Constraint

∧

Encoding in TTT2

Solver Backend

UNSAT

SAT

Proof

Template

Template Constraint∧

Lexicographic Path Order

• parameter is precedence

• -prec gets prec template• prec template specifies a (partial) precedence

Syntax

prec

fun >��

� =�� >=��

�

fun�

�

• parameter is precedence• -prec gets prec template

• prec template specifies a (partial) precedence

Syntax

prec

fun >��

� =�� >=��

�

fun�

�

• parameter is precedence• -prec gets prec template• prec template specifies a (partial) precedence

Syntax

prec

fun >��

� =�� >=��

�

fun�

�

• parameter is precedence• -prec gets prec template• prec template specifies a (partial) precedence

Syntax

prec

fun >��

� =�� >=��

�

fun�

�

0 + y → y

s(x) + y → s(x + y)

Termination Proof without restrictions by TTT2

precedence: + > s ∼ 0

Also correct?

0 > +, s > + or 0 = +

0tick, cross and question mark are clickable hyperlinksJ. Schopf, C. Sternagel (DCS @ UIBK) WST 2018 11/27

0 + y → y

s(x) + y → s(x + y)

Also correct?

0 > +, s > + or 0 = +

0 + y → y

s(x) + y → s(x + y)

Also correct?

0 > +, s > + or 0 = +

./ttt2 -s ’lpo -prec"0 > +"’ add.trs 3

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=lpo&template=0%3E%2B

0 + y → y

s(x) + y → s(x + y)

Also correct?

0 > +, s > + or 0 = +

./ttt2 -s ’lpo -prec"s > +"’ add.trs ?

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=lpo&template=s%3E%2B

0 + y → y

s(x) + y → s(x + y)

Also correct?

0 > +, s > + or 0 = +

./ttt2 -s ’lpo -prec"0 = +"’ add.trs 3

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=lpo&template=0%3D%2B

0 + y → y

s(x) + y → s(x + y)

Also correct?

0 > +, s > + or 0 = +

./ttt2 -s ’lpo -prec"+ > 0 > +"’ add.trs 7

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=lpo&template=%2B%20%3E%200%20%3E%20%2B

0 + y → y

s(x) + y → s(x + y)

Also correct?

0 > +, s > + or 0 = +

./ttt2 -s ’lpo -prec"+ > 0 > +"’ add.trs 7

./ttt2 -s ’lpo -prec"0 = + > s"’ add.trs 3

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=lpo&template=%2B%20%3E%200%20%3E%20%2B

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=lpo&template=0%20%3D%20%2B%20%3E%20s

Knuth-Bendix Order

• parameters are precedence and weights

• -weights accepts a weights template• -w0 takes a value for w0

Syntax

weights

�� fun =

��

�

fun =��

�<=�� >=��

�

weight

Knuth-Bendix Order

• parameters are precedence and weights• -weights accepts a weights template

• -w0 takes a value for w0

Syntax

weights

�� fun =

��

�

fun =��

�<=�� >=��

�

weight

Knuth-Bendix Order

• parameters are precedence and weights• -weights accepts a weights template• -w0 takes a value for w0

Syntax

weights

�� fun =

��

�

fun =��

�<=�� >=��

�

weight

Knuth-Bendix Order

• parameters are precedence and weights• -weights accepts a weights template• -w0 takes a value for w0

Syntax

weights

�� fun =

��

�

fun =��

�<=�� >=��

�

weight

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

./ttt2 -s ’kbo -weights "+ <= 16, + >= 8"’ add.trs 3

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

./ttt2 -s ’kbo -prec "0 = + > s" -weights "+ = s = 3" 3-w0 3’ add.trs

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=kbo&template=&template1=%2B%20%3C%3D%2016%2C%20%2B%20%3E%3D%208&template2=

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=kbo&template=0%20%3D%20%2B%20%3E%20s&template1=%2B%20%3D%20s%20%3D%203&template2=3

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

0 + y → y

s(x) + y → s(x + y)

Examples

8 ≤ w(+) ≤ 16

16 ≤ w(+) and w(+) ≤ 8

0 = + > s, w(+) = 7, w(s) = 3 and w0 = 3

Linear Interpretations• supported by PIs and MIs

• accepted by the -inters flag• sums of linear monomials• underscore denotes arbitrary parts• integer at the end of var denotes the position of the argument

Syntax

inters

fun =��

��

��const

�

var��const

� _��

�

� +��

�

Linear Interpretations• supported by PIs and MIs• accepted by the -inters flag

• sums of linear monomials• underscore denotes arbitrary parts• integer at the end of var denotes the position of the argument

Syntax

inters

fun =��

��

��const

�

var��const

� _��

�

� +��

�

Linear Interpretations• supported by PIs and MIs• accepted by the -inters flag• sums of linear monomials

• underscore denotes arbitrary parts• integer at the end of var denotes the position of the argument

Syntax

inters

fun =��

��

��const

�

var��const

� _��

�

� +��

�

Linear Interpretations• supported by PIs and MIs• accepted by the -inters flag• sums of linear monomials• underscore denotes arbitrary parts

• integer at the end of var denotes the position of the argument

Syntax

inters

fun =��

��

��const

�

var��const

� _��

�

� +��

�

Linear Interpretations• supported by PIs and MIs• accepted by the -inters flag• sums of linear monomials• underscore denotes arbitrary parts• integer at the end of var denotes the position of the argument

Syntax

inters

fun =��

��

��const

�

var��const

� _��

�

� +��

�

Linear Interpretations• supported by PIs and MIs• accepted by the -inters flag• sums of linear monomials• underscore denotes arbitrary parts• integer at the end of var denotes the position of the argument

Syntax

inters

fun =��

��

��const

�

var��const

� _��

�

� +��

�

Polynomial Interpretations

take natural numbers for “const” in the inters template

0 + y → y

s(x) + y → s(x + y)

Examples

[+](x, y) = 2x + y + 1, [s](x) = x + 1, [0] = 0

./ttt2 -s ’poly -inters"+ = 2x0 + x1 + 1, s = x0 + 1, 30 = 0"’ add.trs

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=poly&template=%2B%20%3D%202x0%20%2B%20x1%20%2B%201%2C%20s%20%3D%20x0%20%2B%201%2C%200%20%3D%200

0 + y → y

s(x) + y → s(x + y)

Examples

[+](x, y) = 2x + y + 1, [s](x) = x + 1, [0] = 0

0 + y → y

s(x) + y → s(x + y)

Examples

[+](x, y) = 2x + y + 1, [s](x) = x + 1, [0] = 0

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

where a, b and c are arbitrary

./ttt2 -s ’poly -inters "+ = 2x0 + + 1, s = , 0 = 0"’3add.trs

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

./ttt2 -s ’poly -inters "+ = 2x0 + x1"’ add.trs 3

[+](x, y) = 2x + ay, where a is arbitrary

./ttt2 -s ’poly -inters "+ = 2x0 + + 0"’ add.trs 3

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=poly&template=%2B%20%3D%202x0%20%2B%20_%20%2B%201%2C%20s%20%3D%20_%2C%200%20%3D%200

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=poly&template=%2B%20%3D%202x0%20%2B%20x1

http://colo6-c703.uibk.ac.at/ttt2/web/index.php?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=poly&template=%2B%20%3D%202x0%20%2B%20_%20%2B%200

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

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

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

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

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

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

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

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

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

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

Examples cont’d

[+](x, y) = 2x + ay + 1, [s](x) = bx + c, [0] = 0,

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

Matrix Interpretations

• matrices of natural numbers as “const”

• 0 and 1 for zero-vector and one-vector

Syntax

matrix

[�� nat�

� _��

�

�� ,

��

�

� ;��

�

]��

• matrices of natural numbers as “const”• 0 and 1 for zero-vector and one-vector

Syntax

matrix

[�� nat�

� _��

�

�� ,

��

�

� ;��

�

]��

• matrices of natural numbers as “const”• 0 and 1 for zero-vector and one-vector

Syntax

matrix

[�� nat�

� _��

�

�� ,

��

�

� ;��

�

]��

Example: Syntax

(1 2 34 5 6

)in template syntax for MIs: [1,2,3;4,5,6]

Examples

[0] =(

00

)[s](x) = x +

(11

)[+](x, y) =

(1 10 1

)x +

(1 00 1

)y +

(10

)

./ttt2 -s ’matrix -inters "+ = [1,1;0,1]x0 + [1,0;0,1]x1 3+ [1;0], s = x0 + 1, 0 = 0"’ add.trs

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=matrix2&template=%2B%20%3D%20%5B1%2C1%3B0%2C1%5Dx0%20%2B%20%5B1%2C0%3B0%2C1%5Dx1%20%2B%20%5B1%3B0%5D%2C%20s%20%3D%20x0%20%2B%201%2C%200%20%3D%200

Example: Syntax

(1 2 34 5 6

Examples

[0] =(

00

)[s](x) = x +

(11

)[+](x, y) =

(1 10 1

)x +

(1 00 1

)y +

(10

)

Example: Syntax

(1 2 34 5 6

Examples

[0] =(

00

)[s](x) = x +

(11

)[+](x, y) =

(1 10 1

)x +

(1 00 1

)y +

(10

)

Examples cont’d

[0] =(

a0

)[s](x) = x + m1

[+](x, y) =(

1 bc d

)x + m2y +

(10

)where a, b, c, d, m1 and m2 are arbitrary

./ttt2 -s ’matrix -inters "+ = [1, ; , ]x0 + + [1;0], 3s = x0 + , 0 = [ ;0]’ add.trs

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=matrix2&template=%2B%20%3D%20%5B1%2C_%3B_%2C_%5Dx0%20%2B%20_%20%2B%20%5B1%3B0%5D%2C%20s%20%3D%20x0%20%2B%20_%2C%200%20%3D%20%5B_%3B0%5D

Examples cont’d

[0] =(

a0

)[s](x) = x + m1

[+](x, y) =(

1 bc d

)x + m2y +

(10

)where a, b, c, d, m1 and m2 are arbitrary

./ttt2 -s ’matrix -inters "+ = [1, ; , ]x0 + + [1;0], 3s = x0 + , 0 = [ ;0]’ add.trs

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20%2B(0%2Cy)%20-%3E%20y%0A%20%2B(s(x)%2Cy)%20-%3E%20s(%2B(x%2Cy))%0A)&strategy=matrix2&template=%2B%20%3D%20%5B1%2C_%3B_%2C_%5Dx0%20%2B%20_%20%2B%20%5B1%3B0%5D%2C%20s%20%3D%20x0%20%2B%20_%2C%200%20%3D%20%5B_%3B0%5D

Arbitrary Boolean Structure

• arbitrary boolean combinations of atomic constraints

• special case: comma-separated list of atoms

Syntax

combination

atom��NOT

�� (�� combination )

�� AND�� (

�� combination�� ,

��

)��

�OR�� (

��

)��

�

• arbitrary boolean combinations of atomic constraints• special case: comma-separated list of atoms

Syntax

combination

atom��NOT

�� AND�� (

��

)��

�OR�� (

��

)��

�

• arbitrary boolean combinations of atomic constraints• special case: comma-separated list of atoms

Syntax

combination

atom��NOT

�� AND�� (

��

)��

�OR�� (

��

)��

�

Examples

f should not be bigger than h and g at the same time:

./ttt2 -s ’lpo -prec "NOT(AND(f > g, f > h))"’ 2.60.xml 3

constant part of + should be two and when 0 is zero then s should be thesuccessor function:

./ttt2 -s ’poly -inters "AND(+ = + 2, OR(NOT(0 = 0), 3s = x0 + 1))"’ 4.11.xml

upper triangular matrices with only ones in the diagonals for unary f andbinary g and h:

./ttt2 -s ’poly -inters "f=g=h=[1, , ;0,1, ;0,0,1]x0+ , 3g=h=[1, , ;0,1, ;0,0,1]x1+ "’ 27.xml

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(COMMENT%20TRS_Standard%2FSK90%2F2.60.xml)%0A(RULES%20%0A%20%20f(g(f(a)%2Ch(a%2Cf(a))))%20-%3E%20f(h(g(f(a)%2Ca)%2Cg(f(a)%2Cf(a))))%20%20%20%20%20%20%0A)&strategy=lpo&template=NOT(AND(f%20%3E%20g%2C%20f%20%3E%20h))

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(COMMENT%20TRS_Standard%2FSK90%2F4.11.xml)%0A(VAR%20x%20y)%0A(RULES%20%0A%20%20%2B(x%2C0)%20-%3E%20x%0A%20%20%2B(x%2Cs(y))%20-%3E%20s(%2B(x%2Cy))%0A%20%20%2B(0%2Cs(y))%20-%3E%20s(y)%0A%20%20s(%2B(0%2Cy))%20-%3E%20s(y)%20%20%20%20%20%20%0A)&strategy=poly&template=AND(%2B%20%3D%20_%20%2B%202%2C%20OR(NOT(0%20%3D%200)%2C%20s%20%3D%20x0%20%2B%201))

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(COMMENT%20TRS_Standard%2FDer95%2F27.xml)%0A(VAR%20x%20y)%0A(RULES%20%0A%20%20h(f(x)%2Cy)%20-%3E%20f(g(x%2Cy))%0A%20%20g(x%2Cy)%20-%3E%20h(x%2Cy)%20%20%20%20%20%20%0A)&strategy=matrix3&template=f%20%3D%20g%20%3D%20h%20%3D%20%5B1%2C_%2C_%3B0%2C1%2C_%3B0%2C0%2C1%5Dx0%20%2B%20_%2C%20g%20%3D%20h%20%3D%20%5B1%2C_%2C_%3B0%2C1%2C_%3B0%2C0%2C1%5Dx1%20%2B%20_

Examples

f should not be bigger than h and g at the same time:./ttt2 -s ’lpo -prec "NOT(AND(f > g, f > h))"’ 2.60.xml 3

Examples

constant part of + should be two and when 0 is zero then s should be thesuccessor function:./ttt2 -s ’poly -inters "AND(+ = + 2, OR(NOT(0 = 0), 3s = x0 + 1))"’ 4.11.xml

Examples

upper triangular matrices with only ones in the diagonals for unary f andbinary g and h:./ttt2 -s ’poly -inters "f=g=h=[1, , ;0,1, ;0,0,1]x0+ , 3g=h=[1, , ;0,1, ;0,0,1]x1+ "’ 27.xml

Extension of the Web interface

Web Interface

Web Interface - KBO

URL encoding in the Web interface

• idea: encode content of web interface into URL

• we encode the TRS, the strategy and the templates• encoded into the query string of the URL

Normal URL

http://colo6-c703.uibk.ac.at/ttt2/web/

URL with encoded Query String

http://colo6-c703.uibk.ac.at/ttt2/web/?problem=(VAR%20x%20y)%0A(RULES%0A%20add(0%2Cy)%20-%3E%20y%0A%20add(s(x)%2Cy)%20-%3E%20s(add(x%2Cy))%0A)&strategy=kbo&template=add%20%3E%20s&template1=add%20%3D%20s%20%3D%208&template2=3

• idea: encode content of web interface into URL• we encode the TRS, the strategy and the templates

• encoded into the query string of the URL

Normal URL

• idea: encode content of web interface into URL• we encode the TRS, the strategy and the templates• encoded into the query string of the URL

Normal URL

Conclusion

• template mechanism to check specific proofs

• available for LPO, KBO, PIs and MIs• web interface extension to save the configurations for an example• to have near absolute certainty you should use CeTA to validate the

output of TTT2

Future Work

• templates for other methods like arctic interpretations, the(generalized) subterm criterion, . . .

• which extensions to our templates are most useful for teaching?

Conclusion

• template mechanism to check specific proofs• available for LPO, KBO, PIs and MIs

• web interface extension to save the configurations for an example• to have near absolute certainty you should use CeTA to validate the

output of TTT2

Future Work

Conclusion

• template mechanism to check specific proofs• available for LPO, KBO, PIs and MIs• web interface extension to save the configurations for an example

• to have near absolute certainty you should use CeTA to validate theoutput of TTT2

Future Work

Conclusion

• template mechanism to check specific proofs• available for LPO, KBO, PIs and MIs• web interface extension to save the configurations for an example• to have near absolute certainty you should use CeTA to validate the

output of TTT2

Future Work

Conclusion

output of TTT2

Future Work

Conclusion

output of TTT2

Future Work

Thank you for your attention!

ttt2 with termination templates for teachingwst2018.webs.upv.es/papers/p07_slides.pdf · 2018. 7....

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