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I ( C ) . i = 0 i = 0 i = 0 D ANIELA R ITIRC C OMPLEXITY OF C - - PowerPoint PPT Presentation

Jul 09, 2023 162 likes •697 views

C OMPLEXITY OF C IRCUIT I DEAL M EMBERSHIP T ESTING Daniela Ritirc, Armin Biere, Manuel Kauers Johannes Kepler University Linz, Austria SC-Square Workshop 2017 University of Kaiserslautern, Germany 29. July 2017 D ANIELA R ITIRC C OMPLEXITY OF

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COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

Daniela Ritirc, Armin Biere, Manuel Kauers

Johannes Kepler University Linz, Austria

SC-Square Workshop 2017

University of Kaiserslautern, Germany

  • 29. July 2017

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 2

MOTIVATION & SOLVING TECHNIQUES

Given: a (gate level) multiplier circuit C for fixed-size bitwidth n

FA FA FA FA FA FA FA FA FA 32s5 16s4 8s3 4s2 2s1 1s0 + + + + + a0b0 a0b1 a0b2 a1b0 a1b1 a1b2 a2b0 a2b1 a2b2 (4a2 + 2a1 + 1a0) ∗ (4b2 + 2b1 + 1b0)

Question: For all ai,bi ∈ B:

∑2n−1

i=0 2isi −

  • ∑n−1

i=0 2iai

  • ∑n−1

i=0 2ibi

  • ?

Motivation verify circuits to avoid issuses like Pentium FDIV bug Solving Techniques SAT using CNF encoding Binary Moment Diagrams (BMD) Algebraic reasoning

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 3

MOTIVATION & SOLVING TECHNIQUES

SAT verifying even small multipliers (16 Bit) is challenging (empirically) conjecture [Biere’16]: even simple ring-properties, e.g., x · y = y · x, require exponential sized resolution proofs (for gate-level CNF encoding) recent theoretical result [BeameLiew’17]: polynomial sized resolution proofs for simple ring-properties exist no theoretical nor practical results on general multiplier verification

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 4

MOTIVATION & SOLVING TECHNIQUES

SAT verifying even small multipliers (16 Bit) is challenging (empirically) conjecture [Biere’16]: even simple ring-properties, e.g., x · y = y · x, require exponential sized resolution proofs (for gate-level CNF encoding) recent theoretical result [BeameLiew’17]: polynomial sized resolution proofs for simple ring-properties exist no theoretical nor practical results on general multiplier verification BMD approach not robust requires structural knowledge

  • nly works for simple (clean) multipliers

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 5

IN A NUTSHELL

Multiplier

FA FA FA FA FA FA FA FA FA 32s5 16s4 8s3 4s2 2s1 1s0 + + + + + a0b0 a0b1 a0b2 a1b0 a1b1 a1b2 a2b0 a2b1 a2b2

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 6

IN A NUTSHELL

Multiplier

FA FA FA FA FA FA FA FA FA 32s5 16s4 8s3 4s2 2s1 1s0 + + + + + a0b0 a0b1 a0b2 a1b0 a1b1 a1b2 a2b0 a2b1 a2b2

AIGMULTOPOLY Translation Gröbner basis B = { x − a0 ∗ b0, y − a1 ∗ b1, s0 − x ∗ y,

}

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 7

IN A NUTSHELL

Multiplier

FA FA FA FA FA FA FA FA FA 32s5 16s4 8s3 4s2 2s1 1s0 + + + + + a0b0 a0b1 a0b2 a1b0 a1b1 a1b2 a2b0 a2b1 a2b2

AIGMULTOPOLY Translation Gröbner basis B = { x − a0 ∗ b0, y − a1 ∗ b1, s0 − x ∗ y,

}

Verification

= 0 ✗ = 0 ✓

C A S y s t e m R e d u c t i

  • n

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 8

ALGEBRA

f = 2x + 4y + 3 ∈ Q[x,y] g = y + 1

∈ Q[x,y]

Ring Q[x,y] ring of polynomials with variables x,y and coefficients in Q Polynomial f,g finite sum of monomials

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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ALGEBRA

f = 2x + 4y + 3 ∈ Q[x,y] g = y + 1

∈ Q[x,y]

Monomial constant multiple of a term Term power product xe1ye2 for e1,e2 ∈ N Term order well-defined, x > y > 1 Leading monomial/term/coefficient

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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ALGEBRA

f = 2x + 4y + 3 ∈ Q[x,y] g = y + 1

∈ Q[x,y]

Ideal generated by f,g I = {q1f + q2g | q1,q2 ∈ Q[x,y]} = f,g

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 11

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 Ideal generated by f,g I = {q1f + q2g | q1,q2 ∈ Q[x,y]} = f,g “I contains all elements which evaluate to 0, when f and g evaluate to 0”

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 12

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 13

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

for I: a priori not obvious how to check this for a Gröbner basis G: “easy” reduction method really?

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 14

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

for I: a priori not obvious how to check this for a Gröbner basis G: “easy” reduction method really?

Gröbner basis

every ideal of Q[X] has a Gröbner basis construction algorithm by Buchberger

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 15

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

for I: a priori not obvious how to check this for a Gröbner basis G: “easy” reduction method really?

Gröbner basis

every ideal of Q[X] has a Gröbner basis construction algorithm by Buchberger special case: leading terms of ideal generators have no variables in common

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 16

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 G = {2x + 4y + 3,y + 1} Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

for I: a priori not obvious how to check this for a Gröbner basis G: “easy” reduction method really?

Gröbner basis

every ideal of Q[X] has a Gröbner basis construction algorithm by Buchberger special case: leading terms of ideal generators have no variables in common

G = {f,g} is a Gröbner basis for I

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 17

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 G = {2x + 4y + 3,y + 1} Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

for I: a priori not obvious how to check this for a Gröbner basis G: “easy” reduction method really?

Reduction multivariate version of polynomial division with remainder

divide h by elements of G remainder r contains no term that is a multiple

  • f any of the leading terms of G

Notation: r = Remainder(h,G)

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 18

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 G = {2x + 4y + 3,y + 1} Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ?

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 19

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 G = {2x + 4y + 3,y + 1} Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 9 ∈ I ? Answer: Yes h = 3∗(2x + 4y + 3)+ y2 ∗(y + 1) Remainder(h,G)=0

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 20

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 G = {2x + 4y + 3,y + 1} Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 10 ∈ I ?

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 21

ALGEBRA

I = f,g = 2x + 4y + 3,y + 1 G = {2x + 4y + 3,y + 1} Ideal membership problem Question: h = 6x + y3 + y2 + 12y + 10 ∈ I ? Answer: No h = 3∗(2x + 4y + 3)+ y2 ∗(y + 1)+ 1 Remainder(h,G)=1

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 22

IDEALS ASSOCIATED TO CIRCUITS

Polynomial Representation of Circuit Gates

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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IDEALS ASSOCIATED TO CIRCUITS

Polynomial Representation of Circuit Gates Boolean Gate Polynomials u = ¬v implies 0 = −u + 1− v u = v ∧ w implies 0 = −u + vw u = v ∨ w implies 0 = −u + v + w − vw u = v ⊕ w implies 0 = −u + v + w − 2vw

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 24

IDEALS ASSOCIATED TO CIRCUITS

Polynomial Representation of Circuit Gates Boolean Gate Polynomials u = ¬v implies 0 = −u + 1− v u = v ∧ w implies 0 = −u + vw u = v ∨ w implies 0 = −u + v + w − vw u = v ⊕ w implies 0 = −u + v + w − 2vw Field Polynomials “ u ∈ B ” implies 0 = u(u − 1) 0 = u2 − u

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 25

IDEALS ASSOCIATED TO CIRCUITS

n-Bit Multipliers

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

n ∗ n = 2n 2n inputs: a0,...,an−1,b0,...,bn−1 2n outputs: s0,...,s2n−1

  • ne variable to each internal gate
  • utput: g0,...,gk

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 26

IDEALS ASSOCIATED TO CIRCUITS

n-Bit Multipliers

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

n ∗ n = 2n 2n inputs: a0,...,an−1,b0,...,bn−1 2n outputs: s0,...,s2n−1

  • ne variable to each internal gate
  • utput: g0,...,gk

Values of g0,...,gk and s0,...,s2n−1 are uniquely determined as soon as a0,...,an−1,b0,...,bn−1 are fixed.

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 27

IDEALS ASSOCIATED TO CIRCUITS

Polynomial Circuit Constraints A polynomial p is called a polynomial circuit constraint (PCC) for a circuit C if for every choice of

(a0,...,an−1,b0,...,bn−1) ∈ {0,1}2n

and resulting values g1,...,gk,s0,...,s2n−1 implied by the gates of C the substitution of these values into p gives zero.

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 28

IDEALS ASSOCIATED TO CIRCUITS

Polynomial Circuit Constraints A polynomial p is called a polynomial circuit constraint (PCC) for a circuit C if for every choice of

(a0,...,an−1,b0,...,bn−1) ∈ {0,1}2n

and resulting values g1,...,gk,s0,...,s2n−1 implied by the gates of C the substitution of these values into p gives zero. The set of all PCCs for C is denoted by I(C). I(C) is an ideal.

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 29

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

Examples for PCCs: p0 = s0 − a0b0

and gate

p1 = a2

1 − a1

a1 boolean

p2 = g2

2 − g2

g2 boolean

p3 = s1g4

xor-and constraint

...

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 30

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

Examples for PCCs: p0 = s0 − a0b0

and gate

p1 = a2

1 − a1

a1 boolean

p2 = g2

2 − g2

g2 boolean

p3 = s1g4

xor-and constraint

...

A circuit C is called a multiplier if

2n−1

∑

i=0

2isi −

n−1

∑

i=0

2iai

n−1

∑

i=0

2ibi

  • ∈

I(C).

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 31

IDEALS ASSOCIATED TO CIRCUITS

Problem: Definition of I(C) does not provide a basis

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 32

IDEALS ASSOCIATED TO CIRCUITS

Problem: Definition of I(C) does not provide a basis We can deduce at least some elements of I(C): G = {Gate Polynomials}∪{Field Polynomials for inputs} The ideal generated by G is denoted by J(C).

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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IDEALS ASSOCIATED TO CIRCUITS

Problem: Definition of I(C) does not provide a basis We can deduce at least some elements of I(C): G = {Gate Polynomials}∪{Field Polynomials for inputs} The ideal generated by G is denoted by J(C). Reverse topological order:

  • utput variable of a gate is greater than input variables

→Then G is a Gröbner basis for J(C).

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 34

IDEALS ASSOCIATED TO CIRCUITS

THEOREM

For all acyclic circuits C, we have J(C) = I(C).

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 35

IDEALS ASSOCIATED TO CIRCUITS

THEOREM

For all acyclic circuits C, we have J(C) = I(C). J(C) ⊆ I(C): corresponds to soundness I(C) ⊆ J(C): corresponds to completeness

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 36

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 37

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

−s3 + g1g4, −s2 + g1 + g4 − 2g1g4,

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 38

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

−s3 + g1g4, −s2 + g1 + g4 − 2g1g4, −g4 + g2g3, −s1 + g2 + g3 − 2g2g3,

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 39

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

−s3 + g1g4, −s2 + g1 + g4 − 2g1g4, −g4 + g2g3, −s1 + g2 + g3 − 2g2g3, −g1 + a1b1, −g2 + a0b1, −g3 + a1b0, −s0 + a0b0,

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 40

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

−s3 + g1g4, −s2 + g1 + g4 − 2g1g4, −g4 + g2g3, −s1 + g2 + g3 − 2g2g3, −g1 + a1b1, −g2 + a0b1, −g3 + a1b0, −s0 + a0b0, −a2

1 + a1,−a2 0 + a0,

−b2

1 + b1,−b2 0 + b0

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 41

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

−s3 + g1g4, −s2 + g1 + g4 − 2g1g4, −g4 + g2g3, −s1 + g2 + g3 − 2g2g3, −g1 + a1b1, −g2 + a0b1, −g3 + a1b0, −s0 + a0b0, −a2

1 + a1,−a2 0 + a0,

−b2

1 + b1,−b2 0 + b0

Order: s3 > s2 > g4 > s1 > g1 > g2 > g3 > s0 > a1 > a0 > b1 > b0

⇒ Generators of J(C) form a Gröbner basis

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 42

IDEALS ASSOCIATED TO CIRCUITS

a1 b1 a0 b1 a1 b0 a0 b0 g1 g2 g3 g4 s0 s1 s2 s3

J(C) =

−s3 + g1g4, −s2 + g1 + g4 − 2g1g4, −g4 + g2g3, −s1 + g2 + g3 − 2g2g3, −g1 + a1b1, −g2 + a0b1, −g3 + a1b0, −s0 + a0b0, −a2

1 + a1,−a2 0 + a0,

−b2

1 + b1,−b2 0 + b0

Order: s3 > s2 > g4 > s1 > g1 > g2 > g3 > s0 > a1 > a0 > b1 > b0

⇒ Generators of J(C) form a Gröbner basis

Question: 8s3 + 4s2 + 2s1 + s0 −(2a1 + a0)(2b1 + b0) ∈ J(C)?

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 43

COMPLEXITY

COROLLARY

Checking non-ideal membership over Q[x1,...,xn] even in terms of a given Gröbner basis is NP-hard.

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 44

COMPLEXITY

COROLLARY

Checking non-ideal membership over Q[x1,...,xn] even in terms of a given Gröbner basis is NP-hard. Connection between circuit SAT and ideal membership testing

known (circuit) SAT circuit ideal membership claim

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 45

COMPLEXITY

COROLLARY

Checking non-ideal membership over Q[x1,...,xn] even in terms of a given Gröbner basis is NP-hard. Connection between circuit SAT and ideal membership testing

known (circuit) SAT circuit ideal membership claim NP-complete SAT not constant

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 46

COMPLEXITY

COROLLARY

Checking non-ideal membership over Q[x1,...,xn] even in terms of a given Gröbner basis is NP-hard. Connection between circuit SAT and ideal membership testing

known (circuit) SAT circuit ideal membership claim NP-complete SAT not constant

→ x,x = 0

NP-hard

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 47

COMPLEXITY

COROLLARY

Checking non-ideal membership over Q[x1,...,xn] even in terms of a given Gröbner basis is NP-hard. Connection between circuit SAT and ideal membership testing

known (circuit) SAT circuit ideal membership claim NP-complete SAT not constant

→ x,x = 0

NP-hard Co-NP-complete UNSAT constant 0

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 48

COMPLEXITY

COROLLARY

Checking non-ideal membership over Q[x1,...,xn] even in terms of a given Gröbner basis is NP-hard. Connection between circuit SAT and ideal membership testing

known (circuit) SAT circuit ideal membership claim NP-complete SAT not constant

→ x,x = 0

NP-hard Co-NP-complete UNSAT constant 0

→ 0

Co-NP hard

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 49

COMPLEXITY

NP-hard transform circuit SAT problem into ideal non-membership testing preserves NP-hardness

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 50

COMPLEXITY

NP-hard transform circuit SAT problem into ideal non-membership testing preserves NP-hardness NP

  • pen question:

non-membership in NP (probably not) h in ideal

⇔

h = ∑pi ∗ gi for some pi (membership) h not in ideal

⇔

h = ∑pi ∗ gi for all pi (non-membership) sufficient condition for membership being in NP:

  • r equivalently non-membership in Co-NP

pi can be restricted to have polynomial size (in our situation) but then NP = Co-NP

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 51

CONCLUSION & FUTURE WORK

Conclusion simple and precise mathematical formulation complexity result: circuit verification using computer algebra is hard results part of an upcoming FMCAD’17 paper

with further experimental results and a novel column-wise incremental verification approach

Future Work modular multiplication (32× 32 → 32 multiplier) algebraic specification of other arithmetic operators algebraically verifying ring-properties upper bounds

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

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SLIDE 52

COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING

Daniela Ritirc, Armin Biere, Manuel Kauers

Johannes Kepler University Linz, Austria

SC-Square Workshop 2017

University of Kaiserslautern, Germany

  • 29. July 2017

DANIELA RITIRC COMPLEXITY OF CIRCUIT IDEAL MEMBERSHIP TESTING