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# Lernmaterialien für Electronic System Synthesis an der Karlsruher Institut für Technologie

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TESTE DEIN WISSEN

Prime implicant table

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TESTE DEIN WISSEN

Determine minimal cover based on theorem that minimal cover is prime.

1. Create matrix with prime implicants as columns and minterms as rows
2. 1 if minterm is covered by prime implicant 0 else
3. Multiply with vector x that selects which prime implicants to use (1 if used 0 else)
4. Check that each value in resulting vector is at least 1
5. Try all permutations of x
6. Select x with minimum cardinality (least amount of prime implicants selected)

➔ Brute force approach trying all possible combinations

Lösung ausblenden
TESTE DEIN WISSEN

Metropolis criterium

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TESTE DEIN WISSEN

random(0,1) ≤ min(1, e...)

e^((-(c(s_new)-c(s))/T)

Due to non improvements being accepted with a certain probability has hillclimbing feature.

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TESTE DEIN WISSEN

Property of trees

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TESTE DEIN WISSEN
• No cycles
• All nodes connected
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TESTE DEIN WISSEN

Shannon-Tree minimization

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TESTE DEIN WISSEN
• Rule 1: Remove identical cubes in cofactors and add it to result
• Rule 2: Remove covered cube from cofactors and add it to result, leave covering cube in cofactors
Lösung ausblenden
TESTE DEIN WISSEN

LUT Based FPGA

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TESTE DEIN WISSEN

Too many possibilities for pattern matching. Goal: Minimize number of LUTs or minimum number of stages. Each LUT has k (typical 5) inputs and 1 output. As long as input and output count is kept it can contain arbitrary amount of gates.

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TESTE DEIN WISSEN

Implicant

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TESTE DEIN WISSEN

Cube that is part of the on and don’t-care set

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TESTE DEIN WISSEN

Minterm

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TESTE DEIN WISSEN

Term describing exactly one 1 in a logical function

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TESTE DEIN WISSEN

Minimum cover

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TESTE DEIN WISSEN

Cover with the minimal amount of implicants

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TESTE DEIN WISSEN

On-Set

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TESTE DEIN WISSEN

Set of 1s in a logical function

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TESTE DEIN WISSEN

Properties of BDD

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TESTE DEIN WISSEN
• Nodes represent decision variables, leaves the function value (0 or 1)
• Each node has to childs one for true, one for false case.
• Variables have to be in the same order in all paths
• Functions identical if BDDs are isomorphic
• Linear in time and memory for most function
• Size of BDD depends on arrangement of variables
Lösung ausblenden
TESTE DEIN WISSEN

Don’t-Care-Set

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TESTE DEIN WISSEN

Set of undefined values in a logical function

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TESTE DEIN WISSEN

Essential prime implicant

Lösung anzeigen
TESTE DEIN WISSEN

Includes a 1 that is not included in any other prime implicant

Lösung ausblenden
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Q:

Prime implicant table

A:

Determine minimal cover based on theorem that minimal cover is prime.

1. Create matrix with prime implicants as columns and minterms as rows
2. 1 if minterm is covered by prime implicant 0 else
3. Multiply with vector x that selects which prime implicants to use (1 if used 0 else)
4. Check that each value in resulting vector is at least 1
5. Try all permutations of x
6. Select x with minimum cardinality (least amount of prime implicants selected)

➔ Brute force approach trying all possible combinations

Q:

Metropolis criterium

A:

random(0,1) ≤ min(1, e...)

e^((-(c(s_new)-c(s))/T)

Due to non improvements being accepted with a certain probability has hillclimbing feature.

Q:

Property of trees

A:
• No cycles
• All nodes connected
Q:

Shannon-Tree minimization

A:
• Rule 1: Remove identical cubes in cofactors and add it to result
• Rule 2: Remove covered cube from cofactors and add it to result, leave covering cube in cofactors
Q:

LUT Based FPGA

A:

Too many possibilities for pattern matching. Goal: Minimize number of LUTs or minimum number of stages. Each LUT has k (typical 5) inputs and 1 output. As long as input and output count is kept it can contain arbitrary amount of gates.

Q:

Implicant

A:

Cube that is part of the on and don’t-care set

Q:

Minterm

A:

Term describing exactly one 1 in a logical function

Q:

Minimum cover

A:

Cover with the minimal amount of implicants

Q:

On-Set

A:

Set of 1s in a logical function

Q:

Properties of BDD

A:
• Nodes represent decision variables, leaves the function value (0 or 1)
• Each node has to childs one for true, one for false case.
• Variables have to be in the same order in all paths
• Functions identical if BDDs are isomorphic
• Linear in time and memory for most function
• Size of BDD depends on arrangement of variables
Q:

Don’t-Care-Set

A:

Set of undefined values in a logical function

Q:

Essential prime implicant

A:

Includes a 1 that is not included in any other prime implicant

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