Roderick Bloem Synthesis of Concurrency Constructs with UIFs 1

Synthesis of Concurrency

Constructs using Uninterpreted

FunctionsWork by: Georg Hofferek, Bettina Koenighofer

Roland Jiang, Ashutosh Gupta

Roderick Bloem

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 2

1. Problem

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 3

Concurrency: RSA in Software

RSA decryption: Compute

𝑐𝑑mod 𝑝𝑞 =crt( 𝑐𝑑mod 𝑝, 𝑐𝑑mod 𝑞)

T2::

1. mq = 𝒄𝒅𝐦𝐨𝐝 𝐪

2. finished2 = true

3. if !merged && finished1

4. merged = true

5. mp = 𝐜𝐫𝐭(𝒎𝒑,𝒎𝒒)

T1::

1. mp = 𝒄𝒅𝐦𝐨𝐝 𝐩

2. finished1 = true

3. if !merged && finished2

4. merged = true

5. mp = 𝐜𝐫𝐭(𝒎𝒑,𝒎𝒒)

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 4

T2::

1. mq = 𝒄𝒅𝐦𝐨𝐝 𝐪

2. finished2 = true

3. if !merged && finished1

4. merged = true

5. mp = 𝐜𝐫𝐭(𝒎𝒑,𝒎𝒒)

T1::

1. mp = 𝒄𝒅𝐦𝐨𝐝 𝐩

2. finished1 = true

3. if !merged && finished2

4. merged = true

5. mp = 𝐜𝐫𝐭(𝒎𝒑,𝒎𝒒)

Hard to program, easy to specify:

T1||T2 = T1;T2 T2;T1

Concurrency: RSA in Software

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 5

Complex system:

64-bit datapath

very complex arithmetic logic unit

IF DE EX MEM WB

MEM

REG

ALU

How do I pipeline that?

Concurrency: RSA in Software

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 6

IF DE EX MEM WB

MEM

REG

ALU

That’s trivial!

Concurrency: Pipelined Processor

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 7

IF DE EX MEM WB

MEM

REG

ALU

r1 := mem[1]r2 := r1 + r2

r1 = 1r2 = 2

r1 := mem[1] r1 := mem[1] r1 := mem[1] r1 := 15

mem[1] = 15

r1 = 15r2 = 2

r2 := r1 + r2

stall

15

15

r2 := 15 + 2 r2 := 17 r2 := 17

r1 = 15r2 = 17

forward

Concurrency: Pipelined Processor

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 8

Not so trivial!

IF DE EX MEM WB

MEM

REG

ALU

stall

forward

Concurrency: Pipelined Processor

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 9

=ISA

Pipeline

Concurrency: Pipelined Processor

Specification

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Easy & HardSpecification Implementation

sequential, data hard easy

parallel easy hard

perfect application area for synthesis:

construct synchronization automatically

Cf. ClarkeEmerson82; VechevYahavYorsh10,…

Focus today:

Uninterpreted functions for complex systems

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2. A Solution

Roderick Bloem Synthesis of Concurrency Constructs with UIFs 12

Uninterpreted Functions

Thread1()1. m1= 𝒄𝒅 𝒎𝒐𝒅 𝒑2. 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑1 = 𝑡𝑟𝑢𝑒3. 𝑰𝑭 !𝑚𝑒𝑟𝑔𝑒𝑑 𝑨𝑵𝑫 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑24. 𝑚𝑒𝑟𝑔𝑒𝑑 = 𝑡𝑟𝑢𝑒5. 𝑚1= 𝒄𝒓𝒕(𝒎𝟏, 𝒎𝟐)

Thread2()1. m2= 𝒄𝒅 𝒎𝒐𝒅 𝒒2. 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑2 = 𝑡𝑟𝑢𝑒3. 𝑰𝑭 !𝑚𝑒𝑟𝑔𝑒𝑑 𝑨𝑵𝑫 𝑓𝑖𝑛𝑖𝑠ℎ𝑒𝑑14. 𝑚𝑒𝑟𝑔𝑒𝑑 = 𝑡𝑟𝑢𝑒5. 𝑚1= 𝒄𝒓𝒕(𝒎𝟏, 𝒎𝟐)

Correctness is independent of

semantic of functions

Useuninterpreted

functions

𝒇𝟏(𝒄, 𝒅, 𝒑) 𝒇𝟏(𝒄, 𝒅, 𝒒)

𝒇𝟐(𝒎𝒑, 𝒎𝒒) 𝒇𝟐(𝒎𝒑, 𝒎𝒒)

Spec: 𝑇1 || 𝑇2 = 𝑇1; 𝑇2 ∪ 𝑇2; 𝑇1

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=ISA

Pipeline

Concurrency: Pipelined Processor

Specification

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Synthesis = Solving a Quantified Formula

Lock Synthesis

𝜓 = ∃ 𝑎. ∀ 𝑖𝑛, 𝑜𝑢𝑡. Φ

𝑎 fixes atomic sections

𝑎: Boolean

𝑖, 𝑜 : uninterpreted domain

Pipeline Controller

𝜓 = ∀𝑚𝑒𝑚, 𝑟𝑒𝑔 . ∃𝑠𝑡𝑎𝑙𝑙,𝑓𝑤𝑑 . ∀ 𝑚𝑒𝑚′, 𝑟𝑒𝑔′. Φ

𝑠𝑡𝑎𝑙𝑙, 𝑓𝑤𝑑 fix concurrency

𝑠𝑡𝑎𝑙𝑙, 𝑓𝑤𝑑: Boolean

𝑚𝑒𝑚, 𝑟𝑒𝑔 : uninterpreted

domain

Φ in Quantifier Free FOL with uninterpreted functions

Clearly decidable

Problem: Efficiently find certificates

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Options:

1. Reduce to propositional logic

2. Expand and use interpolation

3. Build a “QBF MT” solver

4. ?

Insert Real Work here

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Simple processors

Lock synthesis for programs without loops

And what we cannot do

Temporal aspects, loops

Combinations with other abstraction techniques?

More examples… Other logics?

What we can do

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Ras’ Ten Commandments

1. Extend the Language

2. Don’t reinvent the wheel

3. Just Say It

4. Don’t assume knowledge of spec language

5. Expect the Unexpected

6. Embrace Ambiguity

7. Interact

8. Invent a Calculus ~

9. Divide and conquer. Abstract and refine.

10. Talk to Ras