1. Georg Hofferek, Ashutosh Gupta, Bettina Könighofer, Jie-Hong Roland Jiang and Roderick Bloem
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Institute for Applied Information Processing and Communications
Graz University of Technology, Austria

2. Motivation: Concurrency Issues
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Motivation: Concurrency Issues
Image Source:
Concurrency needs careful control!
Institute for Applied Information Processing and Communications

3. Example: Concurrent Execution
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Example: Concurrent Execution
Institute for Applied Information Processing and Communications

4. “Black” Boxes  Uninterpreted Functions
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Abstraction
( )
( )
“Black” Boxes  Uninterpreted Functions
Institute for Applied Information Processing and Communications

5. Application Example: Pipelined Processors
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Application Example: Pipelined Processors
Basic elements are the same
use uninterpreted
functions =
Burch & Dill Paradigm
(see Hofferek, Bloem, MemoCODE’11)

6. Overview Motivation: Pipeline Controller
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Overview
Motivation: Pipeline Controller
Synthesis Problem  Quantified Formulas
Interpolation
Single
Multiple
Proof Requirements
Colorable
Local-First
Proof Transformations
Results & Conclusion
Institute for Applied Information Processing and Communications

7. Synthesis Problem  Quantified Formulas
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Synthesis Problem  Quantified Formulas
Ψ=∀ 𝑚𝑒𝑚, 𝑟𝑒𝑔, 𝑝𝑖𝑝𝑒𝑙𝑖𝑛𝑒𝑠𝑡𝑎𝑡𝑒 .
∃ 𝑠𝑡𝑎𝑙𝑙, 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 .
∀ 𝑚𝑒 𝑚 ′ , 𝑟𝑒 𝑔 ′ , 𝑝𝑖𝑝𝑒𝑙𝑖𝑛𝑒𝑠𝑡𝑎𝑡 𝑒 ′ Φ
stall, forward: Boolean control signals
mem, reg, pipelinestate: Uninterpreted domain
Compute Certificates: 𝑠𝑡𝑎𝑙𝑙, 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 =𝒇(𝑚𝑒𝑚, 𝑟𝑒𝑔, 𝑝𝑖𝑝𝑒𝑙𝑖𝑛𝑒𝑠𝑡𝑎𝑡𝑒)
Institute for Applied Information Processing and Communications

8. Craig Interpolant 𝐶𝑁𝐹 Φ = 𝐶 1 ∧ 𝐶 2 ∧ 𝐶 3 ∧…∧ 𝐶 𝑛−1 ∧ 𝐶 𝑛 = ⊥
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Craig Interpolant 9
𝐶𝑁𝐹 Φ = 𝐶 1 ∧ 𝐶 2 ∧ 𝐶 3 ∧…∧ 𝐶 𝑛−1 ∧ 𝐶 𝑛 = ⊥
Interpolant 𝑰:
𝐴→𝐼
𝐼→¬𝐵, in other words: 𝐼∧𝐵= ⊥
𝑉 𝐼 ⊆𝑉 𝐴 ∩ 𝑉(𝐵) A B B I A

9. Expanding Formula for Single Interpolation
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Expanding Formula for Single Interpolation 10
∀ 𝑎 ∃𝑐 ∀ 𝑏 . Φ 𝑎 ,𝑐, 𝑏 is valid
¬Φ 𝑎 , 0, 𝑏 0 ∧¬Φ 𝑎 ,1, 𝑏 is unsatisfiable
Expansion of ∃
Renaming of 𝑏
Negation

10. Certificate via Interpolation
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Certificate via Interpolation 11
¬Φ 𝑎 ,0, 𝑏 0 ∧¬Φ 𝑎 ,1, 𝑏 1 = ⊥
Interpolant 𝑰 𝑎 :
¬Φ 𝑎 , 0, 𝑏 0 →𝐼
𝐼 is 1, whenever 0 not allowed
𝐼→Φ 𝑎 , 1, 𝑏 1
Whenever 𝐼 is 1, 1 is allowed A B
0 not allowed
1 not allowed
Boolean Case: see Jiang et al., ICCAD’09

11. Multiple Control Signals
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Multiple Control Signals 12
Interdependence!
e.g. two signals must have same value
Iterative Resubstitution
Many SMT calls
Increasing “difficulty”
Multiple Coordinated Interpolants
 Only one proof required
 Special requirements towards proof

12. Expansion for Multiple Interpolants
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Expansion for Multiple Interpolants 13
Formula: ∀ 𝑎 ∃ 𝑐 0 , 𝑐 1 ∀ 𝑏 . Φ 𝑎 , 𝑏 , 𝑐 0 , 𝑐 1 =⊤
Expansion:
¬Φ 𝑎 , 𝑏 00 ,0,0 ∧ ¬Φ 𝑎 , 𝑏 10 ,1,0 ∧
¬Φ 𝑎 , 𝑏 01 ,0,1 ∧ ¬Φ 𝑎 , 𝑏 11 ,1,1 = ⊥
“Partitions”: 𝜙 00 , 𝜙 01 , 𝜙 10 , 𝜙 11

13. Definitions: Colorable, Local, Global
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Definitions: Colorable, Local, Global 14
Partitions ≈ Colors:
¬ Φ 00 𝑎 , 𝑏 00 ∧¬ Φ 10 𝑎 , 𝑏 10 ∧¬ Φ 01 𝑎 , 𝑏 01 ∧¬ Φ 11 𝑎 , 𝑏 11
Local Symbols: 𝑏 00 , 𝑏 10 , 𝑏 01 , 𝑏 11 (colored)
Global Symbols: 𝑎 („colorless“)
Colorable: 𝒙=𝒚 , 𝒖=𝒗 , 𝒘=𝒛
Non-colorable: 𝒙=𝒖
Generalization of notions for single interpolation

14. (Reasonable) Assumptions on Proofs
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
(Reasonable) Assumptions on Proofs 15
Pure Resolution Proofs
All internal nodes are resolution nodes
Theory reasoning via tautology clauses
E.g. Transitivity: (𝑎≠𝑏∨𝑏≠𝑐∨𝑎=𝑐)
Leaves:
Clause from one partition
Theory tautology
Root: ⊥
New Literals:
Defined via theory by “existing” ones
veriT Solver [

15. Requirements towards Proof
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Requirements towards Proof 16
Colorability
“No literals or leaves with symbols from two partitions”
Achieved in two steps
Remove non-colorable literals
Split non-colorable leaves
Local-first
“Local literals are resolved before global literals”
Achieved by standard reordering

16. Removing non-colorable Literals
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Removing non-colorable Literals 17
𝑎≠𝑏 ⇒ 𝑎≠𝑥 ∨ 𝑥≠𝑏
Tautology (Transitivity),
“defining” 𝑎≠𝑏
Tautology (Transitivity),
“using” 𝑎≠𝑏
𝑎≠𝑥 ∨ 𝑥≠𝑏 ∨ 𝑎=𝑏
𝑎≠𝑏 ∨ 𝑏≠𝑦 ∨ 𝑎=𝑦
Replace

17. Split Non-Colorable Leaves
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Split Non-Colorable Leaves 18
𝑎≠𝑥 ∨ 𝑥≠𝑦 ∨ 𝑎=𝑦
𝑥≠𝑏 ∨ 𝑏≠𝑦 ∨ 𝑥=𝑦
𝑎≠𝑥 ∨ 𝑥≠𝑏 ∨ 𝑏≠𝑦 ∨ 𝑎=𝑦 𝑥 𝑎 𝑏 𝑦

18. Making Proof Local-First
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Making Proof Local-First 19
Standard Pivot Reordering Techniques
e.g. D’Silva, Kroening, Purandare, and Weissenbacher, VMCAI 2010

19. Computing Multiple Interpolants
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Computing Multiple Interpolants 20
¬𝜑 𝑎 , 𝑏 0 ,0,0 ∧ ¬𝜑 𝑎 , 𝑏 1 ,1,0 ∧¬𝜑 𝑎 , 𝑏 2 ,0,1 ∧ ¬𝜑 𝑎 , 𝑏 3 ,1,1
Local Literals 0
Local Literals 1
Local Literals 2
Local Literals 3
Global Literals ⊥
𝐼 0 𝐼 1

20. Computing Multiple Interpolants
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Computing Multiple Interpolants 21
¬𝜑 𝑎 , 𝑏 0 ,0,0 ∧ ¬𝜑 𝑎 , 𝑏 1 ,1,0 ∧¬𝜑 𝑎 , 𝑏 2 ,0,1 ∧ ¬𝜑 𝑎 , 𝑏 3 ,1,1
Constants
0, 0
Constants
1, 0
Constants
0, 1
Constants
1, 1
Multiplexer
Cf. Pudlaks’ Interpolation Procedure (JSL’97) ⊥
𝐼 0 𝐼 1

21. Experimental Results pipe: Illustrative pipeline example (MemoCODE’11)
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Experimental Results 22
pipe: Illustrative pipeline example (MemoCODE’11)
1.6 seconds instead of 14 hours
proc: Simple 2-stage pipelined processor
2 control signals
28.1 seconds
illu02-08: Scalable illustrative example
2-8 control signals
Mutual interdependence

22. Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Scalability: illu02-08
08: 1270s

23. Conclusion Multiple Coordinated Interpolants Uninterpreted Functions
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Conclusion 24
Multiple Coordinated Interpolants
just one proof
Uninterpreted Functions
Good abstraction
Concurrency issues
Full potential unleashed
No reductions to propositional logic
Improvement: Several orders of magnitude
Future work
Colorable and/or local-first proofs from SMT solver
More theories (e.g. linear arithmetic)

25. Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Appendix
Detailed slide on some issues that were left out of the main presentation for time reasons
Institute for Applied Information Processing and Communications
11/21/2018

26. Resubstitution Expanding 𝑐 0 only:
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Resubstitution
Expanding 𝑐 0 only:
∀ 𝑎 ∃ 𝑐 1 ∀ 𝑏 . Φ 𝑎 , 𝑏 ,0, 𝑐 1 ∨ ∃ 𝑐 1 ∀ 𝑏 . Φ 𝑎 , 𝑏 ,1, 𝑐 1
 Still (mixed) quantifiers
Same full expansion required for first interpolation:
¬Φ 𝑎 , 𝑏 00 ,0,0 ∧¬Φ 𝑎 , 𝑏 01 ,0,1 ∧¬Φ 𝑎 , 𝑏 10 ,1,0 ∧ ¬Φ 𝑎 , 𝑏 11 ,1,1 A B
Institute for Applied Information Processing and Communications

27. A Processor IF DE EX MEM WB REG Tough: 64-bit datapath
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
A Processor 28 IF DE EX
MEM WB
REG
ALU
How do I pipeline that?
Tough:
64-bit datapath
very complex arithmetic logic unit

28. A Pipelined Processor REG MEM IF DE EX MEM WB ALU That’s trivial!
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
A Pipelined Processor 29
REG
MEM
ALU IF DE EX
MEM WB
That’s trivial!

29. A Pipelined Processor REG MEM IF DE EX MEM WB ALU r1 = 15 r2 = 2
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
A Pipelined Processor 30
r1 = 15
r2 = 2
r1 = 1
r2 = 2
r1 = 15
r2 = 17
Instructions:
r1 := mem[1]
r2 := r1 + r2
REG 15
MEM
mem[1] = 15
ALU IF DE EX
MEM WB
r1 := mem[1]
r1 := mem[1]
r1 := mem[1]
r1 := 15
r2 := r1 + r2
r2 :=
r2 := 17
r2 := 17 15
stall
forward

30. A Pipelined Processor IF DE EX MEM WB REG
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
A Pipelined Processor 31 IF DE EX
MEM WB
REG
ALU
stall
forward
Not so trivial!
 Hard to implement  Hard to test  Easy to specify

31. Sufficient Condition: Commutative Diagram
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Sufficient Condition: Commutative Diagram 32
Burch & Dill, for verification
instr1
instr2
instr3
non-
pipelined =
flush
flush =
instr1
instr2
instr3
pipelined
flushed
flushed
Pipelined and non-pipelined processor give same result for any instruction sequence

32. Commutative Diagram in Logic
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Commutative Diagram in Logic 33
Burch & Dill, for verification
instr
non-
pipelined
flush
flush EX
ALU
instr
pipelined
 =
(mem’,reg’) = flush  non-pipe-instr (mem,reg) 
(mem’’,reg’’) = pipe-instr  flush (mem,reg)
 (mem’,reg’) = (mem’’,reg’’)
Pipeline correct iff  valid.

33. Commutative Diagram in Logic
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Commutative Diagram in Logic 34
Burch & Dill, for verification
instr
non-
pipelined
flush
flush EX
ALU
instr
pipelined
 written in logic with uninterpreted functions, arrays, and equality
Part of :
res_ex = ALU(opc_de, arg1_de, arg2_de)

34. Removing non-colorable Literals
Synthesizing Multiple Boolean Functions using Interpolation on a Single Proof
Removing non-colorable Literals 35
𝑎≠𝑏 ⇒ 𝑎≠𝑥 ∨ 𝑥≠𝑏
Tautology (Transitivity)
Tautology (Transitivity)
𝑎≠𝑥 ∨ 𝑥≠𝑏 ∨ 𝑎=𝑏
𝑎≠𝑏 ∨ 𝑏≠𝑦 ∨ 𝑎=𝑦
Replace
𝑎≠𝑥 ∨ 𝑥≠𝑏 ∨ 𝑏≠𝑦 ∨ 𝑎=𝑦