Interpolation. Proof Reduction

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1 Interpolation and Proof Reduction Georg Weissenbacher & R. Bloem, S. Malik, M. Schlaipfer EXCAPE 2014

2 Motivation: Circuit Synthesis designing ICs is hard G. Weissenbacher (TU Wien) Interpolation and Proof Reduction E X C APE / 15

3 Motivation: Circuit Synthesis designing ICs is hard what if we could leave holes (black boxes)?? G. Weissenbacher (TU Wien)? Interpolation and Proof Reduction E X C APE / 15

4 Motivation: Circuit Synthesis designing ICs is hard what if we could leave holes (black boxes)? specify behaviour using input/output relation? G. Weissenbacher (TU Wien)? Interpolation and Proof Reduction E X C APE / 15

5 Motivation: Circuit Synthesis designing ICs is hard what if we could leave holes (black boxes)? specify behaviour using input/output relation? G. Weissenbacher (TU Wien)? then let a program fill in the blanks Interpolation and Proof Reduction E X C APE / 15

6 Synthesis from Relational Specifications Jiang, Lin, Hung ICCAD 09 Given: quantifier-free (propositional) specification R : Inputs Output B Wanted: functional implementation I : Inputs B G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

7 Synthesis from Relational Specifications Jiang, Lin, Hung ICCAD 09 Given: quantifier-free (propositional) specification R : Inputs Output B Wanted: functional implementation I : Inputs B same setting as in logic synthesis onset offset G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

8 Synthesis from Relational Specifications Jiang, Lin, Hung ICCAD 09 Given: quantifier-free (propositional) specification R : Inputs Output B Wanted: functional implementation I : Inputs B same setting as in logic synthesis Hofferek, Bloem MEMOCODE 11; Hofferek et al. FMCAD 13 Given: circuit specification with parameter c: i : Inputs c : B o : Outputs. R( i, c, o) Wanted: functional implementation I : Inputs B G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

9 Synthesis from Relational Specifications Jiang, Lin, Hung ICCAD 09 Given: quantifier-free (propositional) specification R : Inputs Output B Wanted: functional implementation I : Inputs B same setting as in logic synthesis Hofferek, Bloem MEMOCODE 11; Hofferek et al. FMCAD 13 Given: circuit specification with parameter c: i : Inputs c : B o : Outputs. R( i, c, o) Wanted: functional implementation I : Inputs B by reduction to [Jiang, Lin, Hung ICCAD 09] more than one parameter by co-factoring, iterative solving G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

10 Interpolants in Synthesis onset offset G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

11 Interpolants in Synthesis Interpolant separates onset from offset interpolant onset only offset only G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

12 What is a Craig-Robinson Interpolant? Craig-Robinson interpolant for inconsistent (first-order) conjunction A B: A I and I B inconsistent all non-logical symbols in I occur in A as well as in B B I A G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

13 Interpolants and Synthesis R( i, 0) }{{} onset only R( i, 1) }{{} offset only G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

14 Interpolants and Synthesis R( i, 0) }{{} onset only R( i, 1) }{{} offset only G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

15 Interpolants and Synthesis R( i, 0) }{{} onset only R( i, 1) }{{} offset only G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

16 State of the Art: Interpolants from Refutation Proofs A B G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

17 State of the Art: Interpolants from Refutation Proofs A B A 0 C 1 B 1 A 1 C 2 G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

18 State of the Art: Interpolants from Refutation Proofs A B A 0 C 1 [I 1 ] B 1 A 1 C 2 [I 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

19 State of the Art: Interpolants from Refutation Proofs A B A 0 C 1 [I 1 ] B 1 A 1 C 2 [I 2 ] [I 1 I 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

20 Interpolation Systems map proof nodes to partial interpolants root node is an interpolant for A B Intuition: eliminate local literals (, ) as in proof keep shared literals ( ) G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

21 Interpolation Systems map proof nodes to partial interpolants root node is an interpolant for A B Intuition: eliminate local literals (, ) as in proof keep shared literals ( ) In synthesis setting, all inputs are shared Symbols in interpolant determined by unsatisfiable core G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

22 Labelled Interpolation Systems [D Silva et al. VMCAI 10] conceived to fine-tune logical strength of interpolants G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

23 Labelled Interpolation Systems [D Silva et al. VMCAI 10] conceived to fine-tune logical strength of interpolants literals can be labelled individually (,, ) allows us to treat literals as if they were local [D Silva ESOP 10] A(i, j) B(i, j) G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

24 Labelled Interpolation Systems [D Silva et al. VMCAI 10] conceived to fine-tune logical strength of interpolants literals can be labelled individually (,, ) allows us to treat literals as if they were local [D Silva ESOP 10] A(i, j) B(i, j) i. A(i, j) i. B(i, j) G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

25 Labelled Interpolation Systems [D Silva et al. VMCAI 10] conceived to fine-tune logical strength of interpolants literals can be labelled individually (,, ) allows us to treat literals as if they were local [D Silva ESOP 10] A(i, j) B(i, j) A(j) B(j) [I(j)] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

26 Labelled Interpolation: Example x 0 x 1 [0] x 1 x 2 [0] x 0 x 2 [0] x 1 x 2 [1] x 0 x 1 [x 2 ] x 1 [1] x 0 [0] x 0 [x 2 ] [x 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

27 Labelled Interpolation: Example x 0 x 1 [0] x 1 x 2 [0] x 0 x 2 [0] x 1 x 2 [1] x 2 : x 0 x 1 [x 2 ] x 1 [1] x 0 [0] x 0 [x 2 ] [x 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

28 Labelled Interpolation: Example x 0 x 1 [0] x 1 x 2 [0] x 0 x 2 [0] x 1 x 2 [1] x 2 : x 0 x 1 [x 2 ] x 1 [1] x 0 [0] x 0 [x 2 ] [x 2 ] Literals in interpolant dictated by proof (core and structure) G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

29 Smaller Interpolants via Proof Reduction? Numerous techniques for reduction of proof size: Simone et al. HVC 10; Bar-Ilan et al. STTT 11; Gupta ATVA 12;... Most can be viewed as (generalised) clause sub-sumption G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

30 Smaller Interpolants via Proof Reduction? x 0 x 1 [0] x 1 x 2 [0] x 0 x 2 [0] x 1 x 2 [1] x 0 x 1 [x 2 ] x 1 [1] x 0 [0] x 0 [x 2 ] [x 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

31 Smaller Interpolants via Proof Reduction? x 0 x 1 [0] x 1 x 2 [0] x 1 x 0 x 2 [0] x 1 x 2 [1] x 0 x 1 [x 2 ] x 1 [1] x 0 [0] x 0 [x 2 ] [x 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

32 Smaller Interpolants via Proof Reduction? x 0 x 1 [0] x 1 x 2 [0] x 1 x 2 [0] x 1 x 2 [1] x 1 [x 2 ] x 1 [1] x 0 [0] [x 1 x 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

33 Smaller Interpolants via Proof Reduction? x 0 x 1 [0] x 1 x 2 [0] x 1 x 2 [0] x 1 x 2 [1] x 1 : x 1 [x 2 ] x 1 [1] x 0 [0] [x 1 x 2 ] Transformation made -local literal shared ( ) G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

34 Smaller Interpolants via Proof Reduction x 0 x 1 [0] x 1 x 2 [0] x 1 x 0 x 2 [0] x 1 x 2 [1] x 0 x 1 [x 2 ] x 1 [1] x 0 [0] x 0 [x 2 ] [x 2 ] G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

35 Smaller Interpolants via Proof Reduction Our current work: Meet-over-all-paths analysis of refutation (linear time) Identifies constraints for sub-sumption avoids reduction in previous example 1%-5% reduction of # variables (HWMCC; SATC/MUS benchmarks) 10%-20% reduction of proof size G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

36 Smaller Interpolants via Proof Reduction Our current work: Meet-over-all-paths analysis of refutation (linear time) Identifies constraints for sub-sumption avoids reduction in previous example 1%-5% reduction of # variables (HWMCC; SATC/MUS benchmarks) 10%-20% reduction of proof size Future work: More aggressive static analysis of proofs Heuristics for identifying beneficial sub-sumptions Eliminate specific (user-defined) symbols Combine with recent MUS-techniques (Marques-Silva s work) G. Weissenbacher (TU Wien) Interpolation and Proof Reduction EXCAPE / 15

37 More on Proofs and Interpolation SAT Intel Association Application deadline: April 19 ( Generous NSF travel grants for US students G. Weissenbacher (TU Wien) Interpolation and Proof Reduction E X C APE / 15

Propositional Logic: Models and Proofs

Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505