1. Symmetry as a Prelude to Implied Constraints Alan Frisch, Ian Miguel, Toby Walsh University of York

2. Overview Motivation Example: The Fractions Puzzle Proof Planning Example: The Steel Mill Conclusions

3. Why Do We Care About Symmetry? Important for generating implied constraints: –Some ICs can only be generated as a result of breaking symmetry first. Symmetry-breaking a powerful means of reducing search in its own right.

4. The Fractions Puzzle Find 9 distinct non-zero digits, A to I, satisfying: Note: BC is short for 10B + C. Before we break symmetry, we can say very little about this.

5. The Fractions Puzzle (2) The problem contains 3 symmetrical terms. Break symmetry with ordering constraints:

6. Given: Elimination Eliminate then in favour of. Giving: Then:

7. Similarly: Elimination(2) and are both ternary. Original constraint has arity 9. We can go further to get unary constraints such as: G  4

8. Proof Planning Used to guide search for proof in ATP. Patterns in proofs are encapsulated in methods. –Strong preconditions limit applicability. –Prevent combinatorially explosive search.

9. Proof Planning (2) Given a goal to prove: –Select a method. –Check pre-conditions. –Execute post-conditions. –Construct output goal(s). Associated tactic constructs actual proof.

10. Proof Planning for Implied Constraints Easily adaptable to generating logical consequences. Methods encapsulate common patterns in hand-generating implied constraints. Forward chaining rather than goal-directed.

11. Symmetry Method Identify variables/terms which are indistinguishable –Swap variable(s) and compare normalised set of equations. Order indistinguishable variables/terms. –If x, y indistinguishable, impose x  y. Then use other methods to infer implied constraints.

12. Normalisation Lexicographic order on variable names. Constants ordered before variables. Non-atomic subterms compared based on the least variable in each. Becomes:

13. Identifying symmetries is potentially expensive. Heuristics based on structural equivalence. –Identical when explicit variable names replaced by a common `marker’. Becomes: Swap corresponding pairs throughout and test for indistinguishability. Symmetry Method (2)

14. The Slab Design Problem The mill can make  different slab sizes. Given d input orders with: –A colour (route through the mill). –A weight. Pack orders onto slabs, minimising total slab capacity. Constraints: –Capacity: Total weight of orders assigned to a slab cannot exceed slab capacity. –Colour: Each slab can contain at most p of k total colours.

15. An Example 2 3 11111 2 1 Slab Sizes: {1, 3, 4} (  = 3) Orders: {o a, …, o i } (d = 9) Colours: {red, green, blue, orange, brown} (k = 5) p = 2 abcdefghi 2 3 11 1 1 1 1 2 Solution:

16. A Redundant Variable Model Number of slabs is not fixed. –Assume highest order weight does not exceed maximum slab size. Slab variables: {s 1, …, s d }. –Value is size of slab. Solution quality:

17. Slab Variable Symmetry Slab variables are indistinguishable. –Counteract with binary symmetry-breaking constraints: s 1  s 2, s 2  s 3, etc. Existing symmetry method can cope with this. Leads to new implied constraints, e.g. s 1  max(orderWeights)

18. Order Assignment oaoa obob ococ odod s1s1 0011 s2s2 0100 s3s3 1000 s4s4 0000 Rows where slab variables assigned same size are indistinguishable. When s i = s i+1 : Corresponding rows of order A are lexicographically ordered. E.g. 1001  0110.

19. A New Symmetry Method? Breaking this type of symmetry is currently beyond our proof planner. Improvements: –Recognise that symmetries can occur based on partial assignments. –Add symmetry-breaking constraints as implications with partial assignments as preconditions.

20. A New Symmetry Method(2) LHS of: are symmetrical if s 1 = s 2. So: s 1 = s 2  order A [i, 1]  lex order A [i, 2] (since the weight(o i ) are constant).

21. Conclusions Symmetry-breaking is an important first step in generating implied constraints. We can already detect and break some symmetry within the proof planner. Next step is to try and remove (simple) symmetries that might appear during search.