Rippling: Heuristic Guidance for Inductive Proof (I) Alan Bundy. Automated Reasoning Rippling

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1 Automated Reasoning Rippling: Heuristic Guidance for Inductive Proof (I) Alan Bundy Lecture 16, page 1

2 Overview In lecture 14 we introduced induction Inductive proofs can be difficult. Problems are in: controlling rewriting in step case proofs selecting induction rules using lemmas and generalisations. In this lecture we will look at a heuristic known as rippling which can help automatically guide inductive proofs by addressing the 3 problems above. Lecture 16, page 2

3 Rippling Rippling term coined by Aubin, who noticed a pattern in the rewriting of step cases Bundy proposed rippling could be used as a heuristic to guide inductive proof Rippling gives a direction to rewriting process Central principle: aim to have IH embedded within the IC by moving differences between induction hypothesis (IH) and induction conclusion (IC) can then use strong fertilization to solve the goal We emphasise the effect by annotating the differences Lecture 16, page 3

4 Annotated Rewrite Rules H # L H # L (1) X 1 # X 2 = Y 1 # Y 2 X 1 = Y 1 X 2 = Y 2 (2) called wave-rules changing bits in orange boxes (wave-fronts) note outward movement of wave-fronts unchanging bits in red (skeleton) Lecture 16, page 4

5 Terminology (X # Z Erasure (X# Y )@ Z (X#.. ) Wave-front Y Wave-hole Z Skeleton Difference removal uses specialised annotations: Wave fronts: difference between skeleton and goal. Wave holes: identify sub-terms inside the wave front that are similar to parts of the given. Lecture 16, page 5

6 ... Rippling: Preserving Common Structure Induction Rule: P(nil) h:τ. t:list(τ). (P(t) P( h# t )) l:list(τ). P(l) Base Case: P(nil) Step Case: P(t) P( ) h# t... k( P(t) ) Progress Terminating and Preserving common structure Lecture 16, page 6

7 Another View @ z z h# t y t y t y ( h# z h# y) z h# z Before During After Lecture 16, page 7

8 Identifying Differences Mismatching Terms Annotated Goal (IC) Given (IH) Goal (X # Z Z even (s(s(x)) even(x) len(h # T) len(t) H # T) T) (X# Y Z even( s(s( X )) ) len( H # T ) H # T ) Lecture 16, page 8

9 Reducing Differences Examples: even( s(s( X )) ) even(x) (3) len( H # T ) s( len(t) ) (4) H # T ) s( T) ) (5) rot( s( N), H # T ) rot( N, (H # nil) ) (6) Definition requirements: identical skeleton on lhs and rhs terminating based on logically valid rewrite rule Sources for wave-rules: recursive definitions, associative laws, replacement laws, etc. automatically annotated Lecture 16, page 9

10 Making Wave Rules Wave rules can be made from: theorems, lemmas, definitions, equations, etc. There may be many ways to create a wave-rule, e.g: s(x) + Y = s(x + Y) could be transformed into the wave-rules: s( X ) + Y s( X + Y ) s( X + Y ) s(x) + Y Note bidirectionality of rewriting with wave-rules. does not cause non-termination. Lecture 16, page 10

11 Exhaustive Rewriting (without Rippling) Rewrite Rule: (X + Y) + Z X + (Y + Z) Given: a + b = 42 Goal: ((c + d) + a) + b = (c + d) + 42 Rewritings: ((c + d) + a) + b = c + (d + 42) (c + (d + a)) + b = (c + d) + 42 (c + d) + (a +b) = (c + d) + 42 Lecture 16, page 11

12 Selective Rewriting (with Rippling) Wave Rule: (X + Y ) + Z (X +( Y + Z ) Given: a + b = 42 Goal: ((c + d) + ) + b = Ripple: (c + d) + ( ) = Non-Ripples: a (c + d) + 42 a + b (c + d) + 42 ((c + d) + a) c + (d + a ) + b = + b = c + (d + 42 ) (c + d) + 42 Lecture 16, page 12

13 Termination of Rippling To provide rippling with a direction and ensure termination, a measure is used that decreases each time the goal is rewritten Measure is a list of natural numbers the list indicates the number of wave-fronts at each depth in the skeleton term the number of wavefronts is counted from leaf to root Decreasing measure means rewrite expresses a valid ripple step Lecture 16, page 13

14 Wave Measure Definition of wave 0 z 1 0 t y 0 0 Decrease of The wave measure of h# z can be written as a list: [0,1,0] most significant z z h# t y t y [1,0,0] > [0,1,0] > [0,0,1] Lecture 16, page 14 t y

15 Annotated Step Case Z) = Z h# z) = ( h# z by (1) 2 h# z) = h# z by (1) h# z) = h# z by (2) h = h z) = z Fertilization is possible when the wave fronts have been removed from a sub-term matching the IH. Lecture 16, page 15

16 Extending Rippling? Conjecture: x, y. P(x, y) Step Case: P(x,Y) P(, y) s( ) x... P(x,Y) P(, y) s( ) x... k( P(x,y) ) k'( P(x,q(y)) ) Fertilize with Y=y Fertilize with Y=q(y) As long as the skeleton is an instance of the hypothesis, not necessarily an exact copy We can exploit this in our proofs Lecture 16, page 16

17 Always Ripple Out? Rippling wave-fronts outside the skeleton is only one way to enable fertilization To exploit additional universal variables in the conjecture we make them free variables in the IH and arbitrary constants in the IC We call the positions of the arbitrary constants sinks We move the wave-fronts to surround the sinks The wave-fronts at the sinks are then absorbed by the free variables during fertilization To do this, we need to be able to ripple sideways and in! Lecture 16, page 17

18 Updated Rippling Sinks provide alternative wave-front destination, available when free variables are in hypothesis Wave fronts have directions: out in Rippling out tries to remove difference or to move it to the top of the term tree. Eventually strong fertilization or weak fertilization in a sub-term occurs. Rippling in tries to move a difference into a sink which would allow it to be matched by the corresponding universally quantified variable. s( N) In black and white notation write: as s( N) Lecture 16, page 18 as s( N) s( N)

19 Termination of Rippling (II) Our definition of a wave measure needs an update to take into account rippling sideways. Our measure is: a pair of lists of natural numbers that indicate the number of wave fronts outward followed by inward - at each depth of the skeleton term Outward list is formed by counting the number of outward wavefronts from leaf to root Inward list is formed by counting the number of inward wavefronts from root to leaf Measures are compared entry-wise using a lexicographical ordering as if they were a single list starting with the outward elements This restriction allows wave fronts to move from out to in but not vice-versa. Lecture 16, page 19

20 Sideways Rippling: Another View φ φ φ φ η η η η μ ν c 2 ( μ ) ν μ c 3 ( ν ) μ ν c 1 ( x ) y x y x y x c 4 ( y ) Out: [1, 0, 0, 0] Out: [0, 1, 0, 0] Out: [0, 0, 0, 0] Out: [0, 0, 0, 0] In: [0, 0, 0, 0] > In: [0, 0, 0, 0] > In: [0, 0, 1, 0] > In: [0, 0, 0, 1] Before Stage-1 Stage-2 After Lecture 16, page 20

21 Directed Wave-rules rot( s( N), H # T ) rot( N, (H # nil) ) (7) ( H # T ) (H # T (8) ( Y Z ( Z ) (9) Definition of wave-rules (revised): identical skeleton on lhs and rhs can go from to or from to can go from to but not vice-versa matching is as before, but additionally, arrows must match Lecture 16, page 21

22 Rippling Sideways and In rot( s( N), H # T ) rot( N, (H # nil) ) (10) ( H # T ) (H # T (11) ( Y Z ( Y@ Z ) (12) rot(len(t), K) = t rot(len( h# t ), h# k ) = ( h# t ) (4)+(1) rot( s( len(t) ), h# k ) = ( h# t ) (10) rot(len(t), (h # nil) ) = ( h# t ) (12)+(11) rot(len(t), (h # nil) ) = (h # t rot(len(t), (h # nil) ) = (h t Lecture 16, page 22

23 The Preconditions of Rippling 1. The induction conclusion contains a wave-front e.g. rot( s( len(t) ), h# k ) = A wave-rule applies to this wave-front e.g. rot( s( N), H # T ) Any condition is provable e.g. X H X H # T X T 1. Inserted inwards wave-fronts contain a sink or an outwards wave-front e.g. rot(len(t), (h # nil) ) =... Lecture 16, page 23

24 Advantages of Rippling Selective: not exhaustive rewriting. Bi-directional: rewriting. Termination: of any set of wave-rules, despite bidirectionality. Heuristic basis: for choosing lemmas, generalisations and inductions. Lecture 16, page 24

25 A Proof Plan for Induction Induction Strategy Induction Base Symbolic Evaluation Step Ripple Wave Fertilization Preconditions Declarative: Rippling must be possible in step cases. Procedural: Look-ahead to choose induction rule that will permit rippling. Lecture 16, page 25

26 Summary Inductive proof introduces new search problems. But also new opportunities have IH. Move differences to make IH match IC. Proof plan for induction based on rippling. Describes common pattern of proof. Rippling: selective; bidirectional; terminating and offers heuristic solution to special problems (next lecture). Lecture 16, page 26