THE WAY OF THE COMBINATOR

Bill Worzel billwzel@gmail.comEvolution Enterprises http://evolver.biz

GP Theory and Practice 08 May 2014Ann Arbor, MI

THE SKGP

• 15 years ago we developed what was then a novel approach to GP using combinators

• Strongly typed, efficient, powerful, reusable code

• Algorithm can become superlinear in parallel application because of code reuse

COMBINATORS• Applicative algebra, derived from Lambda calculus,

binds left-to-right

• Sxyz -> xz(yz)

• Kxy -> x

• Ix -> x

• Bxyz -> x(yz)

• Cxyz -> xzy

• Yx -> x(Yx)Combinators Functions Quickly

VARIABLE ABSTRACTION

•D.A. Turner showed that removing bound variables using combinators could produce an efficient computing system (Turner 1979, A New Implementation Technique for Applicative Languages, Software–Practice and Experience, vol 9, 31-49 )

• Essentially this used the fact we can create expressions that are variable free using combinators to create a highly efficient computer system (Clarke 1980)

THE SKGP

• Implements programs as graphs using combinators with GP to produce pure functional (combinator) expressions

• Uses strong typing similar to (Yu 1997, 1998)

EVALUATING COMBINATOR EXPRESSION

Example: ‘S(S(K +)(K 1))I’ is a curried function that adds 1 to what it is applied to so S(S(K +)(K 1)I applied to 3 is:

S(S(K +)(K 1))I 3

S(K +)(K 1)3(I 3)

K+3((K 1)3)(I 3)

+K 1 3 (I 3)

+ 1 (I 3)

+ 1 3

4

COMBINATORS FUNCTIONS QUICKLY BECOME COMPLEX

Here is the function for factorial:

def fac = S(S(S(K cond)(S(S(K =)(K 0)))I))(K 1))(S(S(K *)I) (S(K fac)(S(S(K -)I)(K 1))))

Evaluation is left as an “exercise to the reader.”

COMBINATOR GRAPHS

• Combinator expressions represented as graphs

• + 1 2

• fxyz

• Sfgx -> fx(gx)

TYPING GP• The SKGP is strongly typed so that it is always “type coherent”

• Based on Hindley/Milner type system as described in (Yu 1997) but for combinators instead of lambda expressions

• Type is checked during graph creation and resolved at time of mutation and crossover - static typing

• If cannot resolve type, back out and try again by creating a new subtree

• Strongly typed system will always terminate with same type (halting problem?)

ESCAPING THE BOTTLE

• (Daida 2003) describes limitation of standard GP in how trees grow

• Presents evidence that GP can be limited in its search ability without structure altering operators

• Combinators have the property of being ‘structure altering operators’

Daida, unpublished based on Daida 2004 Demonstrating Constraints to Diversity with a

Tunably Difficulty Problem for Genetic Programming

CHURCH-ROSSER THEOREM

• The Church-Rosser Theorem says pure function evaluation can be order independent: Regardless of order of evaluation, result will be the same

• Because of this, each functional piece, when evaluated, can be stored for re-use since order of evaluation does not matter

•Normally this is inefficient since some pieces are not used

• Because GP shares pieces across generations, reuse gives super-linear speed up: you don’t have to recompute each component

REAL WORLD GP USING SKGP• (Briggs 2006) shows such a system benchmarks well

(regression, parity, stack & queue evolution)

•What about real world problems?

• Process control program for manufacturing process

•Modeling chemical kinetics for NASA

• Bladder cancer analysis differentiating nodal metastatic cancer from non-metastatic cancer

• Colon cancer prognostic

FUTURE DIRECTIONS

• Reuse of combinator expressions across generations via caching mechanism

• Solving the ‘Y problem’

• Adding chromosome structure