Reverse Folding

Problem Description

* Input predicates: fold/3, length/1, time/1

* Output predicates: pivot/3

This is a simplification of an important Biological problem. A string (e.g., representing a protein) composed of N consecutive elements (defined by the predicate length(N)) at a fixed unitary distance lays on a 2D (cartesian) plane. Admissible angles are 0 (straight line), .90. (left turn) and +90. (right turn). Different elements must occupy different positions. We refer to each placement of the string as a folding. A folding is represented by a predicate fold(I,X,Y) where I in {1,...,N} is the Ith element of the string, and (X,Y) are its coordinates in the plane. You can assume X and Y in {0,...,2N}.

A pivot move is defined by selecting an element i in {2,...,N-1} and its effect is to turn clockwise or counter-clockwise of 90 degrees the part of the string related to the elements i+1,...,N.

A move at time t is represented by pivot(t,i,clock) or pivot(t,i,anticlock). Exactly one move occurs at a time t.

The goal is to find (if it exists) a sequence of T moves (T is defined by the predicate time(T)) such that lead the initial straight line fold fold(1,N,N). fold(2,N,N+1). fold(2,N,N+2). ... fold(N,N,2N-1). into the fold assigned as input.

Assume the input is:

fold(1,9,9).
fold(2,9,10).
fold(3,9,11).
fold(4,10,11).
fold(5,11,11).
fold(6,11,10).
fold(7,11,9).
fold(8,10,9).
fold(9,10,10).

length(9).
time(4).

This means that the the final form of the protein is

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Starting from the straight line fold, it is possible to reach that folding with the application of the following pivot moves (which is the expected output from the system):

pivot(1,3,clock).
pivot(2,5,clock).
pivot(3,8,clock).
pivot(4,7,clock).

Effects of pivot moves

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Ideas on modeling with variants of ASP can be found in:
A. Dovier, A. Formisano, E. Pontelli. Perspectives on Logic-based Approaches for Reasoning About Actions and Change. To appear in LNCS 6565, Essay in honour of Michael Gelfond (paper draft available on line)