Non-Deterministic Lisp with Dependency-directed Backtracking

Extending functional Lisp with McCarthy’s nondeterministic operator AHFJ yields a language which can concisely express search problems. Dependencydirected backtracking is a powerful search strategy. We describe a non-deterministic Lisp dialect called SCHEMER and show that it can provide automatic dependency-directed backtracking. The resulting language provides a convenient interface to this efficient backtracking strategy. Many problems in Artificial Intelligence involve search. SCHEMER is a Lisp-like language with nondeterminism which provides a natural way to express sea.rch problems. Dependency-directed backtracking is a powerful strategy for solving search problems. We describe how to use dependency-directed backtracking to interpret SCHEMER. This provides SCHEMER programs with the benefits of dependency-directed backtracking automatically. We begin by describing the SCHEMER language. We next provide an overview of dependency-directed backtracking and list its requirements. We then show how to meet these requirements in interpreting SCHEMER. Finally, we argue that SCHEMER with automatic dependency-directed backtracking would be a useful tool for Artificial Intelligence by comparing it with current methods for obtaining dependency-directed backtracking. I. SCHEMER is Scheme with AMB SCHEMER consists of functional Scheme [Rees e-t al. 19861 plus McCarthy’s ambiguous operator AMB [McCarthy 19631 and the special form (FAIL). AMB takes two arguments and non-deterministically returns the value of one of them. Selecting the arguments of the AMB's in an expression determines a possible execution. Each SCHEMER ext Author's current address: Computer Science Department, Stanford University, Stanford, California, 94305. This paper describes research done at the Artificial Intelligence Laboratory at the Massachusetts Institute of Technology, supported in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contracts NoO014-80-C0505 and NOOOl4-86-K-0180, in part by National Science Foundation grant MCS-8117633, and in part by khe IBM Corporation. Ramin Zabih is suppori.ed by a fellowship from the Fannie and John Hertz Foundation. pression is thus associated with a set of possible values. In the program below, the expression (ANY-NUMBER) nondeterministically returns some whole number. (DEFINE (ANY-NUMBER) (AMB 0 (l+ (ANY-NUMBER)))) Similarly, (ANY-PRIME) non-deterministically returns some prime number. (DEFINE (ANY-PRIME) (LET ((NUMBER (ANY-NUMBER))) (IF (PRIME? NUMBER) NUMBER (FAIL)))) ANY-PRIME eliminates certain possible values by evaluating (FAIL). The expression (FAIL) has no possible values. A mathematically precise semantics for SCHEMER is beyond the scope of this paper there are several possible semantics that differ in technical detail [Clinger 1982, Zabih et nl. 19871. Under all these semantics, however, the expression (FAIL) can be used to eliminate possible values; finding a possible value for a SCHEMER expression requires finding an execution that doesn’t evaluate (FAIL). For a given expression there may be a very large number of different ways of choosing the values of AMB expressions. If there are 12 independent binary choices in the computation then there are 2* different combinations of choices, and thus 2* different executions. In certain expressions most combinations of choices result in failure. Finding one or more possible values for a SCHEMER expression requires searching the various possible combinations of choices. Interpreting SCHEMER thus requires search. The semantics of the language do not specify a search strategy. Correct interpreters with different strategies will produce the same possible values for an expression, and can differ only in efficiency. It is straightforward to write 3. SCHEMER interpreter that searches all possible esecutions in a brute force manner by backtracking to the most, recent non-exhausted choice in the event of a failure. Such an interpreter would use simple “chronological” backtra.cking. We describe a more sophisticated SCHEMER interpreter that automatically incorporates dependency analysis and dependency-directed backtracking. This interpreter, originally described in [Za.bih 198’71, allows programmers to gain the efficiency benefits of dependencydirected backtracking automatically for SCHEMER code. Zabih, McAllester, and Chapman 59 From: AAAI-87 Proceedings. Copyright ©1987, AAAI (www.aaai.org). All rights reserved.