Negation as failure
Negation as failure. It is possible to do negation-as-failure using Twelf's logic programming engine, with the use of %deterministic. As an example, we will define logic programs to compute the less-than and greater-than-or-equal-to functions. The less-than function will be defined in a standard way. The greater-than-or-equal-to function will be defined using a negation-as-failure interpretation of the less-function.
Natural numbers and booleans.
nat : type. z : nat. s : nat -> nat. bool : type. true : bool. false : bool.
less is defined inductively in the standard way.
less : nat -> nat -> type. %mode less +D1 +D2. less/z : less z (s _). less/s : less (s N1) (s N2) <- less N1 N2.
We will need a judgment that tests whether a boolean is false in order to use negation-as-failure.
isfalse : bool -> type. %mode isfalse +D. isfalse/i : isfalse false.
We define a logic program that when given two numbers returns true if the first is less than the second, and false otherwise. We use %deterministic to make the logic programming engine commit to the first solution it finds.
less-bool : nat -> nat -> bool -> type. %mode less-bool +D1 +D2 -D3. %deterministic less-bool.
Because the less-bool/true case is first, it will attempt to find a proof that the first number is less than the second.
less-bool/true : less-bool N1 N2 true <- less N1 N2.
Because of the %deterministic declaration, when searching for a proof of less N1 N2 B, only executes when less-bool/true fails. However, because of pattern matching a search for a proof of less N1 N2 false always succeeds.
less-bool/false : less-bool N1 N2 false.
We will now define gte using less-bool. It has only one rule, which makes a call to less-bool N1 N2 B. It is important to make sure that the result B is not directly identified as false so that it executes less-bool in the appropriate order. We use the call to isfalse B to verify that the output really is false.
gte : nat -> nat -> type. %mode gte +D1 +D2. gte/i : gte N1 N2 <- less-bool N1 N2 B <- isfalse B.
We can use a number of %solve declarations to test our less and gte judgments.
%solve deriv : less (s z) (s (s z)). %solve deriv1 : gte (s (s z)) (s z). %solve deriv2 : gte z z. % solve deriv3 : gte (s z) (s (s z)). % should fail
It is important to note that while these definitions work as intended as logic programs, proving appropriate meta-theorems about judgments that use negation as failure is problematic or impossible.
Note from dklee: I had to run home to take care of some stuff. I will finish documenting this later in the evening.
This page is incomplete. We should expand it.