P-log: refinement and a new coherency condition

Abstract

This paper focuses on the investigation and improvement of knowledge representation language P-log that allows for both logical and probabilistic reasoning. We refine the definition of the language by eliminating some ambiguities and incidental decisions made in its original version and slightly modify the formal semantics to better match the intuitive meaning of the language constructs. We also define a new class of coherent (i.e., logically and probabilistically consistent) P-log programs which facilitates their construction and proofs of correctness. There are a query answering algorithm, sound for programs from this class, and a prototype implementation which, due to their size, are not included in the paper. They, however, can be found in the dissertation of the first author.

Appendix

In this Appendix we formally define the probabilistic measure defined by a P-log program satisfying Conditions 1-3 from Section 3 . The definition is very similar to the one from [8], with a small difference related to the addition of rules names to special terms random, truly_random, do and probability atoms and the corresponding changes in axioms, introduced in this paper.

Let π be a P-log program with signature Σ, W be an interpretation of Σ, a be an attribute term of Σ. Consider random selection rule of the form

$$random(rn, a, p) \leftarrow B $$

such that W satisfies B. Let PO(W,rn,a) be the set of constants defined as follows:

$$PO(W,rn,a)= \{y~|~W \text{~satisfies~} p(y) \text{~and~} y \in range(a) \}. $$

We will refer to elements of the set PO(W,rn,a) as possible outcomes of a in W via rule rn, and to every atom a = y s.t. y ∈ PO(W,rn,a) as a possible atom in W via rn. Note that, by Condition 1, there can be at most one rule such that a = y is possible in W vie that rule, so we will sometimes say that a = y is possible in W if there is a rule rn^′ such that a = y is possible in W via rn^′.

Let π be a P-log program and a be a random attribute term of the signature of π. For every possible world W of π and every possible atom a = y in W via some rule random(r,a,p) ← B, such that W⊧truly_random(r,a), we will define the corresponding causal probability P(W,a = y). Whenever possible, the probability of an atom a = y will be directly assigned by pr-atoms of the program and denoted by PA(W,a = y). To define probabilities of the remaining atoms we assume that by default, all values of a given attribute which are not assigned a probability by pr-atoms are equally likely. Their probabilities will be denoted by PD(W,a = y). (PA stands for assigned probability and PD stands for default probability).

More precisely, for each atom a = y possible in W via some rule r or π:

• 1. Assigned probability:

  If π contains pr(r,a = y | B^′) = v, W⊧B^′, then

  $$PA(W, a = y) = v $$

  (note that Condition 2 implies that the probability is uniquely defined).

• 2. Default probability:

  Let

  $$A_{a}(W) =\{ y ~|~ a = y \text{ is possible in } W \text{ and } PA(W, a= y)\ \text{is defined}\}, $$
  $$D_{a}(W) = \{ y ~|~ a = y \text{ is possible in } W\} \setminus A_{a}(W) $$

  and $\alpha _{a}(W) = {\sum }_{y \in A_{a}(W)} PA(W, a = y)$.

  The default probability of a = y in W is defined as follows:

  $$PD(W, a = y) = \frac{1 - \alpha_{a}(W)} {|D_{a}(W)|} $$

• 3. Finally, the causal probability P(W,a = y) of a = y in W is defined by:

  $$P(W, a = y) = \left\{\begin{array}{ll} PA(W, a = y) & \text{ if } y \in A_{a}(W)\\ PD(W, a = y) & \text{ otherwise}. \end{array} \right. $$

Definition 1 (Measure)

• 1. Let W be an interpretation of π. The unnormalized probability, $\hat {\mu }_{\Pi }(W)$, of W induced by π is

  $$\hat{\mu}_{\Pi}(W) = {\prod}_{W(a) = y} P(W, a = y ) $$

  where the product is taken over atoms for which P(W,a = y) is defined.

• 2. Suppose π is a P-log program having at least one possible world with nonzero unnormalized probability. The measure, μ_π(W), of a possible world Winduced by π is the unnormalized probability of W divided by the sum of the unnormalized probabilities of all possible worlds of π, i.e.,

  $$\mu_{\Pi}(W) = \frac{\hat{\mu}_{\Pi}(W)} {{\sum}_{W_{i} \in {\Omega}({\Pi})}\hat{\mu}_{\Pi}(W_{i})} $$

When the program π is clear from the context we may simply write $\hat {\mu }$ and μ instead of $\hat {\mu }_{\Pi }$ and μ_π respectively.

Definition 2 (Probability)

Suppose π is a P-log program having at least one possible world with nonzero unnormalized probability. The probability, P_π(E), of a set E of possible worlds of program π is the sum of the measures of the possible worlds from E, i.e.

$$P_{\Pi}(E) = \sum\limits_{W \in E}\mu_{\Pi}(W). $$

When π is clear from the context we may simply write P instead of P_π.

Definition 3 (Probability of a literal)

The probability with respect to program π of a literal l of π, P_π(l), is the sum of the measures of the possible worlds of π in which l is true, i.e.

$$P_{\Pi}(l) = \sum\limits_{W \models l}\mu_{\Pi}(W). $$

Note that, given that conditions 1-3 are satisfied, the function P_π is defined iff

$$\sum\limits_{W_{i} \in {\Omega}({\Pi})}\hat{\mu}_{\Pi}(W_{i}) \not= 0 $$