2. Department of Computer Science, University of Maryland. College Park, MD 20742 USA. A logic of discovery is introduced. In this logic, true sentences are ...
A Logic of Discovery J¯ anis B¯arzdi¸ n˘s1 and R¯ usi¸ nˇs Freivalds1 and Carl H. Smith2 1
Institute of Math and Computer Science, University of Latvia Rai¸ na bulv¯ aris 29, LV-1459, Riga, Latvia 2 Department of Computer Science, University of Maryland College Park, MD 20742 USA
A logic of discovery is introduced. In this logic, true sentences are discovered over time based on arriving data. A notion of expectation is introduced to reflect the growing certainty that a universally quantified sentence is true as more true instances are observed. The logic is shown to be consistent and complete. Monadic predicates are considered as a special case. In this paper, we consider learning models that arise when the goal of the learning is not a complete explanation, but rather some facts about the observed data. This is consistent with the way science is actually done. We present a logic of discovery. Standard first order logic is extended so that some lines of a proof have associated expectations of truth. This is used to generalize from A(a) for some set of examples a to a formulae ∀xA(x). We show how to prove all true first order formulae over monadic predicates. A expectation level is a rational number between 0 and 1. Intuitively, a 0 indicates no confidence in the answer and a 1 indicates certainty. The proofs that we construct will start with a finite set of assumptions, and continue with a list of statements, each of which follows by the rules of logic from some prior statements or assumptions. Some of the statements will be paired with an expectation level. The basic idea is that traditional data such as “f (3) = 5” can be taken as true instance of some predicate F , in which case we say “F (3, 5) is true.” We consider algorithms that take true instances of predicates and produce, deduce, infer, or whatever, true formulae about the predicate F . The basic idea is to have the proof develop over infinitely many statements. At first, the idea of an infinite prove may appear blasphemous. However, consider for the moment that induction is really infinitely may applications of monus ponens collected into a single axiom. While in the case of induction, it is easy to see where the infinitely many applications of monus ponens will lead to, we consider cases where the outcome is not so clear. Hence, we explicitly list all the components of our infinite arguments. In a proof of assertion A, A would appear on an infinite subset of the lines of the proof, each time with a larger expectation of truth. The intuitive meaning of A/e is that, based on the prior steps of the proof, we have expectation e that the formulae A is true. A system of positive weights is employed to calculate is a recursive the expectations. A weighting function n+1 w(n) = 1. For example, let w(n) = 1/2 . The idea function w such that ∞ n=0 of the weighting system is that some examples are more relevant then others. The weighting system will be used to determine the expectation of a universally quantified sentence, based on known examples. S. Arikawa and H. Motoda (Eds.): DS’98, LNAI 1532, pp. 401–402, 1998. c Springer-Verlag Berlin Heidelberg 1998
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J¯ anis B¯ arzdi¸ n˘s and R¯ usi¸ nˇs Freivalds and Carl H. Smith
Our proof system starts with a standard first order theory. Proofs are built in the standard fashion. We allowed to write “A(t)” as a line of a proof if A(t) can be deduced by the usual axioms from prior lines in the proof, or we are given that A(t) is true as an input to the discovery process. For this, an subsequent examples, we consider formulae of a single variable for notational simplicity. However, we envision formulae of several variables. Our target scenario is the discovery of “features” of some observed phenomenon represented (encoded) as a function over the natural numbers. While the data might look like “f (3) = 5” we use a predicate symbol F for the function and enter “F (3, 5)” as a line of the proof we are building. We add to the standard set of axioms for a first order theory the following e-axiom of expectation axiom for A a formula with n free variables: (n)
A(t1 , . . . , tn ) | (t1 , . . . , tn ) ∈ T (n) ⊆ N (∀x1 , . . . , xn )A(x1 , . . . , xn )/ w(t1 ) · . . . · w(tn ) (t1 ,...,tn )∈T (n)
The set T (n) denotes the set of values for which we know the predicate A to be true for. We illustrate the use of the e-axiom for the n = 1 case. If in some lines of a proof we have A(2) and then later A(4), we would be able to use the e-axiom to obtain (∀x)A(x)/e where e = w(2) + w(4). We introduce the t-axiom, or truth axiom that allows the an expectation of 1 to be added to any sentence provable within the standard first order theory. Definition 1. A formula A is epistemically provable (or eprovable) iff there is an A that is either A or logically equivalent to A for each > 0, there is an e ≥ 1 − such that A /e is provable using the traditional first order theory augmented with the e-axiom and the t-axiom. Theorem 1. Suppose w and w are two different weight functions. Then, for any formulae A, A is eprovable with respect to w iff A is eprovable with respect to w . Theorem 2. First order logic plus the the e-axiom and the t-axiom is sound. Assume that all first order formulae are presented in prenex normal form. Theorem 3. Suppose Σ is a signature containing only monadic predicates. If f is a first order formula over Σ and I is an interpretation of the predicates in Σ, there is an eproof of f iff f is true according to the interpretation I. Theorem 4. For any n ≥ 1, for every true Πn sentence S1 there is an (n/2 ·ω)– proof of S1 and for every true Σn sentence S2 there is an ( n/2 · ω + 2)–proof of S2 .
