Mechanizing Hypothesis Formation: Mathematical Foundations by Petr Hájek, Tomáš Havránek (auth.)

By Petr Hájek, Tomáš Havránek (auth.)

Hypothesis formation is named one of many branches of synthetic Intelligence, the overall query of synthetic IntelligencE' ,"Can desktops think?" is designated to the query ,"Can pcs formulate and justify hypotheses?" quite a few makes an attempt were made to respond to the latter query absolutely. the current publication is one such try. Our objective isn't really to formalize and mechanize the complete area of inductive reasoning. Our final query is: Can pcs formulate and justify medical hypotheses? Can they understand empirical info and method them rationally, utilizing the equipment of recent mathematical common sense and facts to aim to provide a rational snapshot of the saw empirical international? Theories of speculation formation are often referred to as logics of discovery. Plotkin divides a good judgment of discovery right into a common sense of induction: learning the thought of justification of a speculation, and a common sense of advice: learning tools of suggesting average hypotheses. We use this department for the association of the current publication: bankruptcy I is introductory and explains the topic of our good judgment of discovery. the remainder falls into elements: half A - a good judgment of induction, and half B - a good judgment of suggestion.

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If there are infinitely many quantifiers then the function associating the type s. with each i is recursive. 1 A predicate language with predicate = id~ntity contains furthermore an additional binary distinct from P l' ... , P n (the equality predicate) . Formulae are defined inductively, the notion of atomic formulae and the induction step for connectives being as usual. Each expression p. (u 1 , ... ,u) t. 1 atomic formula (and u 1 if ~ = are formulae. If q. is a quantifier 1 CP1' ••• , cp s.

E of f. 1 1 ' ... , Qt. has little chance of belonging to Vor::. V if {IS' ; 1 fi ( 0 1 , ... , 0 t. , <» .. V o} e t . 1 In Chapter IV we shall study random structures: Suppose we have a probability measure P on E, i. e. P maps some subsets of ~ and satisfies the usual conditions. Write a probability space (see Chapter IV, Section 1 for 1: -state dependent structure may be called a L. - random V -structure. The probability measure on E defines various notions of small sets.

B; elementary disjunctions (analogous); formulae in conjunc- tive normal form (multiple (possibly empty) conjunctions of elementary disJunctions); formulae in disjunctive normal form (dis Junctions of elementary conJunctions) . Fact: For each open formula 'f different from Q. there is a logically equivalent formula in conjunctive normal form containing only predicates and variables which occur in ce • The same is true for open formulas different from 1 and disjunctive normal form. 7) iff (/\ B) ~ If is a tautology.

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