Propositional logic. – Propositions are interpreted as true or false. – Infer truth of new propositions. • First order logic. – Contains predicates, quantifiers and. 2 First-Order Logic: Syntax. • We shall now introduce a generalisation of propositional logic called first-order logic. (FOL). This new logic affords us much. or false. In first-order logic the atomic formulas are predicates that assert a relationship among The syntax of first-order logic is defined relative to a signature.
|Language:||English, Spanish, Arabic|
|Genre:||Health & Fitness|
|Distribution:||Free* [*Register to download]|
Techniques in Artificial Intelligence. First-Order Logic. • Propositional logic only deals with “facts”, statements that may or may not be true of the world, e.g. In order to develop the theory and metatheory of first-order logic, we must first define the syntax and semantics of its expressions. The expressions of first-. Propositional logic is declarative: pieces of syntax correspond to facts first- order logic (like natural language) assumes the world contains. • Objects: people .
One bit has already a rough idea of what he expects to dis- represents the possible value of a gene, and a string cover. Also, REGALdoes not deal with the kind of represents a chromosomewhich is the genetic encod- background knowledge mentioned in figure 1. It does ing of an individual. Genetic operators that modify not modify the variables in the predicates. Finally, for chromosomesusing this representation are the muta- dealing with numeric values, the binary representation tion and the crossover.
For instance, if one consid- ents. The aim of this last operator is to combineuseful ers that the Length predicate can take values, the information in order to produce better offsprings. However,the FOLrules which overcomes the previously mentioned crossover operator is constraining the kind of possible limitations.
One first difference is that rules will be representations for learning rules because rules must represented directly in FOLby using the predicates be represented as a string of genes. A rule in propo- and their argnunents as genes. The GAwill be allowed sitional logic is already a string of elements because it to perform other operations than in REGAL,like for always uses the same number of attributes. This is the instance changing a predicate into a more general one reason why most of GAsapplications in rule learning according to the background knowledge, or like chang- use an attribute-based representation Holland ing a constant into a variable.
For the same position in two rules must represent the same representation, the problem is to find a good represen- predicate, if the crossover is to workwell.
This problem tation, i. This covering process user provides a model of the rule to be learned. For comes from the AQalgorithm Michalski et al. AO1main algorithm is a covering algorithm simi- that have not produced yet any offsprings. The system tries b For every individual r of the intermediate popula- find from an example ex the best rule r according to tion, possibly insert r in Pop only if r fL Pop and a rule-evaluation criterion taking into account the co- either: Until the fitness of the best individual observed has is a disjunct of each learned rule: This algorithm has the following prop- 2.
This maintains in the population the diver- sity required to prevent premature convergence that a Chooseex "Pos asa s eed. The difference guaranteed be- b Find ttle best rule r according to a rule- tween individuals is syntactic, and it proved to be a evaluation criterion obtained by generalization good alternative in terms of complexity to the use from ex.
Give T4 as the result. This example is used to determine particular SZ, is sensitive to the choice of the the model of the rule to be learned, and is thus not seeds which will be processed. Furthermore, with another seed example, this model will be different and Genetic operators is not constant as in REGAL. The genetic operators have been adapted to the cov- SIAO1 search algorithm ering principle of the algorithm. To obtain the mation between two individuals.
Let us consider for new generation, the algorithm applies to each individ- instance the following rule r: The algo- encodedas the string of genes represented in figure 2. More precisely, the mutation selects randomly a rel- 1. The chromosomecoding rule r. Since this kind of behavior must be that the predicate and the arguments immediately avoided, it has been impeded. The If the seed contains only one predicate symbol, the corresponding part of the chromosome becomes a restrained one-point crossover becomes useless, and non-coding part, and no mutation will take place therefore the classical one-point crossover applies.
This there having no effects.
In such a case it is possible to add the same arbitrary predicate symbol tation can create an interval, or if such an interval already exists, the mutation can widen it. The fol- before each line of data, with no domaintheory for the lowing operations are thus possible: In , like in manygenetic algorithms, tion maycreate an internal disjunction or generalize this function has been empirically designed, and gives an existing disjunction.
This may result in the fol- lowing operations: Four criterions are taken into account: Because the algorithm value like "yellow" mayalso be changed to "yellow- proceeds mainly by generalization, this criterion is like" according to the background knowledge, critical: Because creating new fore, the fitness function will strongly penalize any inconsistent rule, and it will give a score of 0 to any variables is interesting as long as relations can be detected, this kind of mutation may have an effect rule covering more counter-examples than allowed not only on the gene it is destined to, but on the en- by a noise parameter, tire chromosome: Unlike consistency, this symbol concerned in the chromosomeand then ran- criterion gives a positive contribution to the fitness, domly replaces them by the new variable or leaves growing linearly with the number of positive exam- them unchanged.
This criterion is particularly assuming that "X" is a new variable occurring important in the case of a learning base whose ex- nowhere else in the formula. In 8Z, , the user can fix numeric constant. The offsprings that re- gives a result in [0,1]. This restrained one-point crossover is use- number of examples, TO the total number of counter- ful because it prevents the appearance of the hidden examples, the maximumnoise tolerated Af ,the syn- arguments following an empty predicate.
Michalski T 9 et G. Tecuci Eds , pp Therefore, the quality of an individual r will be: Giordana A. Ann Arbor: University of Michigan Press. Escaping brittleness: Michalski, T.
Mitchell, We have presented in this paper a new learning J. Carbonell et Y. Kodratoff Eds , Morgan Kauf- algorithm based on genetic algorithms that learns mann, Theory and methodology of in- negative examples and from background knowledge.
AO1represents rules in a high level language, and Volume 1, R. Mitchell, J.
Car- is thus able to perform high level operations such as bonell et Y. Kodratoff Eds , Morgan Kaufmann, generalizing a predicate according to the background knowledgeor like changing a constant into a variable.
The variable a is instantiated as "Socrates" in the first sentence and is instantiated as "Plato" in the second sentence. While first-order logic allows for the use of predicates, such as "is a philosopher" in this example, propositional logic does not. Consider, for example, the first-order formula "if a is a philosopher, then a is a scholar".
This formula is a conditional statement with "a is a philosopher" as its hypothesis and "a is a scholar" as its conclusion. The truth of this formula depends on which object is denoted by a, and on the interpretations of the predicates "is a philosopher" and "is a scholar".
Quantifiers can be applied to variables in a formula. The variable a in the previous formula can be universally quantified, for instance, with the first-order sentence "For every a, if a is a philosopher, then a is a scholar". The universal quantifier "for every" in this sentence expresses the idea that the claim "if a is a philosopher, then a is a scholar" holds for all choices of a.
The negation of the sentence "For every a, if a is a philosopher, then a is a scholar" is logically equivalent to the sentence "There exists a such that a is a philosopher and a is not a scholar".
The existential quantifier "there exists" expresses the idea that the claim "a is a philosopher and a is not a scholar" holds for some choice of a. The predicates "is a philosopher" and "is a scholar" each take a single variable. In general, predicates can take several variables.
In the first-order sentence "Socrates is the teacher of Plato", the predicate "is the teacher of" takes two variables. An interpretation or model of a first-order formula specifies what each predicate means and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set.
For example, in an interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the Republic ", the sentence "There exists a such that a is a philosopher" is seen as being true, as witnessed by Plato. Syntax[ edit ] There are two key parts of first-order logic. The syntax determines which collections of symbols are legal expressions in first-order logic, while the semantics determine the meanings behind these expressions.
Alphabet[ edit ] Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is legal. There are two key types of legal expressions: terms, which intuitively represent objects, and formulas, which intuitively express predicates that can be true or false. The terms and formulas of first-order logic are strings of symbols, where all the symbols together form the alphabet of the language.
As with all formal languages , the nature of the symbols themselves is outside the scope of formal logic; they are often regarded simply as letters and punctuation symbols.
It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation.
On the other hand, a non-logical predicate symbol such as Phil x could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate, depending on the interpretation at hand. Occasionally other logical connective symbols are included. Parentheses, brackets, and other punctuation symbols.
The choice of such symbols varies depending on context. An infinite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, Subscripts are often used to distinguish variables: x0, x1, x2, Not all of these symbols are required—only one of the quantifiers, negation and conjunction, variables, brackets and equality suffice.
Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers. Non-logical symbols[ edit ] The non-logical symbols represent predicates relations , functions and constants on the domain of discourse.
It used to be standard practice to use a fixed, infinite set of non-logical symbols for all purposes. A more recent practice is to use different non-logical symbols according to the application one has in mind.
Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a signature.
Consequently, under the traditional approach there is only one language of first-order logic. Because they represent relations between n elements, they are also called relation symbols.