Sección 2ª: Plan de mejoras

Let be a set of propositional symbols and the set of their associated literals, where or A clause C is a generalized

disjunction [9] of literals: and a dual clause is a

generalized conjunction of literals:

Given a propositional theory represented by anordinary formula W, there are algorithms for converting it into a conjunctive normal form (CNF):

defined as a generalized conjunction of clauses, or into a disjunctive normal form (DNF): defined as a generalized disjunction of dual clauses, such that e.g., [23].

Alternatively, a special case of CNF and DNF formula can be the prime implicates and prime implicants, that consist of the smallest sets of clauses (or terms) closed for inference, without any subsumed clauses (or terms), and not containing a literal and its negation. In the sequel, conjunctions and disjunctions of literals, clauses or terms are treated as sets.

A clause C is an implicate [12] of a formula W iff and it is a prime implicate iff for all implicates of W such that we have or syntactically [20], for all literals We define as a conjunction of prime implicates of W such that A term D is an implicant

of a formula W iff and it is a prime implicant iff for all implicants of

W such that we have or syntactically, for all literals

We define as a disjunction of prime implicants of W such that

To transform a formula from one clause form to the other, what we call dual trans- formation (DT), only the distributivity of the logical operators and is needed. In propositional logic, implicates and implicants are dual notions, in particular, an algo- rithm that calculates one of them can also be used to calculate the other [5,24].

To represent these normal forms, we introduce the concept of a quantum, defined as a pair where is a literal and is its set of coordinates that contains the subset of clauses in to which the literal belongs. A quantum is noted to remind that F can be seen as a function The rationale behind the choice of the name quantum is to emphasize that the minimal semantical unity in the proposed model is not the value of propositional symbol, but the value of a propositional symbol with respect to the theory in which it occurs.

Any dual clause in the DNF can be represented by a set of quanta: i.e., D contains at least one lit- eral that belongs to each clause in spanning a path through and no pair of contradictory literals, i.e., if a literal belongs to D, its negation is excluded. A dual clause D is minimal, if the following condition is also satisfied:

This condition states that each literal in D should represent such that

Propositional Reasoning for an Embodied Cognitive Model 167

alone at least one clause in otherwise it would be redundant and could be deleted. The notation is symmetric, i.e., a clause in the CNF can be associated with a set of quanta: such that with no tautological literals allowed. Again the minimality condition for C is expressed by

The quantum notation is an enriched representation of the minimal normal forms, in the sense that the quantum representation explicitly contains the relation between literals in one form and the (dual) clauses in the other form. The CNF and DNF, from a syntactical point of view, are totally symmetric and each one of them contains all the information about the theory, but we propose that the agent should store its theories in both minimal normal forms. We belief that this ‘holographic’ representation can be used in others tasks of the agent, such as verification (as presented in the section 5) and belief changes [4], among others2.

4

Learning

Theories can be learned by perceiving and acting in the environment, while keeping track of the truth value of a specific emotional propositional symbol. This symbol can be either a primitive emotional symbol or an abstract emotional symbol represented by a theory that also contains controllable and uncontrollable symbols, but ultimately depends on some set of primitive emotional symbols. The primitive emotional symbols may also depend on a communication from another agent that can be trustfully used as an oracle to identify its truth value.

The proposed learning mechanism has some analogy with the reinforcement learn- ing method [11], where the agent acts in the environment monitoring a given utility

function. Directly learning the relevant assignments can be thought of as a practical

learning.

Example 2. Consider the robot of example 1. To learn the relation between the primitive emotional symbol Move and the controllable and uncontrollable

symbols, it may randomly act in the world, memorizing the situations in which the Move

symbol is assigned the value true. After, trying all possible truth assignments, it concludes that the propositional symbol Move is satisfied only by the 12 assignments3:

The dual transformation (DT), applied on the dual clauses associated with the good assignments, returns the clauses of the minimal CNF A further application of

2 3

The authors presently investigate others properties of the normal forms.

To simplify the notation, an assignment is noted as a set of literals, where is the number of propositional symbols that appear in the theory, such that represents the assignment if or if and

is the semantic function that maps propositional symbols into truth values.

the dual transformation in this CNF returns the minimal DNF 4. The minimal forms and their relation can be represented by the following sets of quanta:

It should be noted that contains less dual clauses than the original number of assignments, nevertheless each assignment satisfies at least one of this dual clauses. The application of the dual transformation provides a conjunctive characterization of the theory that, because of the local character of the clauses, can be used as a set of rules for decision making.

To formalize the proposed learning mechanism, we define an entailment relation that connect semantically neutral propositional symbols (controllable and uncon- trollable) to emotional symbols. Let be a neutral propositional formula and P an emotional symbol, this entailment relation has the following properties.

If then

If and then

In practice, learning is always incremental, that is, the agent begins with an empty theory and incrementally constructs a sequence of theories such that correctly captures the intended emotional propositional symbol P. According to the properties above, we have that and

The algorithm to obtain represented by its CNF and DNF, and given P, and the assignment is the following:

if and then

where is the literals dual clause such that andDTis the dual trans- formation5.

A similar algorithm may be used to incrementally compute the sequence of theories such that and The theories in this sequence are descriptions of those situations that do not entail the emotional symbol

P. During learning, when the agent has already tried theories entailing P and not entailing it, the theory captures those situations that were not yet expe- rienced by the agent and can be used in the choice of future interactions. Its DNF can

4

5

In fact, this second application is not necessary, because, once the prime implicants are known, there are polynomial time algorithms to calculate the prime implicates [8].

As specified in the Section 3.

Propositional Reasoning for an Embodied Cognitive Model 169

be computed by flipping all literals in If learning is complete, then

Although nothing directly associated with the CNF occurs in the environment, if its contents can be communicated by another agent, then a theory can be taught by stating a CNF that represents it. In this case, the trustful oracle would communicate all the relevant rules that define the theory. This transmission of rules can be thought of as an intellectual learning, because it does not involve any direct experience in the environment.