In this section we describe the formal representation of metacognitive expectation using elements of denotational mathematics proposed by Wang [9].

An expectation represents a hypothesis or projection of current knowledge [10]. A metacognitive expectation with respect to the execution time of a cognitive function that is executed at the object-level has an observation of a metacognitive sensor (), an attribute of the observation that represents the execution time (), the expected value () and the set of rules that generates the expectation (). Formally a metacognitive expectation () about observation () is a 4-tuple, i.e.:

Where:

is an expectation about an observation .

represents an observation from a sensor , with where:

is the instance of a profile of a cognitive function that is running in the object level.

is the current execution time of cognitive function.

is the unit in which the execution time is measured in milliseconds.

is an attribute of the observation ο, with represents the set of attributes of the observation 𝜊 from the sensor . represents the set of attributes of profile of a cognitive function .

CARINA is a meta-cognitive architecture for artificial intelligent agents. CARINA is derived from the MISM Metacognitive Metamodel [11]. CARINA integrates self-regulation and meta-memory with support for the meta-cognitive mechanisms of introspective monitoring and meta-level control. In this sense CARINA assumes a functional approach to philosophy of mind, according to Piccinini, Fodor, and Scheutz [12-14]. According to Machamer, Darden, and Craver [15], the term ‘‘mechanism” includes entities and activities involving entities (i.e., including both static and dynamic aspects). The entities in CARINA are called “cognitive elements”.

In accordance with MISM, CARINA is composed of three types of cognitive elements: structural elements, functional elements, and basic elements. Structural elements are containers into which the functional and basic elements are embedded. The main structural element is the “Cognitive level”. The functional elements are tasks that enable reasoning and decision-making. Basic elements consist of the set of elements that participate and interact in reasoning and meta-reasoning processes. The main functional elements in CARINA are reasoning task and meta-reasoning task. Reasoning tasks (RT) are actions that enable the processing (transformation, reduction, elaboration, storage and retrieval) of information by applying knowledge and decision making in order to meet the objectives of the system. A meta- reasoning task (MT) may be used to explain errors in some reasoning task or it may be used to select between “cognitive algorithms” to perform the reasoning [16].

In CARINA, object-level has stages which are sets of cognitive functions (CF): Perception Stage (Perception CF), Situation Assessment Stage (Situation Assessment CF, Categorization CF, and Recognition CF), Reasoning Stage (Reasoning CF, Belief Maintenance CF, and Decision Making CF), Problem Solving Stage (Planning CF and Prediction CF), Interaction Stage (Interaction CF and Communication CF) and Action Stage (Action CF) as shown in Fig. 1. The meta-level contains a dynamic model of the object-level.

Figure 1 Source: Image Generated by Authors

The meta-level includes the components, knowledge and mechanisms necessary for a system to monitor and control its own learning and reasoning processes. The meta-level of CARINA has two types of metacognition:

Self-regulation: this type of metacognition monitors and controls the reasoning process that takes place at the object-level. Self-regulation contains two stages, which are sets of meta-cognitive functions (MCF): Monitoring Stage (Identification MCF, Detection MCF, Explanation MCF, and Goal Generation MCF) and Control Stage (Strategy Selection MCF).

Our ontology generates a metacognitive expectation of time with respect to the execution of a cognitive function in the metacognitive architecture. The implementation of the ontology was done using Protégé, version 5.2.0. This tool allows good level of details and its interface is user-friendly. WOL-DL (Web Ontology Language - Description Logic) was the formal language used for representation, World Wide Web Consortium (W3C) recommends WOL-DL. In this context the metacognitive problem is noted as ρ. ρ represents how to generate a metacognitive expectation with respect to the execution time of a cognitive function that is executed at the object level. Metacognitive expectation is represented as a hierarchical structure in ontology. First, the taxonomy of terms was created with the main concepts (classes): SemanticMemoryUnit. Creating subclasses Beliefs and Facts from derived concepts refined this hierarchy. Next, the fields of the Observation were defined. Malviya, Mishra, and Sahu define Object Properties as relationships between two instances of classes, whereas data properties describe relationships between instances and data values [25].

Figure 3 Source: Image Generated by Authors

According to the structure of the ontology the classes depend on a super class Thing, the classes contain individuals and the object properties. The structure of the ontology is shown in Table 2 which contains an example of object and data properties of the cognitive ontology.

In the object properties, the domain and the range are created.

Table 2

Source: Author's own elaboration

Table 3

Source: Author's own elaboration

The data property has IDConcept which defines the identifier of the semantic memory unit. In the same way, the properties pertaining to each concept were represented as is shown in Table 4.

Table 4

Source: Author's own elaboration

Ontology Instantiation is implemented as a simulation of the process of triggering expectations in CARINA. The next step is the use of the ontology as a simulation of triggering expectations at meta-level. This work explores the role of different types of memories in the triggering of metacognitive expectations in CARINA. We use production rules for representing the procedural knowledge needed to generate a metacognitive expectation. Table 5 shows a rule named R1. R1 gets the current observation from the sensory memory and generates an expectation for the field running time.

Table 5

Source: Author's own elaboration

The sensorial memory contains the observation flow from the metacognitive sensor. In the ontology an observation has a relation (hasObservationCFP) that points to the profile of a cognitive function, and based on the observation, it looks for the beliefs () in the semantic memory to verify and extract the field of cognitive function profile (OCFP). Since the fact field of the observation must be equal to the beliefs (), then the fields in the ontology should not be equal. One way to identify it is to make a datum of the hasIDConcept property of script type when the value is the same.

As stated by Gordon, Hobbs and Cox the reasoner calculates some expected outcome and compares it with the actual outcome that constitutes the feedback [26]. The observation field (OCFP) and the field () in the belief (BCFP) are extracted and the two script data are matched to obtain the execution time.

hasCFIPRunningTime is the average of the sum of i=1 to n of the field () in equation (3), the field () of the belief about in formula (3), the field () of the belief is automatically extracted in a new relationship called triggerRunningTimeExpectation.

An experiment was conducted to validate the performance of the model and its ability to generate expectations regarding the execution time of cognitive functions that are executed in the object-level of a cognitive agent based on CARINA. The syntactic analysis of a sentence was the cognitive task of the real world used in the experiment. In this experiment a cognitive agent identified the different grammatical categories in which the words of the language are grouped. These categories include the word’s meaning, its grammatical function and the way in which it is structured, combined and modified.

The Left-corner is the algorithm implemented in the cognitive agent to decide if the chain of input symbols is a sentence in a given language. The algorithm determines the syntactic structure of the sentence according to a grammar [27].

We collect data from 200 executions of the cognitive task in CARINA, varying the available resources such as the size of the content of the semantic memory with respect to the domain of the cognitive task. Figure 4 shows the results of CARINA in terms of the expectations created with respect to the observations of the profile of cognitive functions that were executed in each cognitive cycle at object-level. The x-axis represents the 12 cognitive functions that were executed in CARINA to perform the cognitive task. The y-axis represents the execution time in milliseconds.

Figure 4 Source: Image Generated by Authors

The blue bars represent the run-time expectation of 12 cognitive functions that were executed in CARINA to perform the cognitive task. CARINA calculates the run-time values using equation (3). The blue lines represent the observed runtime values at a time . This is a clear example of the functionality of the model, because it generates expectations for each observation and is able to compare them against the observed values in real time.

The scope of this investigation only applies to the generation of execution time expectations. In this sense, the model tests demonstrate the ability to generate expectations and the possibility of detecting anomalies in the expected execution times for cognitive functions. An interesting aspect for future investigations is to analyze the reasoning trace of the function that presents the anomaly, with the goal to determine the cause and possible corrective actions.

Another type of result obtained in this research was in terms of the conceptual advance of an expectation, the formal mathematical representation, the design of ontology and the model as an implementation mechanism. The cognitive ontology for reasoning about expectations in CARINA architecture facilitates the visualization of the expectations generated from the observations of the profiles of cognitive functions, as shown in Figure 5.

Figure 5 Source: Image Generated by Authors

In Figure 5, the new relationship in ontology represents an expectation generated by the CARINA meta-level in the process of introspective monitoring. This result is promising because it will serve as a basis for the implementation of predictive capabilities of introspective monitoring in Carina's meta-level.