Closed World Assumption: Fundamentals and Applications

By Fouad Sabry

What Is Closed World Assumption

In a formal system of logic that is used for the representation of knowledge, the closed-world assumption (often abbreviated as CWA) is the supposition that a statement that is true is also known to be true. Therefore, the inverse of this is true, which is that which cannot currently be verified as being accurate. Raymond Reiter is the author of a logical formalization of this assumption that bears the same name as this assumption. The open-world assumption (OWA), which holds that a lack of knowledge does not automatically entail that something is untrue, is the hypothesis that directly contradicts the closed-world hypothesis. The interpretation of the real semantics of a conceptual statement with the same notations of ideas is determined by the decisions made regarding CWA versus OWA. In most cases, a good formalization of natural language semantics is going to need an explicit revelation of whether the implicit logical underpinnings are based on CWA or OWA. This is because CWA and OWA are two distinct schools of logical thought.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Closed-world assumption

Chapter 2: Frame problem

Chapter 3: Propositional calculus

Chapter 4: Inductive logic programming

Chapter 5: Contradiction

Chapter 6: Intuitionistic logic

Chapter 7: Paraconsistent logic

Chapter 8: Default logic

Chapter 9: Method of analytic tableaux

Chapter 10: Belief revision

(II) Answering the public top questions about closed world assumption.

(III) Real world examples for the usage of closed world assumption in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of closed world assumption' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of closed world assumption.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 26, 2023

Book preview

Chapter 1: Closed-world assumption

In a formal system of logic used for knowledge representation, the closed-world assumption (CWA) is the supposition that a statement that is true is also known to be true. What is false, on the other hand, is what is not now known to be true. The same word also refers to Raymond Reiter's logical formalization of this presumption. The open-world assumption (OWA), which asserts that ignorance does not imply untruth, is the antithesis of the closed-world assumption. The interpretation of the true semantics of a conceptual statement with the same concept notations is determined by decisions on CWA vs. OWA. It is typically impossible to avoid revealing explicitly whether the implicit logical underpinnings are based on CWA or OWA in an effective formalization of natural language semantics.

The closed-world assumption is connected to negation as failure since it equates to accepting as true any premise that cannot be supported by evidence.

When the knowledge base is known to be comprehensive (for example, a corporate database holding records for every employee), the closed-world assumption is used. When the knowledge base is known to be incomplete but a best definite answer must be derived from incomplete information, it is employed. For instance, a query on the individuals who did not edit the article on Formal Logic is typically anticipated to return Sarah Johnson if the database has the following table listing editors who have contributed to a certain article.

In the closed-world assumption, Sarah Johnson is the sole editor who hasn't modified the article on Formal Logic, and the table is presumed to be comprehensive (it shows all editor-article associations). With the open-world assumption, however, it is not expected that the table contains all editor-article tuples, and it is unknown who has not modified the article on Formal Logic. Unknown editors are not shown in the table, and unidentified numbers of the articles that Sarah Johnson edited are also not listed there.

The negation of the literals that are not currently implied by the closed-world assumption is added to the knowledge base as the initial formalization of the assumption in formal logic. If the knowledge base is in Horn form, the outcome of this addition is always consistent, but consistency is not ensured in other cases. As an illustration, the knowledge base

\{English(Fred) \vee Irish(Fred)\}

entails neither English(Fred) nor Irish(Fred) .

The result of adding these two literals' negations to the knowledge base is

\{English(Fred) \vee Irish(Fred), \neg English(Fred), \neg Irish(Fred)\}

which is contradictory.

Alternatively put, Occasionally, this formalization of the closed-world assumption transforms a reliable knowledge base into a flawed one.

The closed-world assumption does not introduce an inconsistency on a knowledge base K exactly when the intersection of all Herbrand models of K is also a model of K ; when a suggestion is made, this condition is equivalent to K having a single minimal model, where a model is minimal if no other model assigns true to a subset of its variables.

Alternative formalizations that do not have this issue have been suggested.

In the description that follows, the considered knowledge base K is assumed to be propositional.

At all times, the formalization of the closed-world assumption is based on adding to K the negation of the formulae that are free for negation for K , i.e, the formulas that are likely to be incorrect.

Alternatively put, the closed-world assumption applied to a knowledge base K generates the knowledge base

{\displaystyle K\cup \{\neg f~|~f\in F\}} .

The set F of formulae that are free for negation in K can be defined in different ways, changing how the closed-world assumption is formalized.

The following are the definitions of f being free for negation in the various formalizations.

CWA (closed-world assumption)

f is a positive literal not entailed by K ; GCWA (generalized CWA)

f is a positive literal such that, for every positive clause c such that K\not \vdash c , it holds K\not \vdash c\vee f ; EGCWA (extended GCWA)

similar to above, but f is a conjunction of positive literals; CCWA (careful CWA)

similar to GCWA, but a positive clause is only taken into account if it is made up of literals (both positive and negative) from two different sets; ECWA (extended CWA)

comparable to CCWA, but f is an arbitrary formula not containing literals from a given set.

On propositional theories, the ECWA and the formalism of circumscription are in agreement.

{End Chapter 1}

Chapter 2: Frame problem

The frame problem in AI refers to a challenge with utilizing first-order logic (FOL) to convey facts about a robot in the real world. It has consequences for cognitive science. Traditional FOL involves the use of numerous axioms that only imply that items in the environment do not change haphazardly in order to represent the state of a robot. For instance, Hayes proposes a block universe with guidelines for how to arrange blocks. A FOL system needs extra axioms in order to draw conclusions about the environment (for example, that a block cannot change position unless it is physically moved). Finding sufficient sets of axioms for a workable description of a robot environment is known as the frame issue..

In relation to the issue of restricting the beliefs that must be updated in response to actions, the frame problem in philosophy started to be more extensively understood. Actions are often defined in the logical context by what they change, with the implicit understanding that everything else (the frame) stays the same.

Even in fairly straightforward domains, the frame problem exists.

An example involving a door, which can be either closed or open, and a lamp, it has an on/off switch, is statically represented by two propositions {\displaystyle \mathrm {open} } and {\displaystyle \mathrm {on} } .

If these circumstances can alter, they are better represented by two predicates {\displaystyle \mathrm {open} (t)} and {\displaystyle \mathrm {on} (t)} that depend on time; These predicates are known as fluents.

a region where the door is shut and the lights are turned off at time 0, Then at time 1, the door opened, the following formulas directly describe in logic:

{\displaystyle \neg \mathrm {open} (0)}{\displaystyle \neg \mathrm {on} (0)}{\displaystyle \mathrm {open} (1)}

The original circumstance is represented by the first two formulas; The result of carrying out the action of opening the door at time 1 is represented by the third formula.

If such a thing required requirements,, like when the door is unlocked, it would have been represented by

{\displaystyle \neg \mathrm {locked} (0)\implies \mathrm {open} (1)}

.

In practice, one would have a predicate {\displaystyle \mathrm {executeopen} (t)} for specifying when an action is executed and a rule

{\displaystyle \forall t.\mathrm {executeopen} (t)\implies \mathrm {open} (t+1)}

for specifying the effects of actions.

More information is provided in the scenario calculus article.

The three formulae mentioned above represent a direct logical expression of what is known, but they are insufficient to derive the proper conclusions. The three formulas above are consistent with the following circumstances, but they are not the only ones. These circumstances describe the expected situation.

In fact, a different set of circumstances that is consistent with the three aforementioned equations is:

The issue with the frame is that listing just the conditions that are altered by the actions does not imply that all other conditions are unaffected.

This problem can be solved by adding the so-called frame axioms, which specifically state that while doing an action, all conditions not affected by it remain unchanged.

For example, since opening the door was the action carried out at time zero, According to a frame axiom, the light's status does not change from time 0 to time 1:

{\displaystyle \mathrm {on} (0)\iff \mathrm {on} (1)}

One such frame axiom is required for every action and condition pair such that the action does not influence the condition, and this poses the frame problem. To formalize a dynamical domain without clearly stating the frame axioms is problematic, to put it another way.

McCarthy offers a solution to this issue by making the assumption that only minor condition changes have taken place; This remedy is formalized within the circumscriptional framework.

The gunshot issue at Yale, however, demonstrates that this answer is not always accurate.

Then different options were suggested, pertaining to prerequisite completion, fluent occlusion, Axioms of successor states, etc; the following explains them.

By the 1980s' conclusion, McCarthy and Hayes' definition of the frame problem was resolved.

after that even, however, the term frame problem was still used, in part to discuss the same issue but in several contexts (e.g, concurrent actions), and partially to discuss the overall issue of representing and using dynamical domains for reasoning.

The frame issue is handled using a variety of formalisms, as seen in the following solutions. The formalisms themselves are not fully explained; instead, condensed versions that fully explain the solution are presented.

Erik Sandewall suggested this approach and developed a formal language for specifying dynamical domains, allowing one to first express a domain in this language and then have it automatically converted into logic. Only the logic statement is demonstrated in this article, and it only uses simplified language without any action names.

This solution's justification aims to illustrate not just the value of conditions over time, but also whether the most recent action can have an impact on them.

Another condition serves as the latter's representative, called occlusion.

If an action has just been carried out that has the consequence of making a condition true or false at a certain time point, the condition is said to be occluded.

Occlusion can be viewed as permission to change: if a condition is occluded, It is released from having to follow the inertia's rule.

In the condensed illustration of the door and the light,, occlusion can be formalized by two predicates {\displaystyle \mathrm {occludeopen} (t)} and {\displaystyle \mathrm {occludeon} (t)} .

A condition can only change value if the matching occlusion predicate is true at the following time point, according to the reasoning.

In turn, Only when a condition-affecting action is taken is the occlusion predicate true.

{\displaystyle \neg \mathrm {open} (0)}{\displaystyle \neg \mathrm {on} (0)}{\displaystyle \mathrm {open} (1)\wedge \mathrm {occludeopen} (1)}{\displaystyle \forall t.\neg \mathrm {occludeopen} (t)\implies (\mathrm {open} (t-1)\iff \mathrm {open} (t))}{\displaystyle \forall t.\neg \mathrm {occludeon} (t)\implies (\mathrm {on} (t-1)\iff \mathrm {on} (t))}

In general, Every action that changes the value of a condition also changes the occlusion predicate's value.

Given this,, {\displaystyle \mathrm {occludeopen} (1)} is true, making the antecedent of the fourth formula above false for t=1 ; therefore, the constraint that

{\displaystyle \mathrm {open} (t-1)\iff \mathrm {open} (t)}

does not hold for t=1 .

Therefore, {\displaystyle \mathrm {open} } can change value, which is also what the third formula enforces.

For this condition to be functional, Only when they are made to be true as a result of an action do occlusion predicates need to be true.

Either circumscription or predicate completion can be used to accomplish this.

It is important to note that occlusion does not always reflect a change, executing the action of opening the door when it was already open (in the formalization above) makes the predicate {\displaystyle \mathrm {occludeopen} } true and makes {\displaystyle \mathrm {open} } true; however, {\displaystyle \mathrm {open} } has not changed value, because it was already true.

Similar to the flowing occlusion solution, this encoding, nonetheless, the extra predicates signify change, not authorization to alter.

For example, {\displaystyle \mathrm {changeopen} (t)} represents the fact that the predicate {\displaystyle \mathrm {open} } will change from time t to t+1 .

The result is, If and only if the matching change predicate is true, a predicate changes.

If and only if an action changes a condition from false to true or vice versa, then a change has occurred.

{\displaystyle \neg \mathrm {open} (0)}{\displaystyle \neg \mathrm {on} (0)}{\displaystyle \neg \mathrm {open} (0)\implies \mathrm {changeopen} (0)}{\displaystyle \forall t.\mathrm {changeopen} (t)\iff (\neg \mathrm {open} (t)\iff \mathrm {open} (t+1))}{\displaystyle \forall t.\mathrm {changeon} (t)\iff (\neg \mathrm {on} (t)\iff \mathrm {on} (t+1))}

A different approach to state that opening the door causes it to open is in the third formula.

Precisely, It says that if the door was previously closed, opening it alters its condition.

The last two conditions state that a condition changes value at time t if and only if the corresponding change predicate is true at time t .

To finish the answer, As few time points as feasible must pass before the change predicates are true, and doing so involves using predicate completion on the rules describing how actions have an effect.

The fact that a condition is true if and only if can be used to ascertain the value of a condition after an action has been completed:

the event causes the circumstance to occur; alternatively

the operation doesn't turn a condition that was previously true into a false one.

These two facts are formalized in logic as a successor state axiom.

For example, if {\displaystyle \mathrm {opendoor} (t)} and {\displaystyle \mathrm {closedoor} (t)} are