1. Contradiction-tolerant TMS
CTMS extends the original truth maintenance approach as follows:
Makes it possible to represent and reason about knowledge which is both, incomplete and uncertain.
Makes it possible to explicitly maintain uncertainty during the reasoning process, similar to systems for handling uncertain knowledge.
Makes it possible to handle different types of inconsistencies without trying to resolve them before the reasoning process is completed.

2. Contradiction-tolerant TMS: motivation
If we assume that human beliefs can be divided into two groups, one containing
beliefs that will never change (“firm beliefs”), and the other containing tentative
beliefs, then in order to represent both types of beliefs we need two types of
propositions:
Certain propositions, which represent firm beliefs. They can be true or false, but we can eliminate false propositions by considering their negations as true propositions. We say that a proposition is true, if the evidence associated with that proposition confirms it.
Uncertain (plausible) propositions, which represent tentative beliefs. Their plausibility (or “degree of belief”) depends on the evidence associated with them. We can view unknown propositions as a special case of uncertain propositions, which have no evidence whatsoever.

3. Contradiction-tolerant TMS: motivation
Note that we cannot use uncertain propositions as premises for either
monotonic or non-monotonic inference rules, where each premise is assumed
to be believed or disbelieved.
One possible way to explicitly account for the uncertainty of the premises is to
“relax'' inference rules by dividing their premises into two groups:
The first group consists of those premises which, if true, constitute the minimal evidence for considering the rule's conclusion as a plausible belief.
The second group consists of premises which, along with the premises of the first group, form the complete evidence for the conclusion.

4. CTMS: motivation (cont.)
Thereby, we get inference rules of the form
(A1, ..., Ai)(Ai+1,...,An)  B
which are interpreted as follows: “If A1, ..., Ai are true beliefs, in spite of the
fact that Ai+1,...,An are not known to be true, B is a plausible belief.''
Note that if Ai+1,...,An are consistent assumptions, we get a semi-normal
default rule
A1, ..., Ai : B,  Ai+1,...,  An B
However, contrary to default rules, CTMS-rules do not require any sort of
consistency check, because it is assumed that the minimal evidence A1, ..., Ai
is enough to support B as a plausible belief, even if its negation is also a
plausible, or even a true belief.

5. CTMS contradictions
Note: Rule conclusion, B, and its negation, B, may be supported by different sets of arguments, and in most cases there may not be an objective criterion according to which these sets of arguments can be compared.
However, if both B and B happen to be true at the same time we get a
contradiction. This can happen, for example, if the system receives its
knowledge (data and rules) from independent sources, and there is no way to
resolve the conflict at the meta-level before entering the contradictory rules or
facts into the system. Such contradictions are called objective contradictions.
Their resolution is not trivial because the system has no reason to prefer one of
the contradictory beliefs over the other one (as it is the case with the plausibility
theory or JTMS).
The only way for the system to cope with objective contradictions is to
appropriately revise its knowledge so as to make it compatible with the
detected contradictions. But this should happen only after the reasoning
process is completed and all the evidence relevant to conflicting beliefs is
collected.

6. Logical specification of CTMS
To formally describe both certain and plausible beliefs, we need two
different but closely related languages, L and Le.
L consists of propositional constants A, B, C, ...(possibly subscripted),
and a special operator,  , such that if A is a wff, then A is a wff too.
We do not assign any logical properties to this operator.
The only function of L is to provide a set of wffs that might become
elements of the system's set of beliefs.

7. Logical specification of CTMS (cont.)
The language Le consists of special objects called endorsed formulas, which we
denote as α, β, γ, … Endorsed formulas have the following form:
α = ALV:(T1,...,Tn)(P1,...,Pm), where:
A  L is one of the system's beliefs; we refer to it as the head of the endorsed formula α.
LV is the label, which shows the type of A. It can be:
T, if A is a logically true belief.
T*, if A is an evidentially true belief (i.e. true with respect to a given context).
P, if A is a plausible belief.
T1,...,Tn  L is a set of true arguments for A. We call it the T-set for A. The only restriction over the T-set is that  A should not belong to it.
P1,...,Pm  L is a set of beliefs upon which A depends, but which are not known to be true. We call them the P-set for A. If later some of these beliefs turns out to be true, then it will become an additional evidence for A, and will be shifted to the corresponding T-set.

8. Knowledge representation in CTMS
Depending on the label and content of the T- and P-sets, we
distinguish between:
T-formulas. These have the label T or T*, and empty P-sets, and represent logically or evidentially true beliefs. T-formulas with both T- and P-sets empty represent input data.
P-formulas. These have the label P, and non-empty P-sets and represent plausible beliefs. P-formulas with both T- and P-sets empty represent beliefs that the CTMS knows nothing about.
Definition. Every finite set of endorsed formulas is called a set of
beliefs (SB).

9. CTMS rules
CTMS-inference rules are expressions of the following two types:
(T1,...,Tn)()  AT , called a T-rule.
Here: T1,...,Tn are called T-premises,
A is the head of the T-formula AT :(T1,...,Tn)(), which is the
conclusion of the T-rule.
(T1,...,Tn)(P1,...,Pm),  AP , called a P-rule.
Here: P1,...,Pm are called P-premises,
A is the head of the P-formula AP : (T1,...,Tn)(P1,...,Pm), which
is the conclusion of the P-rule.
T-rules represent deductive knowledge about the domain, and P-rules
sanction a new belief if the minimal evidence for that belief has been
established.

10. Examples of P-rules Example 1: Consider the following rule
(BirdTweety) (PenguinTweety, OstrichTweety)  FliesTweetyP,
If BirdTweety is true, then we can derive FliesTweety, even though we do not know if Tweety is a penguin or an ostrich.
Example 2: Consider the rule
(PositiveResultOfTheBodyScanner) (Headache, Neurosis,
MentalDisturbances)  DiagnoseBrainTumorP
This rule implies DiagnoseBrainTumor as a possible diagnosis if PositiveResultOfTheBodyScanner is true, even though additional symptoms such as headache, neurosis and mental disturbances, which usually accompany the disease, are not observed in a particular case.

11. If any (or several, or all in the extreme case) of the P-premises of a P-rule
become true, then the degree of belief in the rule's conclusion should
increase. In such cases the so-called duplicate rules are used instead of the
original P-rules.
Definition. Let (T1,...,Tn)(P1,...,Pm),  AP be a P-rule. Then:
(T1,...,Tn, P1,...,Pm)(),  AT* is also an inference rule. It is called
the T-duplicate.
For any i1,...,ik  1,...,m, (T1,...,Tn, Pi1, … , Pik)({P1,...,Pm} \ {Pi1, … ,
Pik}),  AP are also inference rules. They are called P-duplicates.
T- and P-duplicates form a set of plausible reasoning patterns that provide
different degrees of plausibility for the conclusion of the original P-rule,
depending on the current evidence about it. The T-duplicate captures
the case in which all elements of the P-set are heads of T-formulas from SB.
Clearly, the conclusion of such a rule should also be a true formula according
to the evidence which was supposed to be the only evidence relevant to it.
Definition. A pair < SB, R>, where SB is a set of beliefs, and R is a set of
inference rules, is called a CTMS-theory.

12. Consider the rule
(BirdTweety)( PenguinTweety,  OstrichTweety)  FliesTweetyP
If we learn that Tweety is not a penguin, then our belief in flying abilities of
Tweety should increase. If we also learn that Tweety is not an ostrich, then
according to this rule we must conclude that Tweety does fly.
But note that Tweety might also be an emu, a non-flying bird we did not know
about , and thus it is not included in the rule's P-set.
In order to minimize the negative effect of such ignorance we must distinguish
between:
T-formulas inferred by T-rules which are logically true. They are marked with the label T.
T-formulas inferred by T-duplicates which are evidentially true. They are marked with the label T*.

13. Representing contradictions in CTMS
Definition. Let < SB, R > be a CTMS-theory, α = AT:(T1,...,Tn)()  SB
and β =  AT: T1,...,Tn)()  SB. The set {α, β} is called a conflicting set and
it identifies a (logical) contradiction between the two beliefs, A and  A.
Each contradiction is described as an endorsed formula of the form
C XP:(X,  X)(CX), where:
T-set consists of the heads of the conflicting endorsed formulas,
P-set contains the contradiction itself.
Definition. The set of endorsed formulas describing contradictions is called
the contradiction-set, and is denoted by CS.
Note: In the same way we can describe contradictions between semantically
incompatible beliefs. That is, if A and B are incompatible beliefs, and we want
to declare that they can never exist together, we must enter the following
endorsed formula: CABP: (A, B)(CAB).
CTMS maintains contradictions in exactly the same way as all other data.

14. Rule revision
To maintain the consistency of the belief set in the presence of a contradiction,
CTMS revises inference rules whose T-premises depend on the contradictory
belief as follows:
If ((T1,...,Tn)()  AT )  R, and for some j (j=1,n), both Tj and  Tj are heads of T-formulas from SB, then
rev ((T1,...,Tn)()  AT ) = ((T1,...,Tn)(CTj)  AP ),
where CTjP: (Tj,  Tj)(CTj).
That is, the original T-rule is transformed into a P-rule which, in turn, changes the status of its conclusion from T-formula to P-formula. Also, adding the head of contradiction CTjP: (Tj,  Tj)(CTj) to the P-set of the revised rule, points out the culprit for that contradiction.
If ((T1,...,Tn)(P1,...,Pm),  AP)  R, and for some j (j=1,n), both Tj and  Tj are heads of T-formulas from SB, then
rev ((T1,...,Tn)(P1,...,Pm)  AP) = (T1,...,Tn)(P1,...,Pm ,CTj)  AP ),
where CTjP: (Tj,  Tj)(CTj).

15. Example Consider the P-rule
(BirdTweety)( PenguinTweety,  OstrichTweety)  FliesTweetyP
and assume PenguinTweety. Since penguins do not fly, the following T-rule
should belong to the set of rules:
(PenguinTweety)()  FliesTweetyT
According to this rule, CTMS will derive FliesTweetyT: (PenguinTweety)()
On the other hand, according to the P-rule above, CTMS will derive
FliesTweetyP: (BirdTweety)( PenguinTweety,  OstrichTweety)
Note: There is no contradiction between the two derived formulas, because a
contradiction only occurs when both a formula and its negation, are
heads of T-formulas from SB.
Also, resolving the conflict between FliesTweetyP and FliesTweetyT
is easy, because CTMS will always prefer a T-formula over a P-formula.

16. Stable extensions of CTMS theories
Given a CTMS-theory <SB, R>, CTMS is intended to construct an extended theory
<SB+, rev(R)>, containing all beliefs from SB, all beliefs that follow from <SB, R>,
and all contradictions detected during the inference process.
Definition. Let <SB, R> be a CTMS-theory, and α = ALV:(T1,...,Tn)(P1,...,Pm), be an
endorsed formula. We say that α follows from <SB, R> if and only if there exists a
rule (T1,...,Tn)(P1,...,Pm)  ALV  R such that T1,...,Tn are heads of T-formulas from
SB, and P1,...,Pm are heads of P-formulas from SB. We denote the set of all
endorsed formulas that follow from <SB, R> by Cons(SB,R), and the new (extended)
set of beliefs by SB+.
Definition. Let <SB, R> be a CTMS-theory. The set SB+ is called the stable
extension of <SB, R> if the following two conditions are satisfied:
SB+ does not contain new conflicting sets, i.e. all contradictions in SB+ have been inherited from SB.
SB+ is closed with respect to the rules of R, i.e. for every rule (T1,...,Tn)(P1,...,Pm)  ALV  R such that T1,...,Tn are heads of T-formulas
from SB, and P1,...,Pm are heads of P-formulas from SB,
α = ALV:(T1,...,Tn)(P1,...,Pm)  SB+ .

17. CTMS algorithm
Let SBi, CSi, Ri be the current set of beliefs, contradiction-set, and set of
inference rules, respectively.
The construction of the stable extension of the CTMS-theory <SB, R> is
carried out according to the following inductive procedure.
Basic step:
Actualization of the set of beliefs: SB0 = SB.
Actualization of the contradiction-set:
j (AjT:(T1,...,Tn)(),  AjT:(Tl,...,Tm)() SB0), (CAjP: (Aj,  Aj)(CAj) CS0.
Revision of the affected rules:
If CS0 = { }, then R0 = R.
If CS0 ≠ { }, then R0 = rev(R).
Inductive step:
Actualization of the set of beliefs:
SBi = SBi-1  Cons(SBi-1,Ri-1)  CSi-1

18. Actualization of the contradiction-set:
j (AjT:(T1,...,Tn)(),  AjT:(Tl,...,Tm)() SBi , but (CAjP: (Aj,  Aj)(CAj) CSi-1),
(CAjP: (Aj,  Aj)(CAj) CSi.
Revision of the affected rules:
If CSi = { }, then Ri = Ri-1.
If CSi ≠ { }, then Ri = rev(Ri-1).
Revision of the affected beliefs: If there is a formula AjLV:(T1,...,Tn)(P1, …, Pm)  SBi, such that for some j (j=1,n), CTjP: (Tj,  Tj)(CTj)  CSi, then:
If it is a T-formula, then rev(AT: (T1,...,Tn)()) = AP: (T1,...,Tn)(CTj).
If it is a P-formula, then rev(AP: (T1,...,Tn)(P1,...,Pm)) = AP: (T1,...,Tn)(P1,...,Pm, CTj).
Check whether the stable extension has been achieved. If yes, then stop.
Every CTMS-theory <SB, R> has a unique stable extension if the initial sets of
beliefs and rules are finite.

19. More plausible vs less plausible beliefs
The stable extension of a CTMS-theory may contain endorsed formulas with the same
heads but different T- and P-sets, which have been inferred by different T- and P-
duplicates of P-rules, as well as formulas whose heads are contradictory literals.
We call such formulas competitive.
To define which of the competitive endorsed formulas are the most plausible conclusions
of a given CTMS-theory, we introduce an order among competitive endorsed formulas by
means of the following rules:
If α = AT* :(T1,...,Tn)(), β = AT:(T1’ ,..., Ti‘ )(), and T1,...,Tn  T1‘ ,..., Ti’, then α < β.
If the T-sets of α and β are incomparable, then α = β.
If α = AP:(T1,...,Tn)(P1,...,Pm), β = AP:(T1’ ,..., Ti‘ )(P1‘ ,...,Pj‘ ), and T1,...,Tn  T1’ ,..., Ti‘ , then α < β. If the T-sets of α and β are incomparable, then α = β.
If α = AP:(T1,...,Tn)(P1,...,Pm) and β = AT:(T1 ,..., Ti )(), then α < β.
If α =  AP:(T1,...,Tn)(P1,...,Pm) and β = AT:(T1 ,..., Ti )(), and some element of the P-set of α is also an element of the T-set of β, then α < β.
If α =  AT:(T1,...,Tn)() and β = AT:(Tl,...,Ts)(), then α = β.
Each set of competitive endorsed formulas has one or more maximal elements according
to this order. These maximal elements are the most plausible conclusions of the CTMS-
theory.