Paolo Liberatore

Paolo Liberatore

A superredundant clause is a clause that is redundant in the resolution closure of a formula. The converse concept of superirredundancy ensures membership of the clause in all minimal CNF formulae that are equivalent to the given one. This allows for building formulae where some clauses are fixed when minimizing size. An example are proofs of complexity hardness of the problems of minimal formula size. Others are proofs of size when forgetting variables or revising a formula. Most clauses can be made superirredundant by splitting them over a new variable.

Paolo Liberatore

Abductive forgetting is removing variables from a logical formula while maintaining its abductive explanations. It is carried in two alternative ways depending on its intended application. Both differ from the usual forgetting, which maintains consequences rather than explanations. Differently from that, abductive forgetting from a propositional formula may not be expressed by any propositional formula. A necessary and sufficient condition tells when it is. Checking it is $Π^p_3$-complete. A way to guarantee expressibility of abductive forgetting is to switch from propositional to default logic. Another is to introduce new variables.

Paolo Liberatore

Four algorithms for propositional forgetting are compared. The first performs all possible resolutions and deletes the clauses containing a variable to forget. The second forgets a variable at time by resolving and then deleting all clauses that resolve on that variable. The third outputs the result of all possible linear resolutions on the variables to forget. The fourth generates a clause from the points of contradiction during a backtracking search. The latter emerges as the winner, with the second and first having some role in specific cases. The linear resolution algorithm performs poorly in this implementation.

Paolo Liberatore

Natural revision seems so natural: it changes beliefs as little as possible to incorporate new information. Yet, some counterexamples show it wrong. It is so conservative that it never fully believes. It only believes in the current conditions. This is right in some cases and wrong in others. Which is which? The answer requires extending natural revision from simple formulae expressing universal truths (something holds) to conditionals expressing conditional truth (something holds in certain conditions). The extension is based on the basic principles natural revision follows, identified as minimal change and naivety: change mind as little as possible; believe what not contradicted. The extension says that natural revision restricts changes to the current conditions. A comparison with an unrestricting revision shows what exactly the current conditions are. It is not what currently considered true if it contradicts the new information. It includes something more and more unlikely until the new information is at least possible.

Paolo Liberatore

Forgetting a belief acquisition episode may not cause information loss because of the others. Checking whether it does is not obvious, as the contribution of each belief revision is not isolated from the others, and the same information may be given not directly but by deduction. An algorithm for checking whether forgetting reduces information is given for a number of iterated belief revision operators: lexicographic, natural, severe, plain severe, moderate severe, restrained, very radical and full meet revisions. It may take exponential time in the worst case, which is expected given that the problem is coNP-hard, even in the Horn restriction. It is in coNP for homogeneous sequences of lexicographic revisions.

Paolo Liberatore

The belief revision field is opulent in new proposals and indigent in analyses of existing approaches. Much work hinge on postulates, employed as syntactic characterizations: some revision mechanism is equivalent to some properties. Postulates constraint specific revision instances: certain revisions update certain beliefs in a certain way. As an example, if the revision is consistent with the current beliefs, it is incorporated with no other change. A postulate like this tells what revisions must do and neglect what they can do. Can they reach a certain state of beliefs? Can they reach all possible states of beliefs? Can they reach all possible states of beliefs from no previous belief? Can they reach a dogmatic state of beliefs, where everything not believed is impossible? Can they make two conditions equally believed? An application where every possible state of beliefs is sensible requires each state of beliefs to be reachable. An application where conditions may be equally believed requires such a belief state to be reachable. An application where beliefs may become dogmatic requires a way to make them dogmatic. Such doxastic states need to be reached in a way or another. Not in specific way, as dictated by a typical belief revision postulate. This is an ability, not a constraint: the ability of being plastic, equating, dogmatic. Amnesic, correcting, believer, damascan, learnable are other abilities. Each revision mechanism owns some of these abilities and lacks the others: lexicographic, natural, restrained, very radical, full meet, radical, severe, moderate severe, deep severe, plain severe and deep severe revisions, each of these revisions is proved to possess certain abilities.

Paolo Liberatore

Forgetting a specific belief revision episode may not erase information because the other revisions may provide or entail the same information. Whether it does was proved coNP-hard for sequences of two arbitrary lexicographic revisions or arbitrarily long lexicographic Horn revisions. A polynomial algorithm is presented for the case of two lexicographic Horn revision. Heterogeneous sequences, including revisions other than lexicographic, were proved to belong in Delta2. Their previously proved coNP-hardness is enhanced to Dp-hardness.

Paolo Liberatore

Iterated belief revision requires information about the current beliefs. This information is represented by mathematical structures called doxastic states. Most literature concentrates on how to revise a doxastic state and neglects that it may exponentially grow. This problem is studied for the most common ways of storing a doxastic state. All four methods are able to store every doxastic state, but some do it in less space than others. In particular, the explicit representation (an enumeration of the current beliefs) is the more wasteful on space. The level representation (a sequence of propositional formulae) and the natural representation (a history of natural revisions) are more compact than it. The lexicographic representation (a history of lexicographic revision) is even more compact than them.

Paolo Liberatore

Several forms of iterable belief change exist, differing in the kind of change and its strength: some operators introduce formulae, others remove them; some add formulae unconditionally, others only as additions to the previous beliefs; some only relative to the current situation, others in all possible cases. A sequence of changes may involve several of them: for example, the first step is a revision, the second a contraction and the third a refinement of the previous beliefs. The ten operators considered in this article are shown to be all reducible to three: lexicographic revision, refinement and severe withdrawal. In turn, these three can be expressed in terms of lexicographic revision at the cost of restructuring the sequence. This restructuring needs not to be done explicitly: an algorithm that works on the original sequence is shown. The complexity of mixed sequences of belief change operators is also analyzed. Most of them require only a polynomial number of calls to a satisfiability checker, some are even easier.

Paolo Liberatore

Merging beliefs depends on the relative reliability of their sources. When unknown, assuming equal reliability is unwarranted. The solution proposed in this article is that every reliability profile is possible, and only what holds according to all is accepted. Alternatively, one source is completely reliable, but which one is unknown. These two cases motivate two existing forms of merging: maxcons-based merging and arbitration.

Paolo Liberatore

Logical forgetting may take exponential time in general, but it does not when its input is a single-head propositional definite Horn formula. Single-head means that no variable is the head of multiple clauses. An algorithm to make a formula single-head if possible is shown. It improves over a previous one by being complete: it always finds a single-head formula equivalent to the given one if any.

Paolo Liberatore

Logical forgetting is \np-complete even in the simple case of propositional Horn formulae, and may exponentially increase their size. A way to forget is to replace each variable to forget with the body of each clause whose head is the variable. It takes polynomial time in the single-head case: each variable is at most the head of a clause. Some formulae are not single-head but can be made so to simplify forgetting. They are single-head equivalent. The first contribution of this article is the study of a semantical characterization of single-head equivalence. Two necessary conditions are given. They are sufficient when the formula is inequivalent: it makes two sets of variables equivalent only if they are also equivalent to their intersection. All acyclic formulae are inequivalent. The second contribution of this article is an incomplete algorithm for turning a formula single-head. In case of success, forgetting becomes possible in polynomial time and produces a polynomial-size formula, none of which is otherwise guaranteed. The algorithm is complete on inequivalent formulae.

Paolo Liberatore

Forgetting variables from a propositional formula may increase its size. Introducing new variables is a way to shorten it. Both operations can be expressed in terms of common equivalence, a weakened version of equivalence. In turn, common equivalence can be expressed in terms of forgetting. An algorithm for forgetting and checking common equivalence in polynomial space is given for the Horn case; it is polynomial-time for the subclass of single-head formulae. Minimizing after forgetting is polynomial-time if the formula is also acyclic and variables cannot be introduced, NP-hard when they can.

Paolo Liberatore

Forgetting is removing variables from a logical formula while preserving the constraints on the other variables. In spite of being a form of reduction, it does not always decrease the size of the formula and may sometimes increase it. This article discusses the implications of such an increase and analyzes the computational properties of the phenomenon. Given a propositional Horn formula, a set of variables and a maximum allowed size, deciding whether forgetting the variables from the formula can be expressed in that size is $D^p$-hard in $Σ^p_2$. The same problem for unrestricted propositional formulae is $D^p_2$-hard in $Σ^p_3$.

Paolo Liberatore

Merging beliefs requires the plausibility of the sources of the information to be merged. They are typically assumed equally reliable in lack of hints indicating otherwise; yet, a recent line of research spun from the idea of deriving this information from the revision process itself. In particular, the history of previous revisions and previous merging examples provide information for performing subsequent mergings. Yet, no examples or previous revisions may be available. In spite of the apparent lack of information, something can still be inferred by a try-and-check approach: a relative reliability ordering is assumed, the merging process is performed based on it, and the result is compared with the original information. The outcome of this check may be incoherent with the initial assumption, like when a completely reliable source is rejected some of the information it provided. In such cases, the reliability ordering assumed in the first place can be excluded from consideration. The first theorem of this article proves that such a scenario is indeed possible. Other results are obtained under various definition of reliability and merging.

Paolo Liberatore

A common assumption in belief revision is that the reliability of the information sources is either given, derived from temporal information, or the same for all. This article does not describe a new semantics for integration but the problem of obtaining the reliability of the sources given the result of a previous merging. As an example, the relative reliability of two sensors can be assessed given some certain observation, and allows for subsequent mergings of data coming from them.

Paolo Liberatore, Marco Schaerf

While looking for abductive explanations of a given set of manifestations, an ordering between possible solutions is often assumed. The complexity of finding/verifying optimal solutions is already known. In this paper we consider the computational complexity of finding second-best solutions. We consider different orderings, and consider also different possible definitions of what a second-best solution is.