Two types of explanations have been receiving increased attention in the literature when analyzing the decisions made by classifiers. The first type explains why a decision was made and is known as a sufficient reason for the decision, also an abductive explanation or a PI-explanation. The second type explains why some other decision was not made and is known as a necessary reason for the decision, also a contrastive or counterfactual explanation. These explanations were defined for classifiers with binary, discrete and, in some cases, continuous features. We show that these explanations can be significantly improved in the presence of non-binary features, leading to a new class of explanations that relay more information about decisions and the underlying classifiers. Necessary and sufficient reasons were also shown to be the prime implicates and implicants of the complete reason for a decision, which can be obtained using a quantification operator. We show that our improved notions of necessary and sufficient reasons are also prime implicates and implicants but for an improved notion of complete reason obtained by a new quantification operator that we also define and study.

A central quest in explainable AI relates to understanding the decisions made by (learned) classifiers. There are three dimensions of this understanding that have been receiving significant attention in recent years. The first dimension relates to characterizing conditions on instances that are necessary and sufficient for decisions, therefore providing abstractions of instances that can be viewed as the "reasons behind decisions." The next dimension relates to characterizing minimal conditions that are sufficient for a decision, therefore identifying maximal aspects of the instance that are irrelevant to the decision. The last dimension relates to characterizing minimal conditions that are necessary for a decision, therefore identifying minimal perturbations to the instance that yield alternate decisions. We discuss in this tutorial a comprehensive, semantical and computational theory of explainability along these dimensions which is based on some recent developments in symbolic logic. The tutorial will also discuss how this theory is particularly applicable to non-symbolic classifiers such as those based on Bayesian networks, decision trees, random forests and some types of neural networks.

In this paper, we investigate CNF encodings, for which unit propagation is strong enough to derive a contradiction if the encoding is not consistent with a partial assignment of the variables (unit refutation complete or URC encoding) or additionally to derive all implied literals if the encoding is consistent with the partial assignment (propagation complete or PC encoding). We prove an exponential separation between the sizes of PC and URC encodings without auxiliary variables and strengthen the known results on their relationship to the PC and URC encodings that can use auxiliary variables. Besides of this, we prove that the sizes of any two irredundant PC formulas representing the same function differ at most by a factor polynomial in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.

In this paper we investigate CNF formulas, for which the unit propagation is strong enough to derive a contradiction if the formula together with a partial assignment of the variables is unsatisfiable (unit refutation complete or URC formulas) or additionally to derive all implied literals if the formula is satisfiable (propagation complete or PC formulas). If a formula represents a function using existentially quantified auxiliary variables, it is called an encoding of the function. We prove several results on the sizes of PC and URC formulas and encodings. One of them are separations between the sizes of formulas of different types. Namely, we prove an exponential separation between the size of URC formulas and PC formulas and between the size of PC encodings using auxiliary variables and URC formulas. Besides of this, we prove that the sizes of any two irredundant PC formulas for the same function differ at most by a polynomial factor in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.

Prime compilation, i.e., the generation of all prime implicates or implicants (primes for short) of formulae, is a prominent fundamental issue for AI. Recently, the prime compilation for non-clausal formulae has received great attention. The state-of-the-art approaches generate all primes along with a prime cover constructed by prime implicates using dual rail encoding. However, the dual rail encoding potentially expands search space. In addition, constructing a prime cover, which is necessary for their methods, is time-consuming. To address these issues, we propose a novel two-phase method -- CoAPI. The two phases are the key to construct a cover without using dual rail encoding. Specifically, given a non-clausal formula, we first propose a core-guided method to rewrite the non-clausal formula into a cover constructed by over-approximate implicates in the first phase. Then, we generate all the primes based on the cover in the second phase. In order to reduce the size of the cover, we provide a multi-order based shrinking method, with a good tradeoff between the small size and efficiency, to compress the size of cover considerably. The experimental results show that CoAPI outperforms state-of-the-art approaches. Particularly, for generating all prime implicates, CoAPI consumes about one order of magnitude less time.

Proper epistemic knowledge bases (PEKBs) are syntactic knowledge bases that use multi-agent epistemic logic to represent nested multi-agent knowledge and belief. PEKBs have certain syntactic restrictions that lead to desirable computational properties; primarily, a PEKB is a conjunction of modal literals, and therefore contains no disjunction. Sound entailment can be checked in polynomial time, and is complete for a large set of arbitrary formulae in logics K n and KD n . In this paper, we extend PEKBs to deal with a restricted form of disjunction: 'knowing whether.' An agent i knows whether Q iff agent i knows Q or knows not Q; that is, []Q or []not(Q). In our experience, the ability to represent that an agent knows whether something holds is useful in many multi-agent domains. We represent knowing whether with a modal operator, and present sound polynomial-time entailment algorithms on PEKBs with the knowing whether operator in K n and KD n , but which are complete for a smaller class of queries than standard PEKBs.

In many scenarios where the integration of information into a knowledge base (KB) leads to inconsistencies there is a need to change the KB minimally. In belief revision, relevance postulates meet the minimality requirement by restricting the elimination of KB elements to those that are relevant for the incoming information. This paper focuses on two minimality postulates in an ontology revision scenario in which conflicts are caused by ambiguous use of symbols: a relevance postulate and a generalized inclusion postulate which limits the creativity of the operators. Both postulates exploit the (satisfiably) equivalent representation of a first-order logic KB by its prime implicates, which, intuitively, represent the most atomic logical components of the KB. The paper shows that reinterpretation operators (which are ontology revision operators) fulfill both postulates.

Formula compilation by generation of prime implicates or implicants finds a wide range of applications in AI. Recent work on formula compilation by prime implicate/implicant generation often assumes a Conjunctive/Disjunctive Normal Form (CNF/DNF) representation. However, in many settings propositional formulae are naturally expressed in non-clausal form. Despite a large body of work on compilation of non-clausal formulae, in practice existing approaches can only be applied to fairly small formulae, containing at most a few hundred variables. This paper describes two novel approaches for the compilation of non-clausal formulae either with prime implicants or implicates, that is based on propositional Satisfiability (SAT) solving. These novel algorithms also find application when computing all prime implicates of a CNF formula. The proposed approach is shown to allow the compilation of non-clausal formulae of size significantly larger than existing approaches.

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in $n$ Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of $2^{\log^{1-o(1)} n}$. This is the case even when the inputs are restricted to pure Horn 3-CNFs with $O(n^{1+\varepsilon})$ clauses, for some small positive constant $\varepsilon$. Furthermore, we show that even allowing sub-exponential time computation, it is still not possible to obtain constant factor approximations for such problems unless the Exponential Time Hypothesis turns out to be false.