The graphical structure of Probabilistic Graphical The implication problem is the task of determining whether Models (PGMs) encodes the conditional independence a set of CIs termed antecedents logically entail another (CI) relations that hold in the modeled distribution. CI, called the consequent, and it has received considerable Graph algorithms, such as d-separation, attention from both the AI and Database communities use this structure to infer additional conditional [10, 12, 15, 16, 22, 23]. Known algorithms for deriving independencies, and to query whether a specific CIs from the topological structure of the graphical model CI holds in the distribution. The premise of all are, in fact, an instance of implication. Notably, the DAG current systems-of-inference for deriving CIs in structure of Bayesian Networks is generated based on a set PGMs, is that the set of CIs used for the construction of CIs termed the recursive basis [11], and the d-separation of the PGM hold exactly. In practice, algorithms algorithm is used to derive additional CIs, implied by this for extracting the structure of PGMs from set. The d-separation algorithm is a sound and complete data, discover approximate CIs that do not hold exactly method for deriving CIs in probability distributions represented in the distribution. In this paper, we ask how by DAGs [10, 11], and hence completely characterizes the error in this set propagates to the inferred CIs the CIs that hold in the distribution.

Logic-based argumentation is a well-established formalism modelling nonmonotonic reasoning. It has been playing a major role in AI for decades, now. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this paper, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Parson et al., J. Log. Comput., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: first, we consider all possible propositional fragments of the problem within Schaefer's framework (STOC 1978), and then study different parameterizations for each of the fragment. We identify a list of reasonable structural parameters (size of the claim, support, knowledge-base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (paraNP and beyond).

A lattice-theoretic framework is introduced that permits the study of the conditional independence (CI) implication problem relative to the class of discrete probability measures. Semi-lattices are associated with CI statements and a finite, sound and complete inference system relative to semi-lattice inclusions is presented. This system is shown to be (1) sound and complete for saturated CI statements, (2) complete for general CI statements, and (3) sound and complete for stable CI statements. These results yield a criterion that can be used to falsify instances of the implication problem and several heuristics are derived that approximate this "lattice-exclusion" criterion in polynomial time. Finally, we provide experimental results that relate our work to results obtained from other existing inference algorithms.

We formulate a finite axiomatization of the implication problem for inclusion and conditional independence atoms (dependencies) in the dependence logic context. The input of this problem is given by a finite set Σ {φ} consisting of conditional independence atoms and inclusion atoms, and the question to decide is whether the following logical consequence holds Σ φ. (1) Independence logic [12] and inclusion logic [6] are recent variants of dependence logic the semantics of which are defined over sets of assigments (teams) rather than a single assignment as in first-order logic.

The implication problem of probabilistic conditional independencies is investigated in the presence of missing data. Here, graph separation axioms fail to hold for saturated conditional independencies, unlike the known idealized case with no missing data. Several axiomatic, algorithmic, and logical characterizations of the implication problem for saturated conditional independencies are established. In particular, equivalences are shown to the implication problem of a propositional fragment under Levesque's situations, and that of Lien's class of multivalued database dependencies under null values.

Logical inference algorithms for conditional independence (CI) statements have important applications from testing consistency during knowledge elicitation to constraintbased structure learning of graphical models. We prove that the implication problem for CI statements is decidable, given that the size of the domains of the random variables is known and fixed. We will present an approximate logical inference algorithm which combines a falsification and a novel validation algorithm. The validation algorithm represents each set of CI statements as a sparse 0-1 matrix A and validates instances of the implication problem by solving specific linear programs with constraint matrix A. We will show experimentally that the algorithm is both effective and efficient in validating and falsifying instances of the probabilistic CI implication problem.

We investigate the application of Courcelle's Theorem and the logspace version of Elberfeld etal. in the context of the implication problem for propositional sets of formulae, the extension existence problem for default logic, as well as the expansion existence problem for autoepistemic logic and obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu), unless P=NP.

We review and extend earlier work on the logic CFD, a description logic that allows terminological cycles with universal restrictions over functional roles. In particular, we consider the problem of reasoning about concept subsumption and the problem of computing certain answers for a family of attribute-connected conjunctive queries, showing that both problems are in PTIME. We then consider the effect on the complexity of these problems after adding a concept constructor that expresses concept union, or after adding a concept constructor for the bottom class. Finally, we show that adding both constructors makes both problems EXPTIME-complete.

A lattice-theoretic framework is introduced that permits the study of the conditional independence (CI) implication problem relative to the class of discrete probability measures. Semi-lattices are associated with CI statements and a finite, sound and complete inference system relative to semi-lattice inclusions is presented. This system is shown to be (1) sound and complete for saturated CI statements, (2) complete for general CI statements, and (3) sound and complete for stable CI statements. These results yield a criterion that can be used to falsify instances of the implication problem and several heuristics are derived that approximate this "lattice-exclusion" criterion in polynomial time. Finally, we provide experimental results that relate our work to results obtained from other existing inference algorithms.