Using Possibilistic Logic for Modeling Qualitative Decision: ATMS-based Algorithms

Sugeno integrals are well-known qualitative aggregation functions in multiple criteria decision making. They return a global evaluation between the minimum and the maximum of the input criteria values. They can model sophisticated aggregation schemes through a system of priorities that applies to any subset of criteria and can take into account some kind of synergy inside subsets of criteria. Although a given Sugeno integral specifies a particular way of implicitly describing a set of entities reaching some global satisfaction level, it is hard to figure out what is the underlying explicit meaning of such an integral in practice (even if the priority level associated to each subset of criteria has a precise meaning). The paper proposes an answer to this problem. Any capacity on a finite set can be represented by a special possibilistic logic base containing positive prioritised clauses, and conversely any possibilistic logic base can represent a set-function. Moreover, Sugeno integral can be represented by a possibilistic logic base expressing how it behaves (thanks to a mapping between the scale and a set of logical atoms reflecting the different values for each criterion). Viewing a Sugeno integral as a set of prioritized logically expressed goals has not only the advantage to make the contents of a Sugeno integral more readable, but it also prompts Sugeno integrals into the realm of logic, and makes it possible to define entailment between them.

The paper survey recent AI-oriented works in qualitative decision developed by the authors in the framework of possibility theory. Lottery-based and act-based axiomatics underlying pessimistic and optimistic criteria for decision under uncertainty are first briefly restated, when uncertainty and preferences are encoded with an ordinal scale. A logical machinery capable of computing optimal decisions in the sense of these criteria is presented. Then an approach to qualitative decision under uncertainty which does not require a commensurateness hypothesis between the uncertainty and the preference scales is proposed; this approach is closely related to nonmonotonic reasoning, but turns our to be ineffective for practical decision. Lastly, the modeling of preference as prioritized sets of goals, as sets of solutions reaching some given level of satisfaction, or in terms of possibilistic constraints is discussed briefly.

This note corrects a claim made in the above-mentioned paper about the exact representation of a conditional preference network by means of a possibilistic logic base with partially ordered symbolic weights. We provide a counter-example that shows that the possibilistic logic representation is indeed not always exact. This is the basis of a short discussion on the difficulty of obtaining an exact representation.

In the framework of quantitative possibility theory, two representation modes were developed: logical-based representation in terms of quantitative possibilistic bases and graphical-based representation in terms of product-based possibilistic networks. This paper deals with logical and graphical representations of uncertain information using a quantitative possibility theory framework. We first provide a deep analysis of the relationships between these two forms of representational frameworks. Then, in the logical setting, we develop syntactic relations between penalty logic and quantitative possibilistic logic. These translations are useful for different applications and are interesting for taking advantage of each format at the inferential level. We also provide an algorithm for reasoning with quantitative possibilistic logic. The algorithm takes inspiration from the syntactic relations between quantitative possibilistic logic bases and penalty logic. Finally, we provide experimental results that compare the uncertainty propagation algorithms developed for the possibilistic product-based networks and the inference algorithms based on WMAXSAT developed in this paper.

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how it can be applied to logical deduction in first-order logic, modal and conditional logics, circumscription and possibilistic logic.

Automated theorem proving is a method to establish or disprove logical theorems. While these can be theorems in the classical mathematical sense, we are more concerned with logical encodings of properties of algorithms, hardware and software. Especially in the area of hardware verification, propositional logic is used widely in industry. Satisfiability Module Theories (SMT) is a set of logics which extend propositional logic with theories relevant for specific application domains. In particular, software verification has received much attention, and efficient algorithms have been devised for reasoning over arithmetic and data types. Built-in support for theories by decision procedures is often significantly more efficient than reductions to propositional logic (SAT). Most efficient SAT solvers are based on the DPLL architecture, which is also the basis for most efficient SMT solvers. The main shortcoming of both kinds of logics is the weak support for non-ground reasoning, which noticeably limits the applicability to real world systems. The Model Evolution Calculus (ME) was devised as a lifting of the DPLL architecture from the propositional setting to full first-order logic. In previous work, we created the solver Darwin as an implementation of ME, and showed how to adapt improvements from the DPLL setting. The first half of this thesis is concerned with ME and Darwin. First, we lift a further crucial ingredient of SAT and SMT solvers, lemma-learning, to Darwin and evaluate its benefits. Then, we show how to use Darwin for finite model finding, and how this application benefits from lemma-learning.

Measures of quantity of information have been studied extensively for more than fifty years. The seminal work on information theory is by Shannon [67]. This work, based on probability theory, can be used in a logical setting when the worlds are the possible events. This work is also the basis of Lozinskii's work [48] for defining the quantity of information of a formula (or knowledgebase) in propositional logic. But this definition is not suitable when the knowledgebase is inconsistent. In this case, it has no classical model, so we have no “event” to count. This is a shortcoming since in practical applications (e.g. databases) it often happens that the knowledgebase is not consistent. And it is definitely not true that all inconsistent knowledgebases contain the same (null) amount of information, as given by the “classical information theory”. As explored for several years in the paraconsistent logic community, two inconsistent knowledgebases can lead to very different conclusions, showing that they do not convey the same information. There has been some recent interest in this issue, with some interesting proposals. Though a general approach for information theory in (possibly inconsistent) logical knowledgebases is missing. Another related measure is the measure of contradiction. It is usual in classical logic to use a binary measure of contradiction: a knowledgebase is either consistent or inconsistent. This dichotomy is obvious when the only deductive tool is classical inference, since inconsistent knowledgebases are of no use. But there are now a number of logics developed to draw non-trivial conclusions from an inconsistent knowledgebase. So this dichotomy is not sufficient to describe the amount of contradiction of a knowledgebase, one needs more fine-grained measures. Some interesting proposals have been made for this. The main aim of this paper is to review the measures of information and contradiction, and to study some potential practical applications. This has significant potential in developing intelligent systems that can be tolerant to inconsistencies when reasoning with real-world knowledge.

Possibilistic networks are important and efficient tools for reasoning under uncertainty. This paper proposes a new approach for decision making under uncertainty based on possibilistic networks. The qualitative possibilistic decision is viewed as a data fusion problem of two particular possibilistic networks: the first one encodes agent's beliefs and the second one represents the qualitative utility. In this framework, we propose a new algorithm for computing optimistic optimal decisions based on merging these two possibilistic networks. We show that the computation of optimal decisions comes down to compute a normalization degree of the junction tree associated with the graph representing the fusion of agent's beliefs and preferences.

A new fuzzy set theory, C-fuzzy set theory, is introduced in this paper. It is a particular case of the classical set theory and satisfies all formulas of the classical set theory. To add a limitation to C-fuzzy set system, in which all fuzzy sets must be “non-uniform inclusive” to each other, then it forms a family of sub-systems, the Z-fuzzy set family. It can be proved that the Z0-fuzzy set system, one of Z-fuzzy set systems, is equivalent to Zadeh’s fuzzy set system. Analysis shows that 1) Zadeh’s fuzzy set system defines the relations A = B and A ⊆ B between two fuzzy sets A and B as “∀u ∈ U, (μA ∈ (u) = μB(u))” and “∀u ∈ U, (μA(u) ≤ μB(u))” respectively is inappropriate, because it makes all fuzzy sets be “non-uniformly inclusive”; 2) it is also inappropriate to define two fuzzy sets’ union and intersection operations as the max and min of their grades of membership, because this prevents fuzzy set’s ability to correctly reflect different kinds of fuzzy phenomenon in the natural world. Then it has to work around the problem by invent unnatural functions that are hard to understand, such as augmenting max and min for union and intersection to min{a + b, 1} and max{a + b – 1, 0}, but these functions are incorrect on inclusive case. If both pairs of definitions are used together, not only are they unnatural, but also they are still unable to cover all possible set relationships in the natural world; and 3) it is incorrect to define the set complement as 1 – μA(u), because it can be proved that set complement cannot exist in Zadeh’s fuzzy set, and it causes confusion in logic and thinking. And it is seriously mistaken to believe that logics of fuzzy sets necessarily go against classical and normal thinking, logic, and conception. The C-fuzzy set theory proposed in this paper overcomes all of the above errors and shortcomings, and more reasonably reflects fuzzy phenomenon in the natural world. It satisfies all relations, formulas, and operations of the classical set theory. It is consistent with normal, natural, and classical thinking, logic, and concepts.

With each superintuitionistic propositional logic L with a disjunction property we associate a set of modal logics the assertoric fragment of which is L. Each formula of these modal logics is interdeducible with a formula representing a set of rules admissible in L. The smallest of these logics contains only formulas representing derivable in L rules while the greatest one contains formulas corresponding to all admissible in L rules. The algebraic semantic for these logics is described.

In this paper we are interested in developing goal-based normative agent architectures. We ask ourselves the question what a normative goal is. To answer this question we introduce a qualitative normative decision theory based on belief (B) and obligation (O) rules. We show that every agent which makes optimal decisions – which we call a BO rational agent – acts as if it is maximizing the set of normative goals that will be achieved. This is the basis of our design of goal-based normative agents.

Three-valued conditional logic (CL), defined by Guzm ́an and Squier (1990) and based on McCarthy’s noncommutative connectives, axiomatises a short-circuit logic (SCL), that is, a logic that prescribes shortcircuit evaluation of conjunction and disjunction. CL defines more identities than three-valued MSCL (Memorising SCL, which also has a two-valued variant). This follows from the fact that the definable connective that prescribes full left-sequential conjunction is commutative in CL. We observe that CL also has a two-valued variant of which the full left-sequential connectives and negation define a commutative logic that is weaker than propositional logic because the absorption laws do not hold. Next, we show that the original, equational axiomatisation of CL is not independent and give several alternative, independent axiomatisations. Finally, we show that in CL, the full left-sequential connectives and negation define Bochvar’s three-valued logic. The paper ends with an appendix on the use of Prover9 and Mace4.

This paper introduces possibilistic functional dependencies. These dependencies are associated with a particular possibility distribution over possible worlds of a classical database. The possibility distribution reflects a layered view of the database. The highest layer of the (classical) database consists of those tuples that certainly belong to it, while the other layers add tuples that only possibly belong to the database, with different levels of possibility. The relation between the confidence levels associated with the tuples and the possibility distribution over possible database worlds is discussed in detail in the setting of possibility theory. A possibilistic functional dependency is a classical functional dependency associated with a certainty level that reflects the highest confidence level where the functional dependency no longer holds in the layered database. Moreover, the relationship between possibilistic functional dependencies and possibilistic logic formulas is established. Related work is reviewed, and the intended use of possibilistic functional dependencies is discussed in the conclusion.

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity are explicitly stated as open in the literature.

The somatic marker hypothesis is one of the more dominant physiological models of human decision-making and yet is seldom applied to decision-making in financial investment scenarios. This study provides preliminary evidence about the application of the somatic marker hypothesis in investment choices using heart rate and skin conductance response measures. Twenty undergraduate students were split equally into expert (defined by familiarity with investments) and novice (no familiarity) groups, because previous research has associated expertise with cognitive differences in decision making scenarios. Both groups completed the Balloon Analogue Risk Task and Behavioral Investment Allocation Strategy task-a computerized simulation of real trading scenarios- as assessments of investment decision-making in conditions of low versus high uncertainty, as defined by the Bayesian calculation (level of certainty is more than: (1 - (-300%))/(300% - (-300%)) = 66.67% (0.67). Results suggest that, although primary inducers (innate physiological responses) support and guide optimal decision-making in conditions of uncertainty, secondary inducers (physiological responses dependent on memory/experience) moderate this effect; that is, the stressful thoughts that accompany the task restrict optimal decision-making. This study contributes to the current knowledge on why emotions in finance can lead people to suboptimal decisions.