Handling SQL Nulls with Two-Valued Logic

The design of SQL is based on a three-valued logic (3VL), rather than the familiar Boolean logic with truth values true and false, to accommodate the additional truth value unknown for handling nulls. It is viewed as indispensable for SQL expressiveness, but is at the same time much criticized for leading to unintuitive behavior of queries and thus being a source of programmer mistakes. We show that, contrary to the widely held view, SQL could have been designed based on the standard Boolean logic, without any loss of expressiveness and without giving up nulls. The approach itself follows SQL’s evaluation which only retains tuples for which conditions in the WHERE clause evaluate to true. We show that conflating unknown, resulting from nulls, with false leads to an equally expressive version of SQL that does not use the third truth value. Queries written under the two-valued semantics can be efficiently translated into the standard SQL and thus executed on any existing RDBMS. These results cover the core of the SQL 1999 Standard, including SELECT-FROM-WHERE-GROUP BY-HAVING queries extended with subqueries and IN/EXISTS/ANY/ALL conditions, and recursive queries. We provide two extensions of this result showing that no other way of converting 3VL into Boolean logic, nor any other many-valued logic for treating nulls could have possibly led to a more expressive language. These results not only present small modifications of SQL that eliminate the source of many programmer errors without the need to reimplement database internals, but they also strongly suggest that new query languages for various data models do not have to follow the much criticized SQL’s three-valued approach

From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Normality Operators and Classical Collapse

In this paper, we extend the expressive power of the logics K3, LP and FDE with anormality operator, which is able to express whether a for-mula is assigned a classical truth value or not. We then establish classical recapture theorems for the resulting logics. Finally, we compare the approach via normality operator with the classical collapse approach devisedby Jc Beall

On the Relative Expressiveness of Argumentation Frameworks, Normal Logic Programs and Abstract Dialectical Frameworks

We analyse the expressiveness of the two-valued semantics of abstract argumentation frameworks, normal logic programs and abstract dialectical frameworks. By expressiveness we mean the ability to encode a desired set of two-valued interpretations over a given propositional signature using only atoms from that signature. While the computational complexity of the two-valued model existence problem for all these languages is (almost) the same, we show that the languages form a neat hierarchy with respect to their expressiveness.Comment: Proceedings of the 15th International Workshop on Non-Monotonic Reasoning (NMR 2014

FO(FD): Extending classical logic with rule-based fixpoint definitions

We introduce fixpoint definitions, a rule-based reformulation of fixpoint constructs. The logic FO(FD), an extension of classical logic with fixpoint definitions, is defined. We illustrate the relation between FO(FD) and FO(ID), which is developed as an integration of two knowledge representation paradigms. The satisfiability problem for FO(FD) is investigated by first reducing FO(FD) to difference logic and then using solvers for difference logic. These reductions are evaluated in the computation of models for FO(FD) theories representing fairness conditions and we provide potential applications of FO(FD).Comment: Presented at ICLP 2010. 16 pages, 1 figur

Expressive Logics for Coinductive Predicates

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata