Towards Polynomial Approximations of Full Propositional Logic

Chapter 20 - Automated Deduction for Many-Valued Logics

The universe of propositional approximations

We seek a better understanding of why an inferential semantics devised by Tor Sandqvist yields full classical logic, by providing and analysing a direct proof via a suitable maximality construction.

The main purpose of this note is to present “difficult” embeddings of minimal and full intuitionistic logic into classical linear logic, and to prove their soundness and faithfulness. Moreover, it is also pointed out that Girard's translation of intuitionistic logic into classical linear logic is provably equivalent to one of the translations considered in this paper.

We propose a new direct presentation of full (propositional) classical logic by means of proof nets, for which strong normalization and confluence hold true. Our proposal is based on the breaking of symmetry between propositional connectives, combined with a principle of focusing/defocusing.

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

Chapter 1 Knowledge Representation and Classical Logic

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].

Description logics provide powerful languages for representing and reasoning about knowledge of static application domains. The main strength of description logics is that they offer considerable expressive power going far beyond propositional logic, while reasoning is still decidable. There is a demand to bring the power and character of description logics into the description and reasoning of dynamic application domains which are characterized by actions. In this paper, based on a combination of the propositional dynamic logic PDL, a family of description logics and an action formalism constructed over description logics, we propose a family of dynamic description logics DDL(X @) for representing and reasoning about actions, where X represents well-studied description logics ranging from the to the , and X @ denotes the extension of X with the @ constructor. The representation power of DDL(X @) is reflected in four aspects. Firstly, the static knowledge of application domains is represented as RBoxes and acyclic TBoxes of the description logic X. Secondly, the states of the world and the pre-conditions of atomic actions are described by ABox assertions of the description logic X @, and the post-conditions of atomic actions are described by primitive literals of X @. Thirdly, starting with atomic actions and ABox assertions of X @, complex actions are constructed with regular program constructors of PDL, so that various control structures on actions such as the "Sequence", "Choice", "Any-Order", "Iterate", "If-Then-Else", "Repeat-While" and "Repeat-Until" can be represented. Finally, both atomic actions and complex actions are used as modal operators for the construction of formulas, so that many properties on actions can be explicitly stated by formulas. A tableau-algorithm is provided for deciding the satisfiability of DDL(X @)-formulas; based on this algorithm, reasoning tasks such as the realizability, executability and projection of actions can be effectively carried out. As a result, DDL(X @) not only offers considerable expressive power going beyond many action formalisms which are propositional, but also provides decidable reasoning services for actions described by it.

A universal model of a database D and a set Σ of integrity constraints is a database that extends D, satisfies Σ, and is most general in the sense that it contains sound and complete information. Universal models have a number of applications including answering conjunctive queries, and deciding containment of conjunctive queries, with respect to databases with integrity constraints. Furthermore, they are used in slightly modified form as solutions in data exchange. In general, it is undecidable whether a database possesses a universal model, but in the past few years researchers identified various settings where this problem is decidable, and even efficiently solvable.This paper focuses on computing universal models under finite sets of guarded TGDs, non-conflicting keys, and negative constraints. Such constraints generalize inclusion dependencies, and were recently shown to be expressive enough to capture certain members of the DL-Lite family of description logics. The main result is an algorithm that, given a database without null values and a finite set Σ of such constraints, decides whether there is a universal model, and if so, outputs such a model. If Σ is fixed, the algorithm runs in polynomial time. The algorithm can be extended to cope with databases containing nulls; however, in this case, polynomial running time can be guaranteed only for databases with bounded block size.

The Nonassociative Lambek Calculus (NL) represents a logic devoid of the structural rules of exchange, weakening, and contraction, and it does not presume the associativity of its connectives. Its finitary consequence relation is decidable in polynomial time. However, the addition of classical connectives conjunction and disjunction (FNL) makes the consequence relation undecidable. Interestingly, if these connectives are distributive, the consequence relation is decidable in exponential time. This paper provides the proof, that we can merge classical logic with NL (i.e. BFNL) and intuitionistic logic with NL (i.e. HFNL), and still consequence relations are decidable in exponential time.

Encoding deductive argumentation in quantified Boolean formulae

This paper deals with automated deduction for classical and partial logics, especially for the three-valued logic L3, which has been introduced, for example, in the study of natural language semantics. Based on ideas from a Plaisted's Gentzen style system for classical two-valued logic, we present a new tree-structured proof procedure (TMPR) together with a new completeness proof using proof transformation techniques and some improvements including the generation and use of lemmata. TMPR extends SLD-resolution with a Prolog-style backward chaining to full first-order logic by a controlled use of case analysis. This is done without having to extend negative goals needed, for example, for model elimination. A classification of TMPR, model elimination and related calculi in a common tableau framework is given. Thereafter, we present our extension of the TMPR proof procedure to L3 and show its soundness and completeness. As a side result, a TMPR proof system for the four-valued logic L4 is given. Finally, the restriction of TMPR to ‘L3-Horn clauses’ is considered, and, additionally, an idea for similarly extending model elimination and related systems to L3 (and L4) is illustrated.

In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.

Symmetry had been well studied in classical logics and constraint programming since a decade. Early, Krishna-murthy showed that some tricky formulas admit short proofs when augmenting the propositional logic resolution proof system by the symmetry rule. However, in Artificial Intelligence, we usually manipulate incomplete information and need to include uncertainty to reason on knowledge with exceptions and non-monotonicity. Several non classic logics are introduced for that purpose, but as far as we know, symmetry for these frameworks had not been studied yet. Here, we are interested to extend the notion of symmetry to that non classical logics such as preferential logics, X-logics and default logics, then give a new symmetry inference rule for the X-logics and the default logics. Finally, we show how symmetry reasoning is profitable for these logics and how they handle some symmetries that do not exist in classical logics.