Distributivity in Łℵ0 and Other Sentential Logics

Certain distributivity results for ukasiewicz's infinite-valued logic ₀ are proved axiomatically (for the first time) with the help of the automated reasoning program OTTER. In addition, nondistributivity results are established for a wide variety of positive substructural logics by the use of logical matrices discovered with the automated model-finding programs MACE and MAGIC.     

An Update on “Might”

This paper is on the update semantics for might of Veltman (1996). Threeconsequence relations are introduced and studied in an abstract setting.Next we present sequent-style systems for each of the consequence relations.We show the logics to be complete and decidable. The paper ends with asyntactic cut elimination result.     

Alternative Translation Techniques for Propositional and First-Order Modal Logics

We describe and analyze techniques, other than the standard relational/functional methods, for translating validity problems of modal logics into first-order languages. For propositional modal logics we summarize the -as-Pow method, a complete and automatic translation into a weak set theory, and then describe an alternative method, which we call algebraic, that achieves the same full generality of -as-Pow but is simpler and computationally more attractive. We also discuss the relationships between the two methods, showing that -as-Pow generalizes to the first-order case. For first-order modal logics, we describe two extensions, of different degrees of generality, of -as-Pow to logics of rigid designators and constant domains.     

A Philosophical Approach to the Concept of Data Model: Is a Data Model, in Fact, a Model?

The design of the database is crucial to the process of designing almost any Information System (IS) and involves two clearly identifiable key concepts: schema and data model, the latter allowing us to define the former. Nevertheless, the term model is commonly applied indistinctly to both, the confusion arising from the fact that in Software Engineering (SE), unlike in formal or empirical sciences, the notion of model has a double meaning of which we are not always aware. If we take our idea of model directly from empirical sciences, then the schema of a database would actually be a model, whereas the data model would be a set of tools allowing us to define such a schema.The present paper discusses the meaning of model in the area of Software Engineering from a philosophical point of view, an important topic for the confusion arising directly affects other debates where model is a key concept. We would also suggest that the need for a philosophical discussion on the concept of data model is a further argument in favour of institutionalizing a new area of knowledge, which could be called: Philosophy of Engineering.     

Adding a temporal dimension to a logic system

We introduce a methodology whereby an arbitrary logic system L can be enriched with temporal features to create a new system T(L). The new system is constructed by combining L with a pure propositional temporal logic T (such as linear temporal logic with Since and Until) in a special way. We refer to this method as adding a temporal dimension to L or just temporalising L. We show that the logic system T(L) preserves several properties of the original temporal logic like soundness, completeness, decidability, conservativeness and separation over linear flows of time. We then focus on the temporalisation of first-order logic, and a comparison is make with other first-order approaches to the handling of time.     

A Subset-Matching Size-Bounded Cache for Testing Satisfiability in Modal Logics

The implementation of efficient decision procedures for modal logics is a major research problem in automated deduction. Caching the result of intermediate consistency checks has experimentally proved to be a very important technique for attaining efficiency. Current state-of-the-art systems implement caching mechanisms based on hash tables. In this paper we present a data type – that we call bit matrix – for caching the (in)consistency of sets of formulas. Bit matrices have three distinguishing features: (i) they can be queried for subsets and supersets, (ii) they can be bounded in size, and (iii) if bounded, they can easily implement different policies to resolve which results have to be kept. We have implemented caching mechanisms based on bit matrices and hash tables in *SAT. In *SAT, the bit matrix cache is bounded, and keeps the latest obtained (in)consistency results. We experiment with the benchmarks proposed for the modal logic K at the TABLEAUX Non Classical Systems Comparison (TANCS) 2000. On the basis of the results, we conclude that *SAT performances are improved by (i) caching the results of intermediate consistency checks, (ii) using bit matrices instead of hash tables, and (iii) storing a small number of results in the bit matrices.     

A theory of nonmonotonic rule systems I

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some premises being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.Work partially supported by NSF grant RII-8610671 and Kentucky EPSCoR program and ARO contract DAAL03-89-K-0124.Work partially supported by NSF grant DMS-8902797 and ARO contract DAAG629-85-C-0018.Work partially supported by NSF grant DMS-8702473.     

Finiteness in Infinite-Valued Łukasiewicz Logic

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic _m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in _m. We also reduce the notion of logical consequence in to the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic.     

Defaults as restrictions on classical Hilbert-style proofs

Since the earliest formalisation of default logic by Reiter many contributions to this appealing approach to nonmonotonic reasoning have been given. The different formalisations are here presented in a general framework that gathers the basic notions, concepts and constructions underlying default logic. Our view is to interpret defaults as special rules that impose a restriction on the juxtaposition of monotonic Hubert-style proofs of a given logicL. We propose to describe default logic as a logic where the juxtaposition of default proofs is subordinate to a restriction condition . Hence a default logic is a pair (L, ) where properties of the logic , like compactness, can be interpreted through the restriction condition . Different default systems are then given a common characterization through a specific condition on the logicL. We also prove cumulativity for any default logic (L, ) by slightly modifying the notion of default proof. We extend, in fact, the language ofL in a way close to that followed by Brewka in the formulation of his cumulative default system. Finally we show the existence of infinitely many intermediary default logics, depending on and called linear logics, which lie between Reiter's and ukaszewicz' versions of default logic.Work carried out in the framework of the agreement between Italian PT Administration and FUBLaforia, Université Paris VI Pierre et Marie Curie, 4 Place Jussieu,Tour 46, 75252 Paris, France     

A logic for the specification and proof of regular controllable processes of CCS

Summary This work has been motivated by the following general problem: find logics for the specification and proof of programs, described by terms of some algebra with given congruence relation. This relation is supposed to define a satisfactory concept for the behavioural comparison of programs. We require these logics to be adequate with respect to the term language, in the sense that two programs, behaviourally equivalent satisfy the same formulas and conversely. The term language considered is the subset of controllable, regular terms of CCS, on a vocabulary of actions A, with observational congruence. A term is said to be controllable if it is congruent to some term without occurrence of . We obtain an adequate logic whose language of formulas is obtained from constants true, false and ¦Nil¦ by using operators , , fixpoint operators, + and a for aA; the latter can be considered as extensions of the operators + and a for aA of CCS. As a result, controllable CCS terms can be considered as formulas of this logic and the problem of program verification is reduced to the proof of the validity of a formula.     

Informational interpretation of substructural propositional logics

This paper deals with various substructural propositional logics, in particular with substructural subsystems of Nelson's constructive propositional logics N^– and N. Doen's groupoid semantics is extended to these constructive systems and is provided with an informational interpretation in terms of information pieces and operations on information pieces.     

Automated Proof Construction in Type Theory Using Resolution

We provide techniques to integrate resolution logic with equality in type theory. The results may be rendered as follows. A clausification procedure in type theory, equipped with a correctness proof, all encoded using higher-order primitive recursion. A novel representation of clauses in minimal logic such that the -representation of resolution steps is linear in the size of the premisses. A translation of resolution proofs into lambda terms, yielding a verification procedure for those proofs. Availability of the power of resolution theorem provers in interactive proof construction systems based on type theory.     

On the completeness of the Lambek Calculus with respect to relativized Relational Semantics

Recently M. Szabolcs [12] has shown that many substructural logics including Lambek CalculusL are complete with respect to relativized Relational Semantics. The current paper proves that it is sufficient forL to consider a relativization to the relation x dividesy in some fixed semigroupG.Research partially supported by the Russian Fund for Fundamental Research Grant 93-012-590.     

One Connection between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers

The language of standard propositional modal logic has one operator ( or ), that can be thought of as being determined by the quantifiers or , respectively: for example, a formula of the form is true at a point s just in case all the immediate successors of s verify .This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of and .     

Proof strategies in linear logic

Linear logic, introduced by J.-Y. Girard, is a refinement of classical logic providing means for controlling the allocation of resources. It has aroused considerable interest from both proof theorists and computer scientists. In this paper we investigate methods for automated theorem proving in propositional linear logic. Both the bottom-up (tableaux) and top-down (resolution) proof strategies are analyzed. Various modifications of sequent rules and efficient search strategies are presented along with the experiments performed with the implemented theorem provers.     

Proof Reflection in Coq

We formalize natural deduction for first-order logic in the proof assistant Coq, using de Bruijn indices for variable binding. The main judgment we model is of the form d[:], stating that d is a proof term of formula under hypotheses it can be viewed as a typing relation by the Curry–Howard isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, we define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction.     

Fuzzy measurement in the Mishnah and the Talmud

I discuss the attitude of Jewish law sources from the 2nd–:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as impossible reduction. This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to the largest part of a divided whole, or to the first incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of impossible reduction were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious world that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of impossible reduction. Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of impossible reduction could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. Reduction is impossible may thus be reconstructed as there is no core to the intersection of two measures. I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of - _A (upper approximation minus lower approximation), where the problem of impossible reduction applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on -cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as U-uncertainty, defined, for a non-empty fuzzy set A on the set of real numbers, whose -cuts are intervals of real numbers, as U(A) = 1/h(A) ₀ ^h(A) log [1+(A)]d, where h(A) is the largest membership value obtained by any element of A and (A) is the measure of the -cut of A defined by the Lebesge integral of its characteristic function.     

A Typology of Translation Problems for Eurotra Translation Machines

This paper presents a detailed study of Eurotra Machine Translation engines, namely the mainstream Eurotra software known as the E-Framework, and two unofficial spin-offs – the C,A,T and Relaxed Compositionality translator notations – with regard to how these systems handle hard cases, and in particular their ability to handle combinations of such problems. In the C,A,T translator notation, some cases of complex transfer are wild, meaning roughly that they interact badly when presented with other complex cases in the same sentence. The effect of this is that each combination of a wild case and another complex case needs ad hoc treatment. The E-Framework is the same as the C,A,T notation in this respect. In general, the E-Framework is equivalent to the C,A,T notation for the task of transfer. The Relaxed Compositionality translator notation is able to handle each wild case (bar one exception) with a single rule even where it appears in the same sentence as other complex cases.     

On Bounding Solutions of Underdetermined Systems

Sufficient conditions for the existence and uniqueness of a solution x* D (R ⁿ ) of Y(x) = 0 where : R ⁿ R ^m (m n) with C ²(D) where D R ⁿ is an open convex set and Y = (x)⁺ are given, and are compared with similar results due to Zhang, Li and Shen (Reliable Computing 5(1) (1999)). An algorithm for bounding zeros of f (·) is described, and numerical results for several examples are given.