84,896 research outputs found
Post Completeness in Congruential Modal Logics
Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As Humberstone notes, the issue of Post completeness in congruential modal logics is not well understood. The present article shows that in contrast to normal modal logics, the extent of the property of Post completeness among congruential modal logics depends on the background set of logics. Some basic results on the corresponding properties of Post completeness are established, in particular that although a congruential modal logic is Post complete among all modal logics if and only if its modality is truth-functional, there are continuum many modal logics Post complete among congruential modal logics
forall x: Calgary. An Introduction to Formal Logic
forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), translating (formalizing) English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as truth-functional completeness and modal logic. Exercises with solutions are available. It is provided in PDF (for screen reading, printing, and a special version for dyslexics) and in LaTeX source code
On completeness of logic programs
Program correctness (in imperative and functional programming) splits in
logic programming into correctness and completeness. Completeness means that a
program produces all the answers required by its specification. Little work has
been devoted to reasoning about completeness. This paper presents a few
sufficient conditions for completeness of definite programs. We also study
preserving completeness under some cases of pruning of SLD-trees (e.g. due to
using the cut).
We treat logic programming as a declarative paradigm, abstracting from any
operational semantics as far as possible. We argue that the proposed methods
are simple enough to be applied, possibly at an informal level, in practical
Prolog programming. We point out importance of approximate specifications.Comment: 20 page
Completeness in Equational Hybrid Propositional Type Theory
Equational Hybrid Propositional Type Theory (EHPTT) is a combination of
propositional type theory, equational logic and hybrid modal logic. The structures used to
interpret the language contain a hierarchy of propositional types, an algebra (a nonempty
set with functions) and a Kripke frame.
The main result in this paper is the proof of completeness of a calculus specifically
defined for this logic. The completeness proof is based on the three proofs Henkin published
last century: (i) Completeness in type theory (ii) The completeness of the first-order
functional calculus and (iii) Completeness in propositional type theory. More precisely,
from (i) and (ii) we take the idea of building the model described by the maximal consistent
set; in our case the maximal consistent set has to be named, ♦- saturated and extensionally
algebraic-saturated due to the hybrid and equational nature of EHPTT. From (iii), we use
the result that any element in the hierarchy has a name. The challenge was to deal with
all the heterogeneous components in an integrated system.publishe
A Stone-type Duality Theorem for Separation Logic Via its Underlying Bunched Logics
Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because — in addition to elegant abstraction — they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality theorems for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar Boolean BI, and concluding with Separation Logic. Our results encompass all the known existing algebraic approaches to Separation Logic and prove them sound with respect to the standard store-heap semantics. We additionally recover soundness and completeness theorems of the specific truth-functional models of these logics as presented in the literature. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualises the ‘resource semantics’ interpretation underpinning Separation Logic amongst them. As a consequence, theory from those fields — as well as algebraic and topological methods — can be applied to both Separation Logic and the systems of bunched logics it is built upon. Conversely, the notion of indexed resource frame (generalizing the standard model of Separation Logic) and its associated completeness proof can easily be adapted to other non-classical predicate logics
A Complete and Recursive Feature Theory
Various feature descriptions are being employed in logic programming
languages and constrained-based grammar formalisms. The common notational
primitive of these descriptions are functional attributes called features. The
descriptions considered in this paper are the possibly quantified first-order
formulae obtained from a signature of binary and unary predicates called
features and sorts, respectively. We establish a first-order theory FT by means
of three axiom schemes, show its completeness, and construct three elementarily
equivalent models. One of the models consists of so-called feature graphs, a
data structure common in computational linguistics. The other two models
consist of so-called feature trees, a record-like data structure generalizing
the trees corresponding to first-order terms. Our completeness proof exhibits a
terminating simplification system deciding validity and satisfiability of
possibly quantified feature descriptions.Comment: Short version appeared in the 1992 Annual Meeting of the Association
for Computational Linguistic
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