5,113 research outputs found
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
Formalization of Universal Algebra in Agda
In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
Canonical extension and canonicity via DCPO presentations
The canonical extension of a lattice is in an essential way a two-sided
completion. Domain theory, on the contrary, is primarily concerned with
one-sided completeness. In this paper, we show two things. Firstly, that the
canonical extension of a lattice can be given an asymmetric description in two
stages: a free co-directed meet completion, followed by a completion by
\emph{selected} directed joins. Secondly, we show that the general techniques
for dcpo presentations of dcpo algebras used in the second stage of the
construction immediately give us the well-known canonicity result for bounded
lattices with operators.Comment: 17 pages. Definition 5 was revised slightly, without changing any of
the result
A Hoare-like logic of asserted single-pass instruction sequences
We present a formal system for proving the partial correctness of a
single-pass instruction sequence as considered in program algebra by
decomposition into proofs of the partial correctness of segments of the
single-pass instruction sequence concerned. The system is similar to Hoare
logics, but takes into account that, by the presence of jump instructions,
segments of single-pass instruction sequences may have multiple entry points
and multiple exit points. It is intended to support a sound general
understanding of the issues with Hoare-like logics for low-level programming
languages.Comment: 22 pages, the preliminaries have textual overlaps with the
preliminaries in arXiv:1402.4950 [cs.LO] and earlier papers; introduction and
conclusions rewritten, explanatory remarks added; introduction partly
rewritten; 24 pages, clarifying examples adde
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