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Weak Bases of Boolean CoClones
"... Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For classifications where primitive positive definitions are unsu ..."
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Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For classifications where primitive positive definitions are unsuitable, such as sizepreserving reductions, weaker closure operations may be necessary. In this article we consider strong partial clones which can be seen as a more finegrained framework than Post’s lattice where each clone splits into an interval of strong partial clones. We investigate these intervals and give simple relational descriptions, weak bases, of the largest elements. The weak bases have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones is considerably simpler than the full lattice of partial clones.
A note on minors determined by clones of semilattices, arXiv:0809.3234v1
"... Abstract. The Cminor partial orders determined by the clones generated by a semilattice operation (and possibly the constant operations corresponding to the identity or zero elements) satisfy the descending chain condition. 1. Cminors and Cdecompositions Let A be a fixed nonempty base set. An ope ..."
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Abstract. The Cminor partial orders determined by the clones generated by a semilattice operation (and possibly the constant operations corresponding to the identity or zero elements) satisfy the descending chain condition. 1. Cminors and Cdecompositions Let A be a fixed nonempty base set. An operation on A is a map f: An → A for some integer n ≥ 1, called the arity of f. Denote by OA = ⋃ n≥1 AAn the set of all operations on A. The ith nary projection (1 ≤ i ≤ n) is the operation (a1,..., an) ↦ → ai, and it is denoted by x (n) i, or simply by xi when the arity is clear from the context. We say that the ith variable is essential in f: An → A, if there exist elements a1,...,an, b ∈ A such that f(a1,..., ai−1, ai, ai+1,..., an) = f(a1,...,ai−1, b, ai+1,...,an). If the ith variable is not essential in f, then we say that it is inessential in f.
The ubiquity of conservative translations Emil Jeˇrábek ∗
, 2011
"... We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. Th ..."
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We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal. 1
The ubiquity of conservative translations Emil Jeˇrábek ∗
, 2012
"... We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. Th ..."
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We study the notion of conservative translation between logics introduced by Feitosa and D’Ottaviano [7]. We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal. 1