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It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the “bounded width condition”, i.e...
In monadic programming, datatypes are presented as free algebras, generated by data values, and by the algebraic operations and equations capturing some computational effects. These algebras are free in the sense that they satisfy just the equations imposed by their algebraic theory, and remain free of any additional equations. The consequence is that they do not admit quotient types. This is, of...
Brouwer's continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Martín Escardó proved that it is also inconsistent in type theory.
We show that the ellipsoid method for solving semidefinite programs (SDPs) can be expressed in fixed-point logic with counting (FPC). This generalizes an earlier result that the optimal value of a linear program can be expressed in this logic.
Guarded recursion in the sense of Nakano allows general recursive types and terms to be added to type theory without breaking consistency. Recent work has demonstrated applications of guarded recursion such as programming with codata, reasoning about coinductive types, as well as constructing and reasoning about denotational models of general recursive types. Guarded recursion can also be used as...
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