15 research outputs found
Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin
We analyse the complexity of approximate counting constraint satisfactions
problems , where is a set of
nonnegative rational-valued functions of Boolean variables. A complete
classification is known in the conservative case, where is
assumed to contain arbitrary unary functions. We strengthen this result by
fixing any permissive strictly increasing unary function and any permissive
strictly decreasing unary function, and adding only those to :
this is weak conservativity. The resulting classification is employed to
characterise the complexity of a wide range of two-spin problems, fully
classifying the ferromagnetic case. In a further weakening of conservativity,
we also consider what happens if only the pinning functions are assumed to be
in (instead of the two permissive unaries). We show that any set
of functions for which pinning is not sufficient to recover the two kinds of
permissive unaries must either have a very simple range, or must satisfy a
certain monotonicity condition. We exhibit a non-trivial example of a set of
functions satisfying the monotonicity condition.Comment: 37 page
Varieties of unary-determined distributive -magmas and bunched implication algebras
A distributive lattice-ordered magma (-magma)
is a distributive lattice with a binary operation that preserves joins
in both arguments, and when is associative then is an
idempotent semiring. A -magma with a top is unary-determined if
. These
algebras are term-equivalent to a subvariety of distributive lattices with
and two join-preserving unary operations . We obtain simple
conditions on such that is
associative, commutative, idempotent and/or has an identity element.
This generalizes previous results on the structure of doubly idempotent
semirings and, in the case when the distributive lattice is a Heyting algebra,
it provides structural insight into unary-determined algebraic models of
bunched implication logic. We also provide Kripke semantics for the algebras
under consideration, which leads to more efficient algorithms for constructing
finite models. We find all subdirectly irreducible algebras up to cardinality
eight in which is a closure operator, as well as all finite
unary-determined bunched implication chains and map out the poset of
join-irreducible varieties generated by them
Varieties of unary-determined distributive -magmas and bunched implication algebras
A distributive lattice-ordered magma (-magma)
is a distributive lattice with a binary operation that preserves joins
in both arguments, and when is associative then is an
idempotent semiring. A -magma with a top is unary-determined if
. These
algebras are term-equivalent to a subvariety of distributive lattices with
and two join-preserving unary operations . We
obtain simple conditions on such that is associative, commutative,
idempotent and/or has an identity element.
This generalizes previous results on the structure of doubly idempotent
semirings and, in the case when the distributive lattice is a Heyting algebra,
it provides structural insight into unary-determined algebraic models of
bunched implication logic. We also provide Kripke semantics for the algebras
under consideration, which leads to more efficient algorithms for constructing
finite models. We find all subdirectly irreducible algebras up to cardinality
eight in which is a closure operator, as well as all
finite unary-determined bunched implication chains and map out the poset of
join-irreducible varieties generated by them
Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors
It has been known since Ehrhard and Regnier's seminal work on the Taylor
expansion of -terms that this operation commutes with normalization:
the expansion of a -term is always normalizable and its normal form is
the expansion of the B\"ohm tree of the term. We generalize this result to the
non-uniform setting of the algebraic -calculus, i.e.
-calculus extended with linear combinations of terms. This requires us
to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's
techniques rely heavily on the uniform, deterministic nature of the ordinary
-calculus, and thus cannot be adapted; second is the absence of any
satisfactory generic extension of the notion of B\"ohm tree in presence of
quantitative non-determinism, which is reflected by the fact that the Taylor
expansion of an algebraic -term is not always normalizable. Our
solution is to provide a fine grained study of the dynamics of
-reduction under Taylor expansion, by introducing a notion of reduction
on resource vectors, i.e. infinite linear combinations of resource
-terms. The latter form the multilinear fragment of the differential
-calculus, and resource vectors are the target of the Taylor expansion
of -terms. We show the reduction of resource vectors contains the
image of any -reduction step, from which we deduce that Taylor expansion
and normalization commute on the nose. We moreover identify a class of
algebraic -terms, encompassing both normalizable algebraic
-terms and arbitrary ordinary -terms: the expansion of these
is always normalizable, which guides the definition of a generalization of
B\"ohm trees to this setting
Expressing Ecumenical Systems in the ??-Calculus Modulo Theory
Systems in which classical and intuitionistic logics coexist are called ecumenical. Such a system allows for interoperability and hybridization between classical and constructive propositions and proofs. We study Ecumenical STT, a theory expressed in the logical framework of the ??-calculus modulo theory. We prove soudness and conservativity of four subtheories of Ecumenical STT with respect to constructive and classical predicate logic and simple type theory. We also prove the weak normalization of well-typed terms and thus the consistency of Ecumenical STT
Counting small induced subgraphs satisfying monotone properties
Given a graph property , the problem asks, on input a graph and a positive integer , to compute the number of induced subgraphs of size in that satisfy . The search for explicit criteria on ensuring that is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property , the problem cannot be solved in time for any function , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a -completeness result
Counting small induced subgraphs satisfying monotone properties
Given a graph property , the problem asks, on input a graph and a positive integer , to compute the number of induced subgraphs of size in that satisfy . The search for explicit criteria on ensuring that is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property , the problem cannot be solved in time for any function , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a -completeness result