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Orderincompleteness and finite lambda models, extended abstract, in: LICS ’96
 Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society
, 1996
"... Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory ofβ orβηconversion, the partial order is ..."
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Cited by 8 (1 self)
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Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory ofβ orβηconversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the typefree lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal’cev operators, of those lambda theories which cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the typefree lambda calculus. We introduce partial syntactical lambda models, which are derived from Plotkin’s syntactical models of reduction, and we investigate how these models can be used as practical tools for giving finitary proofs of term inequalities. We give a 3element model as an example. 1
On the complexity of some Maltsev conditions
, 2006
"... Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but th ..."
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Cited by 4 (3 self)
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Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of twogenerated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity. 1.
Functionality, polymorphism, and concurrency: a mathematical investigation of programming paradigms
, 1997
"... ..."
A CONGRUENCE IDENTITY SATISFIED BY mPERMUTABLE VARIETIES
, 2005
"... Abstract. We present a new and useful congruence identity satisfied by mpermutable varieties. It has been proved in [L1] that every mpermutable variety satisfies a nontrivial lattice identity (depending only on m). In [L2] we have found another interesting identity: Theorem 1. For m ≥ 3, every m ..."
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Abstract. We present a new and useful congruence identity satisfied by mpermutable varieties. It has been proved in [L1] that every mpermutable variety satisfies a nontrivial lattice identity (depending only on m). In [L2] we have found another interesting identity: Theorem 1. For m ≥ 3, every mpermutable variety satisfies the con] − 1 gruence identity αβh = αγh, for h = m [ m+1 2 Here, [ ] denotes integer part, and βh, γh are defined as usual: βn+1 = β + αγn, β0 = γ0 = 0, γn+1 = γ + αβn. The proof of Theorem 1 consists of two steps. As for the second step, it is an easy application of commutator theory, but in the present note we shall be concerned only with the first step. The first step is a commutatorfree proof of the following Theorem 2. If every subalgebra of A 2 is mpermutable then A satisfies (Xm). More generally: (i) If every congruence of A, thought of as a subalgebra of A 2, is mpermutable then A satisfies (Xm). (ii) If every subalgebra of A 2 generated by m + 1 elements is mpermutable then A satisfies (Xm); actually, A satisfies the stronger version of (Xm) in which α is only supposed to be a compatible relation on A, and δ is any relation on A. In the statement of Theorem 2 we have used: 2000 Mathematics Subject Classification. 08A30, 08B05. Key words and phrases. Congruence mpermutable varieties, congruence identities, HagemannMitschke’s terms.
Space complexity of list Hcolouring: a dichotomy
, 2013
"... The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NPcomplete (FederVardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph ..."
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The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NPcomplete (FederVardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is hard for NL. Our algorithm operates by reducing the lists in a complex manner based on a novel decomposition of an auxiliary digraph, combined with repeated applications of Reingold’s algorithm for undirected reachability (2005). We also prove an algebraic version of this dichotomy: the digraphs without a circular N are precisely those that admit a finite chain of polymorphisms satisfying the HagemannMitschke identities. This confirms a conjecture of Larose and Tesson (2007) for LHOM(H). Moreover, we show that the presence of a circular N can be decided in time polynomial in the size of H. 1
OrderIncompleteness and Finite Lambda Models Extended Abstract
"... Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of ¡ or ¡£ ¢conversion, the partial or ..."
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Many familiar models of the typefree lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of ¡ or ¡£ ¢conversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the typefree lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal’cev operators, of those lambda theories which cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the typefree lambda calculus. We introduce partial syntactical lambda models, which are derived from Plotkin’s syntactical models of reduction, and we investigate how these models can be used as practical tools for giving finitary proofs of term inequalities. We give a 3element model as an example. 1
To appear in Theoretical Computer Science. OrderIncompleteness and Finite Lambda Reduction Models
"... Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of β or βηconversion, the partial order ..."
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Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper provides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of β or βηconversion, the partial order is trivial on term denotations. Equivalently, the open and closed term algebras of the untyped lambda calculus cannot be nontrivially partially ordered. Our second result is a syntactical characterization, in terms of socalled generalized Mal’cev operators, of those lambda theories which cannot be induced by any nontrivially partially ordered model. We also consider a notion of finite models for the untyped lambda calculus, or more precisely, finite models of reduction. We demonstrate how such models can be used as practical tools for giving finitary proofs of term inequalities. 1
Towards a Dichotomy Theorem for the Counting Constraint Satisfaction Problem
"... The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number ..."
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The Counting Constraint Satisfaction Problem (#CSP) can be expressed as follows: given a set of variables, a set of values that can be taken by the variables, and a set of constraints specifying some restrictions on the values that can be taken simultaneously by some variables, determine the number of assignments of values to variables that satisfy all the constraints. The #CSP provides a general framework for numerous counting combinatorial problems including counting satisfying assignments to a propositional formula, counting graph homomorphisms, graph reliability and many others. This problem can be parametrized by the set of relations that may appear in a constraint. In this paper we start a systematic study of subclasses of the #CSP restricted in this way. The ultimate goal of this investigation is to distinguish those restricted subclasses of the #CSP which are solvable in polynomial time from those which are not. We show that the complexity of any restricted #CSP class on a finite domain can be deduced from the properties of polymorphisms of the allowed constraints, similar to that for the decision constraint satisfaction problem. Then we prove that if a subclass of the #CSP is solvable in polynomial time, then constraints allowed by the class satisfy some very restrictive condition: they need to have a Mal’tsev polymorphism, that is a ternary operation m(x, y, z) such that m(x, y, y) = m(y, y, x) = x. This condition uniformly explains many existing complexity results for particular cases of the #CSP, including the dichotomy results for the problem of counting graph homomorphisms, and it allows us to obtain new results.
VARIETIES WHOSE TOLERANCES ARE HOMOMORPHIC IMAGES OF THEIR CONGRUENCES
, 2012
"... The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsevlike condition, we characterize varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove ..."
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The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsevlike condition, we characterize varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence npermutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.