127 research outputs found
Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond
Ruitenburg\u2019s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately periodic if f fixes all the generators but one. More precisely, there is N 65 0 such that f^N+2 = f^N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators
Completeness for Flat Modal Fixpoint Logics
This paper exhibits a general and uniform method to prove completeness for
certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form
\gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the
language L\sharp (\Gamma) is obtained by adding to the language of polymodal
logic a connective \sharp\_\gamma for each \gamma \epsilon. The term
\sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the
least fixed point of the functional interpretation of the term \gamma(x,
\varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma,
construct an axiom system which is sound and complete with respect to the
concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We
prove two results that solve this problem. First, let K\sharp (\Gamma) be the
logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint
axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma.
Provided that each indexing formula \gamma satisfies the syntactic criterion of
being untied in x, we prove this axiom system to be complete. Second,
addressing the general case, we prove the soundness and completeness of an
extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an
effective procedure that, given an indexing formula \gamma as input, returns a
finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded
by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever
\Gamma is finite
An infinitary model of linear logic
In this paper, we construct an infinitary variant of the relational model of
linear logic, where the exponential modality is interpreted as the set of
finite or countable multisets. We explain how to interpret in this model the
fixpoint operator Y as a Conway operator alternatively defined in an inductive
or a coinductive way. We then extend the relational semantics with a notion of
color or priority in the sense of parity games. This extension enables us to
define a new fixpoint operator Y combining both inductive and coinductive
policies. We conclude the paper by sketching the connection between the
resulting model of lambda-calculus with recursion and higher-order
model-checking.Comment: Accepted at Fossacs 201
On the equivalence of game and denotational semantics for the probabilistic mu-calculus
The probabilistic (or quantitative) modal mu-calculus is a fixed-point logic
de- signed for expressing properties of probabilistic labeled transition
systems (PLTS). Two semantics have been studied for this logic, both assigning
to every process state a value in the interval [0,1] representing the
probability that the property expressed by the formula holds at the state. One
semantics is denotational and the other is a game semantics, specified in terms
of two-player stochastic games. The two semantics have been proved to coincide
on all finite PLTS's, but the equivalence of the two semantics on arbitrary
models has been open in literature. In this paper we prove that the equivalence
indeed holds for arbitrary infinite models, and thus our result strengthens the
fruitful connection between denotational and game semantics. Our proof adapts
the unraveling or unfolding method, a general proof technique for proving
result of parity games by induction on their complexity
DDK dependent regulation of TOP2A at centromeres revealed by a chemical genetics approach
In eukaryotic cells the CDC7/DBF4 kinase, also known as DBF4-dependent kinase (DDK), is required for the firing of DNA replication origins. CDC7 is also involved in replication stress responses and its depletion sensitises cells to drugs that affect fork progression, including Topoisomerase 2 poisons. Although CDC7 is an important regulator of cell division, relatively few substrates and bona-fide CDC7 phosphorylation sites have been identified to date in human cells. In this study, we have generated an active recombinant CDC7/DBF4 kinase that can utilize bulky ATP analogues. By performing in vitro kinase assays using benzyl-thio-ATP, we have identified TOP2A as a primary CDC7 substrate in nuclear extracts, and serine 1213 and serine 1525 as in vitro phosphorylation sites. We show that CDC7/DBF4 and TOP2A interact in cells, that this interaction mainly occurs early in S-phase, and that it is compromised after treatment with CDC7 inhibitors. We further provide evidence that human DBF4 localises at centromeres, to which TOP2A is progressively recruited during S-phase. Importantly, we found that CDC7/DBF4 down-regulation, as well S1213A/S1525A TOP2A mutations can advance the timing of centromeric TOP2A recruitment in S-phase. Our results indicate that TOP2A is a novel DDK target and have important implications for centromere biology
On the Proof Theory of Regular Fixed Points
International audienceWe consider encoding finite automata as least fixed points in a proof theoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to proof-theory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
On the word problem for SP-categories, and the properties of two-way communication
International audienceThe word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as processes which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting conversions. As the equality classes under these permuting conversions are finite, it is easy to see that equality between cut-free terms (even in the presence of the additive units) is decidable. Unfortunately, this does not yield a tractable decision algorithm as these equivalence classes can contain exponentially many terms. However, the rather special properties of these free categories -- and, thus, of two-way communication -- allow one to devise a tractable algorithm for equality. We show that, restricted to cut-free terms s,t : X --> A, the decision procedure runs in time polynomial on |X||A|, the product of the sizes of the domain and codomain type
Evidence for the progression through S-phase in the ectopic cell cycle re-entry of neurons in Alzheimer disease
Aberrant neuronal re-entry into the cell cycle is emerging as a potential
pathological mechanism in Alzheimer disease (AD). However, while cyclins,
cyclin dependent kinases (CDKs), and other mitotic factors are ectopically
expressed in neurons, many of these proteins are also involved in other
pathological and physiological processes, generating continued debate on
whether such markers are truly indicative of a bona fide cell cycle
process. To address this issue, here we analyzed one of the minichromosome
maintenance (Mcm) proteins that plays a role in DNA replication and becomes
phosphorylated by the S-phase promoting CDKs and Cdc7 during DNA synthesis.
We found phosphorylated Mcm2 (pMcm2) markedly associated with neurofibrillary
tangles, neuropil threads, and dystrophic neurites in AD but not in
aged-matched controls. These data not only provide further evidence for
cell cycle aberrations in AD, but the cytoplasmic, rather than nuclear,
localization of pMcm2 suggests an abnormal cellular distribution of this
important replication factor in AD that may explain resultant cell cycle
stasis and consequent neuronal degeneration
Infinets: The parallel syntax for non-wellfounded proof-theory
Logics based on the µ-calculus are used to model induc-tive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. While traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. The present paper investigates a discrepancy found in such proof systems , between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (µMLL ∞) and study their correctness and sequentialization. Inductive and coinductive reasoning is pervasive in computer science to specify and reason about infinite data as well as reactive properties. Developing appropriate proof systems amenable to automated reasoning over (co)inductive statements is therefore important for designing programs as well as for analyzing computational systems. Various logical settings have been introduced to reason about such inductive and coinductive statements, both at the level of the logical languages modelling (co)induction (such as Martin Löf's inductive predicates or fixed-point logics, also known as µ-calculi) and at the level of the proof-theoretical framework considered (finite proofs with explicit (co)induction rulesà la Park [23] or infinite, non-wellfounded proofs with fixed-point unfold-ings) [6-8, 4, 1, 2]. Moreover, such proof systems have been considered over classical logic [6, 8], intuitionistic logic [9], linear-time or branching-time temporal logic [19, 18, 25, 26, 13-15] or linear logic [24, 16, 4, 3, 14]
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