arXiv subjects

Pedro Sánchez Terraf

Publications and source records attributed to Pedro Sánchez Terraf.

At least 19 recordsLinked to original sources

arXiv2026

The complexity of bisimilarity on pointmass processes

We assess the descriptive complexity of *bisimilarity* or "equality of behavior" on a family of Markov decision processes over uncountable standard Borel spaces, namely *nondeterministic labelled Markov processes* (NLMP). We show that bisimilarity is analytic for processes with a uniform assignment of finitely-supported measures to each state and label. More finely, we obtain that bisimilarity on the space of countable Kripke frames (or labelled transition systems) is classifiable by countable structures. We show that bisimilarity of well-founded ("terminating") processes is Borel. We also provide a lower complexity bound by reducing the relation of eventual equality of binary sequences $E_0$ to the former. As a consequence, there is no countable fragment of basic modal logic with denumerable conjunctions that characterizes bisimilarity for processes of small rank. We finally apply the previous Borel definability to study the well-founded part of discrete uniform processes over uncountable spaces.

cs.LO↗

arXiv2026

The associative-poset point of view on right regular bands

We present two results on the relation between the class of right regular bands (RRBs) and their underlying *associative posets*. The first one is a construction of a left adjoint to the forgetful functor that takes an RRB $(P,\cdot)$ to the corresponding $(P,\leq)$. The construction of such a left adjoint is actually done in general for any class of relational structures $(X,R)$ obtained from a variety, where $R$ is defined by a finite conjunction of identities. The second result generalizes the "inner" representations of direct product decompositions of semilattices studied by the second author to RRBs having at least one commuting element.

math.LO↗

arXiv2024

Chain Bounding, the leanest proof of Zorn's lemma, and an illustration of computerized proof formalization

We present an exposition of the *Chain Bounding Lemma*, which is a common generalization of both Zorn's Lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest ones that we are aware of. As a by-product, we show that for every poset $P$ and function $f$ from the powerset of $P$ into $P$, there exists a maximal well-ordered chain whose family of initial segments is appropriately closed under $f$. We also provide an introduction to the process of "computer formalization" of mathematical proofs by using *proofs assistants*. As an illustration, we verify our main results with the Lean proof assistant.

math.LO↗

arXiv2024

A classification of bisimilarities for general Markov decision processes

We provide a fine classification of bisimilarities between states of possibly different labelled Markov processes (LMP). We show that a bisimilarity relation proposed by Panangaden that uses direct sums coincides with "event bisimilarity" from his joint work with Danos, Desharnais, and Laviolette. We also extend Giorgio Bacci's notions of bisimilarity between two different processes to the case of nondeterministic LMP and generalize the game characterization of state bisimilarity by Clerc et al. for the latter.

cs.LO↗

arXiv2024

Definability of band structures on posets

The idempotent semigroups (bands) that give rise to partial orders by defining $a \leq b \iff a \cdot b = a$ are the "right-regular" bands (RRB), which are axiomatized by $x\cdot y \cdot x = y \cdot x$. In this work we consider the class of "associative posets", which comprises all partial orders underlying right-regular bands, and study to what extent the ordering determines the possible "compatible" band structures and their canonicity. We show that the class of associative posets in the signature $\{ \leq \}$ is not first-order axiomatizable. We also show that the Axiom of Choice is equivalent over $\mathit{ZF}$ to the fact that every tree with finite branches is associative. We study the smaller class of "normal" posets (corresponding to right-normal bands) and give a structural characterization.

math.LO↗

arXiv2023

The formal verification of the ctm approach to forcing

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC}+\neg\mathit{CH}$ and $\mathit{ZFC}+\mathit{CH}$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $Ω\subseteq\mathcal{R}$ and defined $\mathcal{F}:\mathcal{R}\to\mathcal{R}$ such that for every $Φ\subseteq\mathcal{R}$ and $M$-generic $G$, $M\models \mathit{ZC} \cup \mathcal{F}\text{``}Φ\cup Ω$ implies $M[G]\models \mathit{ZC} \cup Φ\cup \{ \neg \mathit{CH} \}$, where $\mathit{ZC}$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.

math.LO↗

arXiv2022

The Zhou Ordinal of Labelled Markov Processes over Separable Spaces

There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$. When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner property. The second author proved the existence of an LMP $\mathbb{S}$ with $\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$ over separable metrizable spaces having arbitrary large countable $\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of $\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal.

cs.LO↗

arXiv2021

Semipullbacks of labelled Markov processes

A labelled Markov process (LMP) consists of a measurable space $S$ together with an indexed family of Markov kernels from $S$ to itself. This structure has been used to model probabilistic computations in Computer Science, and one of the main problems in the area is to define and decide whether two LMP $S$ and $S'$ "behave the same". There are two natural categorical definitions of sameness of behavior: $S$ and $S'$ are bisimilar if there exist an LMP $ T$ and measure preserving maps forming a diagram of the shape $ S\leftarrow T \rightarrow{S'}$; and they are behaviorally equivalent if there exist some $ U$ and maps forming a dual diagram $ S\rightarrow U \leftarrow{S'}$. These two notions differ for general measurable spaces but Doberkat (extending a result by Edalat) proved that they coincide for analytic Borel spaces, showing that from every diagram $S\rightarrow U \leftarrow{S'}$ one can obtain a bisimilarity diagram as above. Moreover, the resulting square of measure preserving maps is commutative (a semipullback). In this paper, we extend the previous result to measurable spaces $S$ isomorphic to a universally measurable subset of a Polish space with the trace of the Borel $σ$-algebra, using a version of Strassen's theorem on common extensions of finitely additive measures.

math.PR↗

arXiv2020

Formalization of Forcing in Isabelle/ZF

We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC, we construct a proper generic extension and show that the latter also satisfies ZFC. In doing so, we remodularized Paulson's ZF-Constructibility library.

cs.LO↗

arXiv2019

Mechanization of Separation in Generic Extensions

We mechanize, in the proof assistant Isabelle, a proof of the axiom-scheme of Separation in generic extensions of models of set theory by using the fundamental theorems of forcing. We also formalize the satisfaction of the axioms of Extensionality, Foundation, Union, and Powerset. The axiom of Infinity is likewise treated, under additional assumptions on the ground model. In order to achieve these goals, we extended Paulson's library on constructibility with renaming of variables for internalized formulas, improved results on definitions by recursion on well-founded relations, and sharpened hypotheses in his development of relativization and absoluteness.

cs.LO↗

arXiv2018

First steps towards a formalization of Forcing

We lay the ground for an Isabelle/ZF formalization of Cohen's technique of forcing. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize the definition of forcing notions as preorders with top, dense subsets, and generic filters. We formalize a version of the principle of Dependent Choices and using it we prove the Rasiowa-Sikorski lemma on the existence of generic filters. Given a transitive set $M$, we define its generic extension $M[G]$, the canonical names for elements of $M$, and finally show that if $M$ satisfies the axiom of pairing, then $M[G]$ also does. We also prove $M[G]$ is transitive.

cs.LO↗

arXiv2017

The Lattice of Congruences of a Finite Line Frame

Let $\mathbf{F}=\left\langle F,R\right\rangle $ be a finite Kripke frame. A congruence of $\mathbf{F}$ is a bisimulation of $\mathbf{F}$ that is also an equivalence relation on F. The set of all congruences of $\mathbf{F}$ is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that $\mathbf{F}$ is a finite line frame. We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound. Through these descriptions we show that for every nontrivial congruence $ρ$, the interval $[\mathrm{Id_{F},ρ]}$ embeds into the lattice of divisors of a suitable positive integer. We also prove that any two congruences with a nontrivial upper bound permute.

math.LO↗

arXiv2015

Yet Another Proof of Glivenko's Theorem

In this short note we give an alternative proof of Glivenko's Theorem, stating that a formula $ϕ$ is provable in classical propositional logic if and only if $\neg\negϕ$ is provable in intuitionistic propositional logic. We work in the natural deduction system by Gentzen, and the key lemma shows that in any proof one needs only one application of reductio ad absurdum.

math.LO↗

arXiv2015

Stochastic Nondeterminism and Effectivity Functions

This paper investigates stochastic nondeterminism on continuous state spaces by relating nondeterministic kernels and stochastic effectivity functions to each other. Nondeterministic kernels are functions assigning each state a set o subprobability measures, and effectivity functions assign to each state an upper-closed set of subsets of measures. Both concepts are generalizations of Markov kernels used for defining two different models: Nondeterministic labelled Markov processes and stochastic game models, respectively. We show that an effectivity function that maps into principal filters is given by an image-countable nondeterministic kernel, and that image-finite kernels give rise to effectivity functions. We define state bisimilarity for the latter, considering its connection to morphisms. We provide a logical characterization of bisimilarity in the finitary case. A generalization of congruences (event bisimulations) to effectivity functions and its relation to the categorical presentation of bisimulation are also studied.

cs.LO↗

arXiv2014

Bisimilarity is not Borel

We prove that the relation of bisimilarity between countable labelled transition systems is $Σ_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and nondeterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes.

math.LO↗

arXiv2010

Unprovability of the Logical Characterization of Bisimulation

We quickly review labelled Markov processes (LMP) and provide a counterexample showing that in general measurable spaces, event bisimilarity and state bisimilarity differ in LMP. This shows that the logic in Desharnais [*] does not characterize state bisimulation in non-analytic measurable spaces. Furthermore we show that, under current foundations of Mathematics, such logical characterization is unprovable for spaces that are projections of a coanalytic set. Underlying this construction there is a proof that stationary Markov processes over general measurable spaces do not have semi-pullbacks. ([*] J. Desharnais, Labelled Markov Processes. School of Computer Science. McGill University, Montréal (1999))

cs.LO↗

arXiv2010

Bisimulations for Nondeterministic Labeled Markov Processes

We extend the theory of labeled Markov processes with internal nondeterminism, a fundamental concept for the further development of a process theory with abstraction on nondeterministic continuous probabilistic systems. We define nondeterministic labeled Markov processes (NLMP) and provide three definition of bisimulations: a bisimulation following a traditional characterization, a state based bisimulation tailored to our "measurable" non-determinism, and an event based bisimulation. We show the relation between them, including that the largest state bisimulation is also an event bisimulation. We also introduce a variation of the Hennessy-Milner logic that characterizes event bisimulation and that is sound w.r.t. the other bisimulations for arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable $\lor$. We then introduce a finitary sublogic that characterize all bisimulations for image finite NLMP whose underlying measure space is also analytic. Hence, in this setting, all notions of bisimulation we deal with turn out to be equal. Finally, we show that all notions of bisimulations are different in the general case. The counterexamples that separate them turn to be non-probabilistic NLMP.

cs.LO↗

arXiv2010

Factor Congruences in Semilattices

We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum (maximum) element, these generalized ideals turn into ordinary (dual) ideals.

math.LO↗