Pedro Sánchez Terraf

Profesor Asociado en FaMAF - UNC - Argentina


Buscáme en arXiv

En revistas (y manuscritos recientes)

[1] Mechanization of Separation in Generic Extensions. arXiv e-prints, arXiv:1901.03313 (enero 2019). 1901.03313. Junto a E. Gunther y M. Pagano. [ bib | arXiv ]
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.

[2] Semipullbacks of labelled Markov processes. ArXiv e-prints, 1706.02801 (junio 2017). 1706.02801. Junto a J. Pachl. [ bib | arXiv ]
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 STS'; and they are behaviorally equivalent if there exist some U and maps forming a dual diagram SUS'.

These two notions differ for general measurable spaces but Edalat proved that they coincide for analytic Borel spaces, showing that from every diagram SUS' 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 Edalat's 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.

[3] The Lattice of Congruences of a Finite Linear Frame. Journal of Logic and Computation, 27: 2653–2688 (abril 2017). 1504.01789. Junto a C. Areces, M. Campercholi, y D. Penazzi. [ bib | DOI | arXiv ]
Let F=F,R be a finite Kripke frame. A congruence of F is a bisimulation of F that is also an equivalence relation on F. The set of all congruences of F is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that F is a finite linear 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 [IdF,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.

[4] Bisimilarity is not Borel. Mathematical Structures in Computer Science, FirstView (12 2015). [ bib | DOI | http ]
We prove that the relation of bisimilarity between countable labelled transition systems is Σ11-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.

[5] Stochastic nondeterminism and effectivity functions. Journal of Logic and Computation (2015). 1405.7141. Junto a E.-E. Doberkat. [ bib | DOI | arXiv ]
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.

[6] Bisimulations for non-deterministic labelled Markov processes. Mathematical Structures in Comp. Sci., 22 (1): 43–68 (febrero 2012). Junto a P. R. D'Argenio y N. Wolovick. [ bib | DOI | http ]
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 . 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.

[7] Unprovability of the logical characterization of bisimulation. Information and Computation, 209 (7): 1048–1056 (2011). [ bib | DOI | http ]
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 the work by 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.

[8] Boolean factor congruences and property (*). Int. J. Algebra Comput., 21 (6): 931–950 (2011). [ bib | DOI ]
Summary: A variety V has Boolean factor congruences (BFC) if the set of factor congruences of any algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal'tsev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in R. Willard's work [J. Algebra 132, No. 1, 130–153 (1990; Zbl 0737.08006)].

[9] Factor congruences in semilattices. Revista de la Unión Matemática Argentina, 52 (1): 1–10 (2011). 0809.3822. [ bib | arXiv | http ]
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.

[10] Existentially definable factor congruences. Acta Scientiarum Mathematicarum (Szeged), 76 (1–2): 49–53 (2010). [ bib ]
A variety V has definable factor congruences if and only if factor congruences can be defined by a first-order formula Φ having central elements as parameters. We prove that if Φ can be chosen to be existential, factor congruences in every algebra of V are compact.

[11] Varieties with definable factor congruences. Trans. Amer. Math. Soc., 361: 5061–5088 (2009). Junto a D. Vaggione. [ bib | DOI ]
We study direct product representations of algebras in varieties. We collect several conditions expressing that these representations are definable in a first-order-logic sense, among them the concept of Definable Factor Congruences (DFC). The main results are that DFC is a Mal'cev property and that it is equivalent to all other conditions formulated; in particular we prove that V has DFC if and only if V has 0 & 1 and Boolean Factor Congruences. We also obtain an explicit first order definition Φ of the kernel of the canonical projections via the terms associated to the Mal'cev condition for DFC, in such a manner it is preserved by taking direct products and direct factors. The main tool is the use of central elements, which are a generalization of both central idempotent elements in rings with identity and neutral complemented elements in a bounded lattice.

[12] Directly indecomposables in semidegenerate varieties of connected po-groupoids. Order, 25 (4): 377–386 (2008). [ bib | DOI ]
We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In a previous work by Vaggione and the author it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general.

[13] Compact factor congruences imply Boolean factor congruences. Algebra univers., 51: 207–213 (2004). Junto a D. Vaggione. [ bib | DOI ]
We prove that any variety V in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in V the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.

En conferencias con referato

[1] First steps towards a formalization of Forcing. Electronic Notes in Theoretical Computer Science (julio 2018). 1807.05174. Accepted. Presented at Logical and Semantic Frameworks, with Applications (LSFA 2018). Junto a E. Gunther y M. Pagano. [ bib | arXiv ]
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 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.

[2] A theory for the semantics of stochastic and non-deterministic continuous systems. En A. Remke, M. Stoelinga, editores, Stochastic Model Checking. Rigorous Dependability Analysis Using Model Checking Techniques for Stochastic Systems, tomo 8453 de Lecture Notes in Computer Science, págs. 67–86. Springer Berlin Heidelberg (2014). Junto a C. E. Budde, P. R. D'Argenio, y N. Wolovick. [ bib | DOI | http ]
The description of complex systems involving physical or biological components usually requires to model involved continuous behavior induced by variables such as time, distances, speed, temperature, alkalinity of a solution, etc. Often, such variables can be quantified probabilistically to better understand the behavior of the complex systems. For example, the arrival time of events may be considered a Poisson process or the weight of an individual may be assumed to be distributed according to a log-normal distribution. However, it is also common that the uncertainty on how these variables behave make us prefer to leave out the choice of a particular probability and rather model it as a purely non-deterministic decision, as it is the case when a system is intended to be deployed in a variety of very different computer or network architectures. Therefore, the semantics of these systems needs to be represented by a variant of probabilistic automata that involves continuous domains on the state space and the transition relation. In this paper, we provide a survey to the theory of such kind of models. We present the theory of the so called labeled Markov processes (LMP) and its extension with internal non-determinism (NLMP). We show that in these complex domains, the bisimulation relation can be understood in different manners. We show the relation among these different definitions and try to understand the boundaries among them through examples. We also study variants of Hennessy-Milner logic that provides logical characterizations of some of these bisimulations.

[3] Nondeterministic labeled Markov processes: Bisimulations and logical characterization. En QEST, Sixth International Conference on the Quantitative Evaluation of Systems, págs. 11–20. IEEE Computer Society (2009). Junto a P. R. D'Argenio, N. Wolovick, y P. Celayes. [ bib ]

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