# Logic Seminar

### Organizers: Laurențiu Leuștean, Andrei Sipoș

The logic seminar features talks on
mathematical logic,
philosophical logic and
logical aspects of computer science.
All seminars, except where otherwise indicated,
will be at 10:00 in Hall 202,
Faculty of Mathematics and Computer Science,
University of Bucharest.
to the mailing list,

our group

## Talks in 2015-2016

### Thursday, June 9, 2016

Denisa Diaconescu (University of Bucharest)
Skolemization for Substructural Logics

Abstract: The usual Skolemization procedure, which removes strong quantifiers by introducing new function symbols, is in general not sound for first-order substructural logics. However, in this talk, we will show that first-order substructural logics with a semantics satisfying certain witnessing conditions admit a "parallel" Skolemization procedure where a strong quantifier is removed by introducing a finite disjunction or conjunction (as appropriate) of formulas with multiple new function symbols.

### Thursday, June 2, 2016

Daniela Cheptea (University of Bucharest)
Reticulation of hoops

Abstract: Hoops are ordered structures introduced by B. Bosbach under the name of complemented semigroups. We present the reticulation of a hoop, which is a bounded distributive lattice with the property that there exists a homeomorphism between its prime spectrum and the prime spectrum of the hoop. We give an axiomatic definition of the reticulation and we prove that the reticulation is unique modulo a lattice isomorphism. Furthermore, we define a covariant functor between the category of bounded distributive lattices and the category of bounded hoops, which allows us to obtain new properties from the properties of lattices.

### Thursday, May 19, 2016

Andrei Sipoș (IMAR & University of Bucharest)
Proof mining and families of mappings

Abstract: We present the general metatheorems of proof mining for Hilbert spaces and a concrete case on how they might be applied - specifically, the convergence theorem of Lopez-Acedo and Xu for a finite families of pseudo-contractive self-mappings of a convex set in a Hilbert space, for which we shall compute a rate of asymptotic regularity.

### Thursday, April 14, 2016

Traian Şerbănuţă (University of Bucharest)
Partial membership equational logic

Abstract: This talk will review (Partial) Membership Equational Logic as introduced by Meseguer [2] and will present some development and results about a class of algebraic specifications (presentations) which admit the same set of consequences regardless whether they are interpreted in the total or in the partial membership equational logic [1].

References:
[1] J. Meseguer, G. Roşu, A Total Approach to Partial Algebraic Specification, ICALP 2002: 572-584 LNCS.
[2] J. Meseguer, Membership algebra as a logical framework for equational specification, WADT 1997: 18-61.

### Thursday, April 7, 2016

Ioana Leuştean (University of Bucharest)
Finitely presented structures in Łukasiewicz logic

Abstract: In this talk we will present Łukasiewicz logic and its extension obtained by adding a scalar multiplication. We will analyze the connections between logic, algebra and the theory of polyhedra, with a special emphasis on finitely presented structures.

### Thursday, March 31, 2016

Mircea Dumitru (University of Bucharest)
Abstract Free Logic for Fictionalism II

### Thursday, March 24, 2016

Mihai Prunescu (IMAR)
A many-sorted approach to the Special Theory of Relativity III

### Thursday, March 17, 2016

Mircea Dumitru (University of Bucharest)
Abstract Free Logic for Fictionalism

Abstract: Fictionalism is a fashionable and timely doctrine in many quarters of contemporary philosophy. It has fueled and channeled important debates in metaphysics (ontology), philosophy of language and philosophical logic, for its having a genuine explanatory virtue: even if it is extremely hard to buy into the full ontological existence of, say, unobservable things, or abstract things, or fictional objects, or nonfactual (and merely possible) things, or even moral values, one could, nevertheless, endorse forms of meaningful discourse which are about those sui-generis objects. Various kinds of fictionalism will help us in this regard: the things on which we think in those forms of discourse have to be accepted by us, even if they do not qualify ontologically, semantically, or epistemologically as being truth-apt or as truth-makers or truth-bearers. Against this background, my paper aims at disentangling certain logical principles that govern the meaningful fictional discourse on fictional objects. The ontological thesis concerning fictional objects that I endorse is that fictional objects are essentially objects of reference, i.e. objects created through a story or a narrative and introduced via a cluster of descriptions. The main point that I am going to make in my paper is this: in order to articulate the logical principles which govern the meaningful discourse on fictional objects what we need is a sort of free logic. The issue is: what kind? Now, a major motivation for developing free logics systems has always been to provide a basis for theories of definite descriptions. Having in view the essential connection between any given fictional object term and the cluster of descriptions through which the former is introduced, I argue that the kind of logic we need for fictionalism and fictional objects discourse is a positive free logic with free descriptions.

### Thursday, March 10, 2016

Mihai Prunescu (IMAR)
A many-sorted approach to the Special Theory of Relativity II

### Thursday, March 2, 2016

Mihai Prunescu (IMAR)
A many-sorted approach to the Special Theory of Relativity

Abstract: We deduce the Lorentz and the Galilean transformation rules in a many-sorted fi rst-order structure where moving points and moving frames are abstract sorts. If we suppose that the set of speeds allowed for the relative movement of frames is an interval, then both transformation rules can be proven if one states orientation, compatibility with the relative speed, symmetry, and an axiom about the existence or non-existence of an invariant speed. The case generally known as Spherical Relativity (a < 0) is inconsistent with the supposition that the set of speeds is an interval. If we allow more general convex sets as sets of speeds, there are models of the Spherical Relativity as well. In this case the class of ground fields is exactly the class of non-archimedean Euclidean ordered fields.

### Thursday, February 25, 2016

Laura Franzoi (University of Bucharest and University of Trieste)
Distribution calculus: maxitive probabilities vs. additive probabilities

Abstract: Some incongruences met in fuzzy arithmetic are dispelled when one uses systematically maxitive possibilities, and so multi-valued logic, rather than fuzzy sets, and imports time-honoured notions used in random arithmetic, i.e. in distribution calculus for random variables, priviliging Pitt's "implicit" definition of random variables w.r. to Kolmogorov's "explicit" definition, equivalent but not so convenient to be exported into a maxitive theory. In particular, we focus on the notion of irrelevance, due to the talker jointly with A. Sgarro, which remarkably expedites computations w.r. to a more standard approach to fuzzy arithmetic.

### Thursday, February 18, 2016, Hall 220

Andrei Sipoș (IMAR & University of Bucharest)
Applied Proof Theory V

### Thursday, February 11, 2016, Hall 220

Andrei Sipoș (IMAR & University of Bucharest)
Applied Proof Theory IV

### Thursday, February 4, 2016, Hall 220

Andrei Sipoș (IMAR & University of Bucharest)
Applied Proof Theory III

### Thursday, January 28, 2016, Hall 220

Andrei Sipoș (IMAR & University of Bucharest)
Applied Proof Theory II

### Thursday, January 21, 2016, Hall 220

Andrei Sipoș (IMAR & University of Bucharest)
Applied Proof Theory I

Abstract: We aim to provide a comprehensive introduction to proof mining, an area of applied logic that has as its goal the extraction of "quantitative information" (i.e. realizers and bounds) from proofs of an apparently non-constructive nature. We begin with a description of the specific logical systems that are usually worked with in this endeavour.

### Thursday, January 14, 2016, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic VII

### Thursday, January 7, 2016, 12:00-14:00, Hall Spiru Haret

12:00
Claudia Chiriță (Royal Holloway University of London)
Free Jazz and Service-Oriented Improvisations

Abstract: Building on a concept of many-valued institution (called RL-institution) in which the truth spaces are residuated lattices, we study free jazz improvisations in the context of Service-Oriented Computing. We define an RL-institution of graphical notations for Free Jazz, and describe how it can be used to capture music fragments as specifications over this logic. We then model musical improvisations by means of processes of service-oriented discovery, selection and binding.

11:100
Ionuț Țuțu (Royal Holloway University of London)
Multiple-parameterized behavioural specifications

Abstract: Behavioural specification, based on Horst Reichel's notion of behavioural satisfaction, is nowadays one of the most promising algebraic specification paradigms. In this talk, we review a recently proposed axiomatic framework of structured behavioural specifications and put forward a dedicated theory of pushout-style parameterization that supports generic behavioural s pecifications with multiple parameters and implicit sharing. We explain when - and also how - it is possible to simultaneously or sequentially instantiate multiple-parameterized behavioural specifications, and discuss some of the additional conditions that need to be imposed in order to guarantee that the results of the two instantiation procedures are isomorphic.

### Thursday, December 10, 2015, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic VI

### Thursday, December 3, 2015, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic V

### Thursday, November 26, 2015, Hall 220

10:00
Jacek Malinowski (Polish Academy of Sciences)
Theory of Logical Consequence - past, present and future

Abstract: The aim of the lecture is to give a general review of old and more recent results, as well as, possible future developments in the theory of logical consequence operation.

• I start from comparing three frameworks for logical investigations: Tarski's logical consequence operation, Gentzen's sequents and Hilbert-style proofs.
• In the second part I will give a review of results in the theory of logical consequence operation concerning logical matrices, algebraic semantics, implicative logics and equivalential logic.
• Last part concerns some generalizations of logical consequence and logical matrices. The former gives us an interesting and promising conceptual framework for non-monotonic logic. The later gives a general semantics for logic of rejection.

11:15
Serafina Lapenta (University of Salerno)
Convex MV-algebras

(joint work with Tommaso Flaminio)

Abstract: The notions of convexity plays a central rôle in logic and mathematics. Starting from a seminal idea of Brown [1], we propose an axiomatic approach to convex combinations in the realm of MV-algebras [2]. More in detail, we will expand the language of MV-algebras by an uncountable family of binary operations $cc_\alpha(\cdot, \cdot)$ (one for every $\alpha\in [0,1]$) axiomatized so to capture the basic properties of convex combinations in $[0,1]$. The so resulting algebras are called convex MV-algebras (or CMV-algebras for short).
CMV-algebras form a variety. Our first result shows that CMV-algebras are termwise equivalent to Riesz MV-algebras [3] and, consequently, the variety of CMV-algebras is generated by the standard CMV-algebra, that is the standard MV-algebra where the operators $cc_\alpha$ are interpreted in the usual way: for each $x,y,\alpha\in [0,1]$, $cc_\alpha(x,y)$ is $\alpha x+(1-\alpha)y$.
States of MV-algebras [4] are analogous to finitely additive probabilities on boolean algebras and, for every MV-algebra ${\bf A}$, its states form a subset of $[0,1]^A$ which coincide with the topological closure of the convex hull of the MV-homomorphisms of ${\bf A}$ in the standard MV-algebra $[0,1]_{MV}$. Thanks to this characterization of the states space, we will show that each state of a finitely dimensional MV-algebra $[0,1]^X$ (with $X$ finite) has a faithful representation in the free CMV-algebra $|X|$-generated.

References:
[1] N. P. Brown, Topological Dynamical Systems Associated to $\Pi_1$-factors, preprint arXiv:1010.1214.
[2] R. Cignoli, I. M. L. D'Ottaviano, D. Mundici, Algebraic Foundations of Many-valued Reasoning, Trends in Logic Vol 8, Kluwer, Dordrecht, 2000.
[3] A. Di Nola, I. Leuștean, Łukasiewicz logic and Riesz Spaces, Soft Computing, Soft Comp. 18(12) (2014) 2349-2363. arXiv:1309.1575v1
[4] D. Mundici, Averaging the Truth-value in Łukasiewicz Logic. Studia Logica 55(1), 113--127, 1995.

### Thursday, November 19, 2015, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic IV

### Thursday, November 12, 2015, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic III

### Thursday, November 5, 2015, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic II

### Thursday, October 29, 2015, Hall 220

Traian Şerbănuţă (University of Bucharest)
Matching Logic

Abstract: Matching Logic, recently introduced by Grigore Roşu, is a logic for specifying and reasoning about structure by means of patterns and pattern matching. Its syntax is an extension of the FOL syntax; however, its semantics is interpreted over multi-algebras instead of standard first order models. The talk will introduce Matching Logic by means of examples, present known results about the logic, and discuss its relation to FOL and Reynolds' Separation Logic.

### Thursday, October 22, 2015, Hall 220

Mihai Prunescu (IMAR)
Abstract: In the reference book "$p$-adic numbers and their functions" (1973) Kurt Mahler described the continuous functions $f: \mathbb{Z}_p \rightarrow \mathbb{Q}_p$. He proved that $f$ is continuous according to the $p$-adic topology if and only if it has a series expansion $f(x) = \sum\limits_{n=0}^\infty a_n \binom{x}{n}$, where $a_n \rightarrow 0$ according to the same topology.