This page concerns the Lectures on Logic and its Mathematical Aspects (LLAMA) seminar series at the Institute for Logic, Language & Computation (ILLC) of the University of Amsterdam.
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It is well-known that for any regular language there exists a unique minimal DFA accepting it. Establishing a similar result for NFA is more difficult, but of great practical importance, as NFA can be exponentially more succinct than DFA. The main issue is that a regular language can be accepted by several size-minimal NFA that are not isomorphic. A number of sub-classes of NFA have been identified in the literature to tackle this issue, which all admit canonical representatives. In this paper we provide a general categorical framework based on distributive laws that unifies constructions of canonical NFA and unveils new ones.
I present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in system analysis and verification. Coalgebraic generality allows to cover not only classical relational systems but also, e.g. various forms of weighted systems and furthermore to flexibly combine existing system types. Under assumptions on the type functor that allow representing its coalgebras in terms of nodes and edges, the algorithm runs in time O(m log n) for input coalgebras with n states and m edges. The generic complexity result and the possibility of combining system types yields a toolbox for efficient partition refinement algorithms. Instances of our generic algorithm match the run-time of the best known algorithms for unlabelled transition systems, Markov chains, deterministic automata, Segala systems, and for color refinement. For weighted tree automata, the instance of the generic algorithm even improves the best run-time known. The algorithm is implemented in a tool (CoPaR) in full generality and allows the composition of existing systems types at run-time.
We consider Horn formula equations, which are special existential second-order formulas. Our main result states that Horn formula equations are valid iff there are solutions expressible in least fixed-point logic. These canonical solutions correspond to the least and greatest solution in the Horn and dual-Horn case, respectively. As applications and corollaries of the fixed-point theorem we obtain new results and simpler proofs of existing results in second-order quantifier elimination, program verification and proof theory.
This is joint work with Stefan Hetzl, see also https://arxiv.org/abs/2109.04633.
Recently, Baltag & van Benthem introduced a logic of functional dependence LFD with atomic dependence statements and dependence quantifiers that is interpreted over generalised assignment models. Indeed, the logic is closely related to the logic of Cylindrical Relativized Set Algebras (CRS); CRS quantifiers and dependence quantifiers are interdefinable. It follows that LFD can be seen as an extension of CRS with dependence atoms. Decidability was shown by Baltag & van Benthem by a detour via quasi-models. However, this essentially involved an infinite construction, leaving the finite model property (fmp) as an open problem. We solve this by showing that LFD has the fmp, using Herwig's theorem on extending partial isomorphisms. Furthermore, we introduce dependence bisimulations and prove a van Benthem-characterization theorem, characterizing LFD as the largest fragment FOL invariant under this notion.
(These results are the culmination of a project I did under the supervision of Alexandru Baltag.)
https://link.springer.com/article/10.1007/s10992-020-09588-z
https://arxiv.org/abs/2109.08313
I will first explain STV counting and the parlous state of computer-counting code implemented by various Election Commissions from around Australia. I will then explain how we used Coq to specify a "vanilla" version of STV as a proof-calculus and used it to extract a computer program which not only counts votes according to this specification but also produces a certificate during the count. The specification of the certificate is derived from the counting rules. We have proved, in Coq, that if the certificate is correct with respect to its specification, then the result it encapsulates must be correct with respect to the relevant specification of STV. The certificate is designed so that an average third-year computer science student could write a computer program to check the correctness of the certificate. In particular, each political party could hire their own programmer to easily scrutiny the count produced by any computer program, including our own, that produces such certificates. The only caveat is that we require the publication of all ballots.
In this talk, we propose a general method to "decode" (propositional) intuitionistic logic and various intermediate logics as (dynamic) epistemic logics of knowing how. Our approach is inspired by some scattered ideas hidden in the vast literature of math, philosophy, CS, and linguistics about intuitionistic logic in the past 100 years, which echoed Heyting's initial (but forgotten?) conception about intuitionistic truth as "knowing how to prove". The core technical idea is to use a bundled know-how modality to unify the formalized Brouwer–Heyting–Kolmogorov interpretation and the Kripke semantics of intuitionistic logic. As we will see, this approach can reveal the hidden complicated dynamic-epistemic information behind the innocent-looking connectives but makes intuitionistic logic intuitive. With the natural but precise epistemic readings of intuitionistic formulas, various technical results about intuitionistic logic and intermediate logics become transparent, if not trivial. If time permits, we show how to decode inquisitive logic and some version of dependence logic as epistemic logics in order to demonstrate the power of this general approach, which could also be a meeting point of various research themes at ILLC.
(The talk is partly based on the joint work with Haoyu Wang and Yunsong Wang)
I will present recent results establishing decidability and complexity upper bounds for many axiomatic extensions of FLec and FLew. The substructural logics FLec and FLew are obtained from the sequent calculus for intuitionistic propositional logic (FLecw) by omitting the weakening rule and contraction rule respectively. This work can be seen as extending Kripke's decidability argument (1959) and Urquhart's upper bound (1999) for FLec. A notable consequence is a first upper bound for the basic fuzzy logic MTL.
Joint work with A. R. Balasubramanian and Timo Lang. See https://arxiv.org/abs/2104.09716 for details.
While modal logics are robustly decidable, first order modal logics (FOML) are notoriously undecidable, in the sense that modal extensions of decidable fragments of first order logic are undecidable. For example, the two-variable fragment of FOML is undecidable.
Despite such discouragement, this century has seen several positive results on new fragments, such as the monodic fragment, bundled fragment, term-modal logic etc. We give an overview of this exciting area of research, including our own results.
The results presented are joint with Anantha Padmanabha.
Various problems arising in the context of example-driven approaches to conjunctive query discovery turns out to be intimately related to basic structural properties of the homomorphism lattice of finite structures, such as density, or the existence of duals. I will discuss some of these connections, and highlight relevant results from recent work together with Victor Dalmau. See also https://arxiv.org/abs/2008.06824.
Cyclic and non-well-founded proofs have turned out to be very useful tools in the proof theory of modal fixpoint logics. However, most of the known results in this area only treat a particular modal fixpoint logic at once. This is in contrast to basic modal logic, where much more general proof-theoretic methods exist. In this talk I will outline a first attempt to combine these existing general methods with non-well-founded proof theory in order to uniformly treat multiple modal fixpoint logics.
More precisely, I will build on a technique presented by Ori Lahav (LICS 2013) for constructing cut-free hypersequent calculi for basic modal logics characterised by frames satisfying so-called 'simple' first-order conditions. I investigate the generalisation of this technique to a relatively inexpressive modal fixpoint language, namely unimodal logic with the master modality (a.k.a. the reflexive-transitive closure modality). Soundness is obtained for all simple frame conditions, completeness only for those that will be called 'equable'.
Talks from 2020 and earlier can also be found on the Algebra|Coalgebra seminar website.
The seminar is currently organized by Bahareh Afshari, Benno van den Berg, Nick Bezhanishvili, Tobias Kappé and Yde Venema. If you're interested in giving a talk at the seminar or have any questions regarding the seminar please contact Tobias at t.w.j.kappe@uva.nl.
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