**Financial support through the following projects:**

UID/MAT/00297/2019

UID/MAT/00297/2013

PEst-OE/MAT/UI0297/2014

PEst-OE/MAT/UI0297/2011

Orador: Wilfried Sieg (Carnegie Mellon University)

Título do mini-curso: "Thinking, Computing, Thinking"

Local: Sala de seminários, ed. VII, FCT-UNL

(NB: O mini-curso é constituído por 4 palestras **independentes**.)

**16/10/2017 (2a), 14h (Seminário): "Mechanical procedures - What is the concept of computation?"**

The Church-Turing Thesis asserts that particular mathematical notions are adequate to represent informal notions of effective calculability or mechanical decidability. I first sketch contexts that called for such adequate mathematical notions, namely, problems in mathematics (e.g., Hilbert’s 10th problem), decision problems in logic (e.g., the Entscheidungsproblem for first-order logic), and the precise characterization of formality (for the general formulation of Gödel’s incompleteness theorems).** **

The classical approach to the effective calculability of number theoretic functions led, through Gödel and Church, to a notion of computability in logical calculi and metamathematical absoluteness theorems. The classical approach to the mechanical decidability of problems concerning syntactic configurations led, through Turing and Post, to a notion of computability in formal calculi (canonical systems) and metamathematical representation theorems.** **

Particular features of canonical systems motivate the formulation of an abstract concept of a computable dynamical system. This concept articulates finiteness and locality conditions that are satisfied by the standard concrete notions of computation. A representation theorem can be established: Turing machines can simulate the computations of any concrete system falling under the abstract concept. I sketch a generalization of this approach to obtain computable parallel dynamical systems. Some applications will conclude my discussion.

**17/10/2017 (3a), 14h: "Natural deduction in bi-directional ways"**

**20/10/2017 (6a), 13h: "Natural formalization - The Cantor-Bernstein Theorem derived in ZF"**

**23/10/2017 (2a), 14h: "Automated search for Gödel's Theorems"**