Concept

Interpretability logic

Interpretability logics comprise a family of modal logics that extend provability logic to describe interpretability or various related metamathematical properties and relations such as weak interpretability, Π1-conservativity, cointerpretability, tolerance, cotolerance, and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr Hájek, Konstantin Ignatiev, Giorgi Japaridze, Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser, and Domenico Zambella. The language of ILM extends that of classical propositional logic by adding the unary modal operator and the binary modal operator (as always, is defined as ). The arithmetical interpretation of is “ is provable in Peano arithmetic (PA)”, and is understood as “ is interpretable in ”. Axiom schemata:

All classical tautologies

Rules of inference:

“From and conclude ”
“From conclude ”. The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov. The language of TOL extends that of classical propositional logic by adding the modal operator which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of is “ is a tolerant sequence of theories”. Axioms (with standing for any formulas, for any sequences of formulas, and identified with ⊤):
All classical tautologies

Rules of inference:

“From and conclude ”
“From conclude ”. The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze.

Source officielle

https://en.wikipedia.org/wiki/Interpretability_logic

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All classical tautologies

Rules of inference:

“From and conclude ”
“From conclude ”. The completeness of ILM with respect to its arithmetical interpretation was independently proven by Alessandro Berarducci and Vladimir Shavrukov. The language of TOL extends that of classical propositional logic by adding the modal operator which is allowed to take any nonempty sequence of arguments. The arithmetical interpretation of is “ is a tolerant sequence of theories”. Axioms (with standing for any formulas, for any sequences of formulas, and identified with ⊤):
All classical tautologies

Rules of inference:

“From and conclude ”
“From conclude ”. The completeness of TOL with respect to its arithmetical interpretation was proven by Giorgi Japaridze.

Source officielle

https://en.wikipedia.org/wiki/Interpretability_logic

À propos de ce résultat

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Alexandre Alahi is currently an Assistant Professor at EPFL. He spent five years at Stanford University as a Post-doc and Research Scientist after obtaining his Ph.D. from EPFL. His research enables machines to perceive the world and make decisions in the context of transportation problems and smart environments. He has worked on the theoretical challenges and practical applications of socially-aware Artificial Intelligence, i.e., systems equipped with perception and social intelligence. He was awarded the Swiss NSF early and advanced researcher grants for his work on predicting human social behavior. He won the CVPR Open Source Award (2012) for his work on Retina-inspired image descriptors, and the ICDSC Challenge Prize (2009) for his sparsity-driven algorithm that has tracked more than 100 million pedestrians to date. His research has been covered internationally by BBC, abc, PBS, Euronews, Wall street journal, and other national news outlets around the world. Alexandre has also co-founded multiple startups such as Visiosafe, and won several startup competitions. He was elected as one of the Top 20 Swiss Venture leaders in 2010.

Antoine Bosselut

Antoine Bosselut is an assistant professor at EPFL. He leads the EPFL NLP group, which conducts research on natural language processing (NLP) systems that can model, represent, and reason about human and world knowledge. Prior to joining EPFL, he was a postdoctoral researcher at Stanford University working in the SNAP and NLP groups and a young investigator on the Mosaic project at the Allen Institute for AI . He completed his PhD at the University of Washington.

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Concepts associés (1)

Interpretability

In mathematical logic, interpretability is a relation between formal theories that expresses the possibility of interpreting or translating one into the other. Assume T and S are formal theories. Slightly simplified, T is said to be interpretable in S if and only if the language of T can be translated into the language of S in such a way that S proves the translation of every theorem of T. Of course, there are some natural conditions on admissible translations here, such as the necessity for a translation to preserve the logical structure of formulas.