Abstract Syntax and Variable Binding
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
Electronic Notes in Theoretical Computer Science (ENTCS)
Second-Order and Dependently-Sorted Abstract Syntax
LICS '08 Proceedings of the 2008 23rd Annual IEEE Symposium on Logic in Computer Science
Term Equational Systems and Logics
Electronic Notes in Theoretical Computer Science (ENTCS)
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
WoLLIC'11 Proceedings of the 18th international conference on Logic, language, information and computation
Discrete generalised polynomial functors
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
An algebraic presentation of predicate logic
FOSSACS'13 Proceedings of the 16th international conference on Foundations of Software Science and Computation Structures
Multiversal Polymorphic Algebraic Theories: Syntax, Semantics, Translations, and Equational Logic
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.