# Master's Thesis: Completeness Results for Higher-Order Equational Logic

Mark Kaminski, Advisor: Gert Smolka

## Higher-Order Equational Logic

The simply typed lambda-calculus, introduced by Church [13], is nowadays considered one of the most important formal frameworks in mathematical logic and computer science, both in its own right and as a basis for more expressive calculi. Fundamental properties of the pure simply typed lambda-calculus, the simplest version of the calculus without constants or axioms, include decidability of deductive equality, essentially proven by Turing [17], and deductive completeness with respect to general and standard set-theoretic semantics, first shown by Friedman [16].

The equational proof system of the simply typed lambda-calculus with constants can be used to investigate the logical consequences of arbitrary sets of equational axioms in the same way as first-order equational reasoning is used to study algebraic theories (see [33]). Therefore, in the same sense as first-order equational reasoning from algebraic axioms is called first-order equational logic, equational reasoning in the simply typed lambda-calculus from higher-order axioms may be called higher-order equational logic.

The theory of beta-eta-conversion in the pure simply typed lambda-calculus, i.e. the set of all constant-free equations derivable by a finite number of beta-eta-conversion steps, can then be seen as a particular theory of higher-order equational logic, namely the one generated by the empty set of axioms. Of course, higher-order equational logic allows us to specify many more interesting theories, like equational formulations of Church's higher-order logic [13] or fragments thereof, Gödel's T [21] or Scott's PCF [41] (see also [40,38,42,30,45]).

## Abstract

We present several results concerning deductive completeness of the simply typed lambda-calculus with constants and equational axioms.

First, we prove deductive completeness of the calculus with respect to standard semantics for axioms containing neither free nor bound occurrences of higher-order variables. Using this result, we analyze some fundamental deductive and semantic properties of axiomatic systems without higher-order variables and compare them to those of established logical frameworks like first-order logic and Church's higher-order logic.

Second, we present a finite higher-order equational formulation of Henkin's Propositional Type Theory (PTT) [25] and prove its deductive completeness. We introduce a simple criterion which allows to reduce deductive completeness of systems with axiomatically defined constants to completeness of simpler axiomatic systems, and present an application of this criterion to our formulation of PTT.

Third, we prove the simply typed lambda-calculus both with and without eta-conversion complete with respect to general semantics. The result holds for systems with arbitrary axioms and constants.

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