Interaction trees are an active development in representing effectful and impure pro- grams in the Coq proof assistant. Examples of programs they can represent are programs that use: mutable state, concurrency and general recursion. Besides representing these programs we also want to reason about and verify these programs using separation logic. That is the purpose of this thesis. More technically speaking interaction trees are new way to do shallow embeddings in the Coq proof assistant. They are a coinductive variant of the free monad and come with the usual constructions of events and event handlers. The aim of interaction trees is to represent impure programs and potentially non-terminating programs in their environment. Interaction trees are, in contrast to relational operational semantics, executable by interpretation or program extraction. Interaction trees come with a framework for reasoning about their behavior based on equivalency up to weak bisimulation. An open problem is to reason about interaction trees utilizing a separation logic rather than weak bisimulation. We developed Pothos as a solution to this problem. Pothos has an Iris based concurrent separation logic for interaction trees. We address the problem in a non-extensible setting, with mutable state, non-termination and concur- rency as our chosen effects. Pothos inherits all the executable properties from interaction trees and includes a novel relation of Iris’s step-index with coinductive types. We have proven our logic to be sound and include a case study of a spin lock library. The case study shows that our logic is both non-trivial and can utilize the standard Iris patterns for concurrency.

Context-free grammars (CFGs) provide a well-known formalism for the specification of programming languages. They describe the structure of a program in terms of parse trees. One major issue of CFGs is ambiguity, where one sentence can sometimes have multiple different parse trees. Some formalisms like SDF3 or YACC allow annotating a grammar with disambiguation rules, such as priority or associativity. Disambiguation rules filter out certain parse trees, making a grammar less ambiguous. Giving a formal semantics for these disambiguation rules is still an ongoing research topic. In this thesis we verify an existing semantics for these rules by Souza Amorim and Visser (2019) for a subset of expression grammars. These grammars may contain infix, prefix, and postfix expressions. We verify the semantics by proving that it is both safe and complete. Safety states adding disambiguation rules will not change the underlying language of the grammar, meaning each sentence in the language will have at least one valid parse tree that does not get filtered out. Completeness guarantees that a grammar is unambiguous, meaning that each sentence in the language will have at most one valid parse tree that does not get filtered out. We have mechanized the proofs in the Coq Proof Assistant, increasing the confidence in their correctness. As part of the proofs, we also provide a verified implementation for disambiguation rules.