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P4 is a domain-specific language for programming and specifying packet-processing systems. It is based on an elegant design with high-level abstractions like parsers and match-action pipelines that can be compiled to efficient implementations in software or hardware. Unfortunately, like many industrial languages, P4 has developed without a formal foundation. The P4 Language Specification is a 160-page document with a mixture of informal prose, graphical diagrams, and pseudocode. The P4 reference implementation is a complex system, running to over 40KLoC of C++ code. Clearly neither of these artifacts is suitable for formal reasoning. This paper presents a new framework, called Petr4, that puts P4 on a solid foundation. Petr4 consists of a clean-slate definitional interpreter and a calculus that models the semantics of a core fragment of P4. Throughout the specification, some aspects of program behavior are left up to targets. Our interpreter is parameterized over a target interface which collects all the target-specific behavior in the specification in a single interface. The specification makes ad-hoc restrictions on the nesting of certain program constructs in order to simplify compilation and avoid the possibility of nonterminating programs. We captured the latter intention in our core calculus by stratifying its type system, rather than imposing unnatural syntactic restrictions, and we proved that all programs in this core calculus terminate. We have validated the interpreter against a suite of over 750 tests from the P4 reference implementation, exercising our target interface with tests for different targets. We established termination for the core calculus by induction on the stratified type system. While developing Petr4, we reported dozens of bugs in the language specification and the reference implementation, many of which have been fixed.