In this paper, we introduce a novel logic language and theorem prover for robotic task planning. Our language, which we call Linear Planning Logic (LPL), is a fragment of linear logic whose resource-conscious semantics are well suited for reasoning with dynamic state, while its structure admits efficient theorem provers for automatic plan construction. LPL can be considered as an extension of Linear Hereditary Harrop Formulas (LHHF), whose careful design allows the minimization of nondeterminism in proof search, providing a sufficient basis for the design of linear logic programming languages such as Lolli. Our new language extends on the expressivity of LHHF, while keeping the resulting nondeterminism in proof search to a minimum for efficiency. This paper introduces the LPL language, presents the main ideas behind our theorem prover on a smaller fragment of this language and finally provides an experimental illustration of its operation on the problem of task planning for the hexapod robot RHex.