We present a sequent-based deductive system for automatically proving entailments in separation logic by using mathematical induction. Our technique, called mutual explicit induction proof, is an instance of Noetherian induction. Specifically, we propose a novel induction principle on a well-founded relation of separation logic model, and follow the explicit induction methods to implement this principle as inference rules, so that it can be easily integrated into a deductive system. We also support mutual induction, a natural feature of implicit induction, where the goal entailment and other entailments derived during the proof search can be used as hypotheses to prove each other. We have implemented a prototype prover and evaluated it on a benchmark of handcrafted entailments as well as benchmarks from a separation logic competition.

The symbolic-heap fragment of separation logic has been actively developed and advocated for verifying the memory-safety property of computer programs. At present, one of its biggest challenges is to effectively prove entailments containing inductive heap predicates. These entailments are usually proof obligations generated when verifying programs that manipulate complex data structures like linked lists, trees, or graphs.To assist in proving such entailments, this paper introduces a lemma synthesis framework, which automatically discovers lemmas to serve as eureka steps in the proofs. Mathematical induction and template-based constraint solving are two pillars of our framework. To derive the supporting lemmas for a given entailment, the framework firstly identifies possible lemma templates from the entailment's heap structure. It then sets up unknown relations among each template's variables and conducts structural induction proof to generate constraints about these relations. Finally, it solves the constraints to find out actual definitions of the unknown relations, thus discovers the lemmas. We have integrated this framework into a prototype prover and have experimented it on various entailment benchmarks. The experimental results show that our lemma-synthesisassisted prover can prove many entailments that could not be handled by existing techniques. This new proposal opens up more opportunities to automatically reason with complex inductive heap predicates.

Techniques for proving termination and non-termination of imperative programs are usually considered as orthogonal mechanisms. In this paper, we propose a novel mechanism that analyzes and proves both program termination and non-termination at the same time. We first introduce the concept of second-order termination constraints and accumulate a set of relational assumptions on them via a Hoare-style verification. We then solve these assumptions with case analysis to determine the (conditional) termination and nontermination scenarios expressed in some specification logic form. In contrast to current approaches, our technique can construct a summary of terminating and non-terminating behaviors for each method. This enables modularity and reuse for our termination and non-termination proving processes. We have tested our tool on sample programs from a recent termination competition, and compared favorably against state-of-the-art termination analyzers.

Techniques for proving termination and non-termination of imperative programs are usually considered as orthogonal mechanisms. In this paper, we propose a novel mechanism that analyzes and proves both program termination and non-termination at the same time. We first introduce the concept of second-order termination constraints and accumulate a set of relational assumptions on them via a Hoare-style verification. We then solve these assumptions with case analysis to determine the (conditional) termination and nontermination scenarios expressed in some specification logic form. In contrast to current approaches, our technique can construct a summary of terminating and non-terminating behaviors for each method. This enables modularity and reuse for our termination and non-termination proving processes. We have tested our tool on sample programs from a recent termination competition, and compared favorably against state-of-the-art termination analyzers.

We introduce a new dynamic analysis technique to discover invariants in separation logic for heap-manipulating programs. First, we use a debugger to obtain rich program execution traces at locations of interest on sample inputs. These traces consist of heap and stack information of variables that point to dynamically allocated data structures. Next, we iteratively analyze separate memory regions related to each pointer variable and search for a formula over predefined heap predicates in separation logic to model these regions. Finally, we combine the computed formulae into an invariant that describes the shape of explored memory regions.We present SLING, a tool that implements these ideas to automatically generate invariants in separation logic at arbitrary locations in C programs, e.g., program pre and postconditions and loop invariants. Preliminary results on existing benchmarks show that SLING can efficiently generate correct and useful invariants for programs that manipulate a wide variety of complex data structures.