This paper presents a program analysis method that generates program summaries involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. These techniques are able to generate all polynomial invariants for a restricted class of programs, but cannot be applied to programs outside of this class—for instance, programs with nested loops, conditional branching, unstructured control flow, etc. There currently lacks approaches to apply these prior methods to the case of general programs. This paper bridges that gap. Instead of restricting the kinds of programs we can handle, our method abstracts every loop into a model that can be solved with prior techniques, bringing to bear prior work on solvable polynomial maps to general programs. While no method can generate all polynomial invariants for arbitrary programs, our method establishes its merit through a monotonicty result. We have implemented our techniques, and tested them on a suite of benchmarks from the literature. Our experiments indicate our techniques show promise on challenging verification tasks requiring non-linear reasoning.

Abstract We present a method for synthesizing recursive functions that satisfy both a functional specification and an asymptotic resource bound. Prior methods for synthesis with a resource metric require the user to specify aconcreteexpression exactly describing resource usage, whereas our method uses big-Onotation to specify theasymptoticresource usage. Our method can synthesize programs with complex resource bounds, such as a sort function that has complexity $$O(n\log (n))$$ O(nlog(n)) . Our synthesis procedure uses a type system that is able to assign an asymptotic complexity to terms, and can track recurrence relations of functions. These typing rules are justified by theorems used in analysis of algorithms, such as the Master Theorem and the Akra-Bazzi method. We implemented our method as an extension of prior type-based synthesis work. Our tool,SynPlexity, was able to synthesize complex divide-and-conquer programs that cannot be synthesized by prior solvers.

Abstract This paper is a tutorial on algebraic program analysis. It explains the foundations of algebraic program analysis, its strengths and limitations, and gives examples of algebraic program analyses for numerical invariant generation and termination analysis.

This paper is the confluence of two streams of ideas in the literature on generating numerical invariants, namely: (1) template-based methods, and (2) recurrence-based methods.

We consider the problem of automatically establishing that a given syntax-guided-synthesis (SyGuS) problem is unrealizable (i.e., has no solution). We formulate the problem of proving that a SyGuS problem is unrealizable over a finite set of examples as one of solving a set of equations: the solution yields an overapproximation of the set of possible outputs that any term in the search space can produce on the given examples. If none of the possible outputs agrees with all of the examples, our technique has proven that the given SyGuS problem is unrealizable. We then present an algorithm for exactly solving the set of equations that result from SyGuS problems over linear integer arithmetic (LIA) and LIA with conditionals (CLIA), thereby showing that LIA and CLIA SyGuS problems over finitely many examples are decidable. We implement the proposed technique and algorithms in a tool called Nay. Nay can prove unrealizability for 70/132 existing SyGuS benchmarks, with running times comparable to those of the state-of-the-art tool Nope. Moreover, Nay can solve 11 benchmarks that Nope cannot solve.

We consider the problem of automatically establishing that a given syntax-guided-synthesis (SyGuS) problem is unrealizable (i.e., has no solution). Existing techniques have quite limited ability to establish unrealizability for general SyGuS instances in which the grammar describing the search space contains infinitely many programs. By encoding the synthesis problem’s grammar G as a nondeterministic program $$P_G$$ , we reduce the unrealizability problem to a reachability problem such that, if a standard program-analysis tool can establish that a certain assertion in $$P_G$$ always holds, then the synthesis problem is unrealizable. Our method can be used to augment existing SyGuS tools so that they can establish that a successfully synthesized program q is optimal with respect to some syntactic cost—e.g., q has the fewest possible if-then-else operators. Using known techniques, grammar G can be transformed to generate the set of all programs with lower costs than q—e.g., fewer conditional expressions. Our algorithm can then be applied to show that the resulting synthesis problem is unrealizable. We implemented the proposed technique in a tool called nope. nope can prove unrealizability for 59/132 variants of existing linear-integer-arithmetic SyGuS benchmarks, whereas all existing SyGuS solvers lack the ability to prove that these benchmarks are unrealizable, and time out on them.

Algebraic program analyses compute information about a program’s behavior by first (a) computing a valid path expression —i.e., a regular expression that recognizes all feasible execution paths (and usually more)—and then (b) interpreting the path expression in a semantic algebra that defines the analysis. There are an infinite number of different regular expressions that qualify as valid path expressions, which raises the question “ Which one should we choose? ” While any choice yields a sound result, for many analyses the choice can have a drastic effect on the precision of the results obtained. This paper investigates the following two questions: (1) What does it mean for one valid path expression to be “better” than another ? (2) Can we compute a valid path expression that is “better,” and if so, how ? We show that it is not satisfactory to compare two path expressions E 1 and E 2 solely by means of the languages that they generate . Counter to one’s intuition, it is possible for L ( E 2 ) ⊊ L ( E 1 ), yet for E 2 to produce a less-precise analysis result than E 1 —and thus we would not want to perform the transformation E 1 → E 2 . However, the exclusion of paths so as to analyze a smaller language of paths is exactly the refinement criterion used by some prior methods. In this paper, we develop an algorithm that takes as input a valid path expression E , and returns a valid path expression E ′ that is guaranteed to yield analysis results that are at least as good as those obtained using E . While the algorithm sometimes returns E itself, it typically does not: (i) we prove a no-degradation result for the algorithm’s base case—for transforming a leaf loop (i.e., a most-deeply-nested loop); (ii) at a non-leaf loop L , the algorithm treats each loop L ′ in the body of L as an indivisible atom, and applies the leaf-loop algorithm to L ; the no-degradation result carries over to (ii), as well. Our experiments show that the technique has a substantial impact: the loop-refinement algorithm allows the implementation of Compositional Recurrence Analysis to prove over 25% more assertions for a collection of challenging loop micro-benchmarks.

This paper investigates the problem of reasoning about non-linear behavior of simple numerical loops. Our approach builds on classical techniques for analyzing the behavior of linear dynamical systems. It is well-known that a closed-form representation of the behavior of a linear dynamical system can always be expressed using algebraic numbers, but this approach can create formulas that present an obstacle for automated-reasoning tools. This paper characterizes when linear loops have closed forms in simpler theories that are more amenable to automated reasoning. The algorithms for computing closed forms described in the paper avoid the use of algebraic numbers, and produce closed forms expressed using polynomials and exponentials over rational numbers. We show that the logic for expressing closed forms is decidable, yielding decision procedures for verifying safety and termination of a class of numerical loops over rational numbers. We also show that the procedure for computing closed forms for this class of numerical loops can be used to over-approximate the behavior of arbitrary numerical programs (with unrestricted control flow, non-deterministic assignments, and recursive procedures).

Automatic generation of non-linear loop invariants is a long-standing challenge in program analysis, with many applications. For instance, reasoning about exponentials provides a way to find invariants of digital-filter programs, and reasoning about polynomials and/or logarithms is needed for establishing invariants that describe the time or memory usage of many well-known algorithms. An appealing approach to this challenge is to exploit the powerful recurrence-solving techniques that have been developed in the field of computer algebra, which can compute exact characterizations of non-linear repetitive behavior. However, there is a gap between the capabilities of recurrence solvers and the needs of program analysis: (1) loop bodies are not merely systems of recurrence relations---they may contain conditional branches, nested loops, non-deterministic assignments, etc., and (2) a client program analyzer must be able to reason about the closed-form solutions produced by a recurrence solver (e.g., to prove assertions). This paper presents a method for generating non-linear invariants of general loops based on analyzing recurrence relations. The key components are an abstract domain for reasoning about non-linear arithmetic, a semantics-based method for extracting recurrence relations from loop bodies, and a recurrence solver that avoids closed forms that involve complex or irrational numbers. Our technique has been implemented in a program analyzer that can analyze general loops and mutually recursive procedures. Our experiments show that our technique shows promise for non-linear assertion-checking and resource-bound generation.