We present a method for certifying that the values computed by an imperative program will be bounded by polynomials in the program’s inputs. To this end, we introduce mwp-matrices and define a semantic relation | = C: M where C is a program and M is an mwp-matrix. It follows straightforwardly from our definitions that there exists M such that | = C:M holds iff every value computed by C is bounded by a polynomial in the inputs. Furthermore, we provide a syntactical proof calculus and define the relation ⊢ C:M to hold iff there exists a derivation in the calculus where C:M is the bottom line. We prove that ⊢ C:M implies | = C:M. By means of exhaustive proof search, an algorithm can decide if there exists M such that the relation ⊢ C:M holds, and thus, our results yield a computational method. Categories and Subject Descriptors: D.2.4 [Software engineering]: Software/Program Verification; F.2.0 [Analysis of algorithms and problem complexity]: General; F.3.1 [Logics and meanings of programs]: Specifying and Verifying and Reasoning about Programs

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple “core ” programming language— an imperative language with bounded loops, and arithmetics limited to addition and multiplication— it is possible to decide precisely whether a program has certain growth-rate properties, namely polynomial (or linear) bounds on computed values, or on the running time. This work emphasized the role of the core language in mitigating the notorious undecidability of program properties, so that one deals with decidable problems, while allowing the application of the technique to programs in a more realistic language. This is done by over-approximating the semantics of the concrete program. A natural and intriguing problem was whether more elements can be added to the core language, improving the precision of approximation, while keeping the growth-rate properties decidable. In particular, the method presented could not handle a command that resets a variable to zero. This paper shows how to handle resets. The analysis is given in a logical style (proof rules), and the complexity of the algorithm is shown to be PSPACE. The problem is shown PSPACE-complete (in contrast, without resets, the problem was PTIME). The analysis algorithm evolved from the previous solution in an interesting way: focus was shifted from proving a bound to disproving it, and the algorithm works top-down rather than bottom-up. 1

Abstract. We present a new method for inferring complexity properties for imperative programs with bounded loops. The properties handled are: polynomial (or linear) boundedness of computed values, as a function of the input; and similarly for the running time. It is well known that complexity properties are undecidable for a Turing-complete programming language. Much work in program analysis overcomes this obstacle by relaxing the correctness notion: one does not ask for an algorithm that correctly decides whether the property of interest holds or not, but only for “yes ” answers to be sound. In contrast, we reshaped the problem by defining a “core ” programming language that is Turing-incomplete, but strong enough to model real programs of interest. For this language, our method is the first to give a certain answer; in other words, our inference is both sound and complete. The essence of the method is that every command is assigned a “complexity certificate”, which is a concise specification of dependencies of output values on input. These certificates are produced by inference rules that are compositional and efficiently computable. The approach is inspired by previous work by Niggl and Wunderlich and by Jones and Kristiansen, but use a novel, more expressive kind of certificates.

We investigate the decidability of the feasibility problem for imperative programs with bounded loops. A program is called feasible if all values it computes are polynomially bounded in terms of the input. The feasibility problem is representative of a group of related properties, like that of polynomial time complexity. It is well known that such properties are undecidable for a Turing-complete programming language. They may be decidable, however, for fragmentary languages, that are Turing-incomplete. But if such fragments are expressive enough, they do pose a challenge for analysis. We are interested in tracing the edge of decidability for the feasibility problem and similar problems. In previous work, we proved that such problems are decidable for an interesting fragment where loops were indefinite (that is, may exit before completing the given iteration count). In this paper, we consider definite loops. A second language feature that we vary, is the kind of assignment statements. With ordinary assignment, we prove undecidability of a very tiny language fragment. We also prove undecidability with weak assignment, where the modified variable may receive any value bounded by the given expression, even zero. But we prove decidability with max assignments, where the assigned variable never decreases its value.