Alessio Mansutti

Michael Benedikt, Dmitry Chistikov, Alessio Mansutti

We investigate expansions of Presburger arithmetic, i.e., the theory of the integers with addition and order, with additional structure related to exponentiation: either a function that takes a number to the power of $2$, or a predicate for the powers of $2$. The latter theory, denoted $\mathrm{PresPower}$, was introduced by Büchi as a first attempt at characterizing the sets of tuples of numbers that can be expressed using finite automata; Büchi's method does not give an elementary upper bound, and the complexity of this theory has been open. The former theory, denoted as $\mathrm{PresExp}$, was shown decidable by Semenov; while the decision procedure for this theory differs radically from the automata-based method proposed by Büchi, Semenov's method is also non-elementary. And in fact, the theory with the power function has a non-elementary lower bound. In this paper, we show that while Semenov's and Büchi's approaches yield non-elementary blow-ups for $\mathrm{PresPower}$, the theory is in fact decidable in triply exponential time, similarly to the best known quantifier-elimination algorithm for Presburger arithmetic. We also provide a $\mathrm{NExpTime}$ upper bound for the existential fragment of $\mathrm{PresExp}$, a step towards a finer-grained analysis of its complexity. Both these results are established by analyzing a single parameterized satisfiability algorithm for $\mathrm{PresExp}$, which can be specialized to either the setting of $\mathrm{PresPower}$ or the existential theory of $\mathrm{PresExp}$. Besides the new upper bounds for the existential theory of $\mathrm{PresExp}$ and $\mathrm{PresPower}$, we believe our algorithm provides new intuition for the decidability of these theories, and for the features that lead to non-elementary blow-ups.

Jorge Gallego-Hernández, Enrico Lipparini, Alessio Mansutti

We propose a framework for solving quantifier-free formulas from (undecidable) extensions of non-linear real arithmetic (NRA) with transcendental functions, such as exponential and trigonometric ones. The framework extends the Model Constructive Satisfiability calculus (MCSAT), and leverages procedures for NRA and methods from real analysis. At its core, our procedure abstracts the input formula to NRA, and lets MCSAT and an NRA plugin incrementally build a partial model of the abstracted formula. A Transcendental Real Arithmetic plugin, acting as an intermediary between MCSAT and the NRA plugin, ensures the consistency of the partial model and is responsible for refining the abstracted formula. We implemented our procedure in the Yices2 SMT solver for the sine and exponential functions, and conducted an extensive empirical evaluation that shows that our prototype outperforms state-of-the-art solvers on both SAT and UNSAT instances.

Michael Benedikt, Alessio Mansutti

We study fitting problems, sometimes called ``training problems'', where we have a finite sample consisting of inputs and outputs, and we want to know whether there is a function in a certain class that could produce these outputs, exactly or approximately, on the given inputs. We focus on the computational and descriptive complexity of fitting for logically-defined classes in common decidable structures, like the real ordered field and Presburger arithmetic, and also for broader classes defined via combinatorial or model-theoretic properties. We isolate the complexity of these fitting problems, with particular attention to cases where we can use queries in a natural query language over the sample to determine whether a sample is fittable.

Louis Rustenholz, Manuel V. Hermenegildo, Pedro Lopez-Garcia et al.

We present the theory underpinning a complexity analysis tool (under development) that aims at automating tedious parts of the analysis of complex algorithms originating from the field of automated reasoning. Examples are given by super-exponential quantifier elimination procedures in real and integer arithmetics. Our tool implements the following pipeline. * Together with the algorithm to be analysed, the user (expert, e.g. the algorithm designer) can provide key metrics to track, and lemmas to improve the analysis. In pen-and-paper proofs, these correspond to the "non-tedious" and "creative" parts of the complexity analysis, that require human ingenuity. * The second step consists in the extraction of (generalised) recurrence equations. Here, we rely on a novel higher-order abstract interpretation technique, built on the concept of operator semantics. It enables (optimal) abstract compilation of symbolic programs to different kinds of purely numerical recursive representations, such as recurrence equations on interval-valued functions or numerical logic programs. * Finally, our tool solves the recurrence equations. We propose to go beyond the direct usage of computer algebra systems (CAS), and use pre/postfixpoint-based techniques to discover and verify candidate bounds on the solution. This approach makes use, in turn, of recent progress in SMT solvers, and can also be improved by techniques originating in termination analysis research.

Alessio Mansutti, Marino Miculan

The spi-calculus is a formal model for the design and analysis of cryptographic protocols: many security properties, such as authentication and strong confidentiality, can be reduced to the verification of behavioural equivalences between spi processes. In this paper we provide an algorithm for deciding hedged bisimilarity on finite processes, which is equivalent to barbed equivalence (and coarser than framed bisimilarity). This algorithm works with any term equivalence satisfying a simple set of conditions, thus encompassing many different encryption schemata.