Faster Smarter Proof by Induction in Isabelle/HOL

We present sem_ind, a recommendation tool for proof by induction in Isabelle/HOL. Given an inductive problem, sem_ind produces candidate arguments for proof by induction, and selects promising ones using heuristics. Our evaluation based on 1,095 inductive problems from 22 source files shows that sem_ind improves the accuracy of recommendation from 20.1% to 38.2% for the most promising candidates within 5.0 seconds of timeout compared to its predecessor while decreasing the median value of execution time from 2.79 seconds to 1.06 seconds.

Verification and refutation of C programs based on k-induction and invariant inference

DepthK is a source-to-source transformation tool that employs bounded model checking (BMC) to verify and falsify safety properties in single- and multi-threaded C programs, without manual annotation of loop invariants. Here, we describe and evaluate a proof-by-induction algorithm that combines k-induction with invariant inference to prove and refute safety properties. We apply two invariant generators to produce program invariants and feed these into a k-induction-based verification algorithm implemented in DepthK, which uses the efficient SMT-based context-bounded model checker (ESBMC) as sequential verification back-end. A set of C benchmarks from the International Competition on Software Verification (SV-COMP) and embedded-system applications extracted from the available literature are used to evaluate the effectiveness of the proposed approach. Experimental results show that k-induction with invariants can handle a wide variety of safety properties, in typical programs with loops and embedded software applications from the telecommunications, control systems, and medical domains. The results of our comparative evaluation extend the knowledge about approaches that rely on both BMC and k-induction for software verification, in the following ways. (1) The proposed method outperforms the existing implementations that use k-induction with an interval-invariant generator (e.g., 2LS and ESBMC), in the category ConcurrencySafety, and overcame, in others categories, such as SoftwareSystems, other software verifiers that use plain BMC (e.g., CBMC). Also, (2) it is more precise than other verifiers based on the property-directed reachability (PDR) algorithm (i.e., SeaHorn, Vvt and CPAchecker-CTIGAR). This way, our methodology demonstrated improvement over existing BMC and k-induction-based approaches.

Arithmetic, philosophical issues in

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.

From Mathematical Induction to Discrete Time

Proof by induction involves a chain of implications in which the stages are well ordered. A chain of cause and effect in nature also involves a chain of implications. For this chain to “imply” or bring about its effects in a logical sense, it also has to be organized into a well ordering of stages (which are the points or quanta of time). This means that time must be quantized rather than continuous. An argument from relativity implies that space is quantized as a consequence.

Renormalization group approach for Lorentz-violating scalar field theory at all loop orders

We compute, both explicitly, at least, up to next-to-leading order and in a proof by induction for all loop levels, the critical exponents for thermal Lorentz-violating O([Formula: see text]) self-interacting scalar field theory. They are evaluated in a massless theory renormalized at arbitrary external momenta, where a reduced number of Feynman diagrams is needed. The results are presented and shown to be identical to that found previously in distinct theories renormalized at different renormalization schemes. Finally, we give both mathematical explanation and physical interpretation for them based on coordinates redefinition techniques and symmetry ideas, respectively.