ABSTRACT. Financial applications such as Landing and Payment protocols, and their realization in decentralized financial (DeFi) applications in Blockchains, comprise a unique domain where bugs in the code may be exploited by anyone to steal assets. This situation provides unique opportunities for formal verification to enable “move fast and break nothing”. Formal verification can be used to detect errors early in the development process and guarantee correctness when a new version of the code is deployed.I will describe an attempt to automatically verify DeFis and identify potential bugs. The approach is based on breaking the verification of DeFis into decidable verification tasks. Each of these tasks is solved via a decision procedure which automatically generates a formal proof or a test input showing a violation of the specification. In order to overcome undecidability, high level properties are expressed as ghost states and static analysis used to infer how low level programs update ghost states.

ABSTRACT. ksmt is a CDCL-style calculus for solving non-linear constraints over real numbers involving polynomials and transcendental functions. In this paper we investigate properties of the ksmt calculus and show that it is a delta-complete decision procedure. We also propose an extension with local linearisations, which allow for efficient treatment of non-linear constraints.

ABSTRACT. The problem of invariant checking in parametric systems -- which are required to operate correctly regardless of the number and connections of their components -- is gaining increasing importance in various sectors, such as communication protocols and control software. Such systems are typically modeled using quantified formulae, describing the behaviour of an unbounded number of (identical) components, and their automatic verification often relies on the use of decidable fragments of first-order logic in order to effectively deal with the challenges of quantified reasoning.

In this paper, we propose a fully automatic technique for invariant checking of parametric systems which does not rely on quantified reasoning. Parametric systems are modeled with array-based transition systems, and our method iteratively constructs a quantifier-free abstraction by analyzing, with SMT-based invariant checking algorithms for non-parametric systems, increasingly-larger finite instances of the parametric system. Depending on the verification result in the concrete instance, the abstraction is automatically refined by leveraging canditate lemmas from inductive invariants, or by discarding previously computed lemmas.

We implemented the method using a quantifier-free SMT-based IC3 as underlying verification engine. Our experimental evaluation demonstrates that the approach is competitive with the state of the art, solving several benchmarks that are out of reach for other tools.

ABSTRACT. Sentential Calculus with Identity (SCI) is an extension of classical propositional logic, featuring a new connective of identity between formulas. In SCI two formulas are said to be identical if they share the same denotation. In the semantics of the logic, truth values are distinguished from denotations, hence the identity connective is strictly stronger than classical equivalence. In this paper we present a sound, complete, and terminating algorithm deciding the satisfiability of SCI-formulas, based on labelled tableaux. To the best of our knowledge, it is the first implemented decision procedure for SCI which runs in NP, i.e., is complexity-optimal. The obtained complexity bound is a result of dividing derivation rules in the algorithm into two sets: decomposition and equality rules, whose interplay yields derivation trees with branches of polynomial length with respect to the size of the investigated formula. We describe an implementation of the procedure and compare its performance with implementations of other calculi for SCI} (for which, however, the termination results were not established). We show possible refinements of our algorithm and discuss the possibility of extending it to other non-Fregean logics.

ABSTRACT. The material presented in this paper contributes to establishing a basis deemed essential for substantial progress in Automated Deduction. It identifies and studies global features in selected problems and their proofs which offer the potential of guiding proof search in a more direct way. The studied problems are of the wide-spread form of "axiom(s) and rule(s) imply goal(s)". The features include various proof structures and the well-known concept of lemmas. For their elaboration both human and automated proofs of selected theorems are taken into a close comparative consideration. The study at the same time accounts for a coherent and comprehensive formal reconstruction of historical work by Łukasiewicz, Meredith and others. First experiments resulting from the study indicate novel ways of lemma generation to supplement automated first-order provers of various families, strengthening in particular their ability to find short proofs.

ABSTRACT. We present novel reductions of the propositional modal logics KB, KD, KT, K4 and K5 to Separated Normal Form with Sets of Modal Levels. The reductions result in smaller formulae than the well-known reductions by Kracht and allow us to use the local reasoning of the prover \KSP to determine the satisfiability of modal formulae in these logics. We show experimentally that the combination of our reductions with the prover KSP performs well when compared with a specialised resolution calculus for these logics and with the in-built reductions of the first-order prover SPASS.

ABSTRACT. Isabelle is a generic theorem prover with a fragment of higher-order logic as a metalogic for defining object logics. Isabelle also provides proof terms. We formalize this metalogic and the language of proof terms in Isabelle/HOL, define an executable (but inefficient) proof term checker and prove its correctness w.r.t. the metalogic. We integrate the proof checker with Isabelle and run it on a range of logics and theories to check the correctness of all the proofs in those theories.

ABSTRACT. The entailment problem in Separation Logic, between separated conjunctions of equational, spatial and predicate atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called establishment and restrictedness, respectively. Both classes are shown to be in 2EXPTIME by two independent proofs and a many-one reduction of established to restricted entailment problems has been given. In this paper, we strictly generalize the restricted class, by distinguishing the conditions that apply only to the left and the right hand side of entailments, respectively. We provide a many-one reduction of this generalized class, called safe, to the established class. Together with the reduction of established to restricted entailment problems, this new reduction closes the loop and shows that the three classes of entailment problems (respectively established, restricted and safe) form a single, unified, 2EXPTIME-complete class.

ABSTRACT. Subformula linking is an interactive theorem proving technique that was initially proposed for (classical) linear logic. It is based on truth and context preserving rewrites of a conjecture that are triggered by a user indicating directed edges between subformulas, which can be done without tactics or proof languages. The system guarantees that a true conjecture can always be rewritten to a known, usually trivial, theorem. In this work we extend this technique to intuitionistic logic and type theory in three steps. First, we extend subformula linking to intuitionistic first-order logic. Next, we show how to integrate simply typed lambda-terms as the term language of this logic. Finally we rely on a simple embedding of intuitionistic type theory into this logic, and discuss hiding the embedding from the end user's perspective.

ABSTRACT. We  present  an efficient proof search procedure  for Intuitionistic Propositional Logic  which involves the use of an incremental SAT-solver. Basically, it is obtained by adding a restart operation to the system intuit by Claessen and Rosén,thus we call our implementation intuitR. We gain some remarkable advantages: derivations have a simple structure; countermodels are in general small; using a standard benchmarks suite, we outperform intuit and other state-of-the-art provers.

ABSTRACT. Attestation logics have been used for specifying systems with policies involving different principals. Cyberlogic is an attestation logic used for the specification of Evidential Transactions (ETs). In such transactions, evidence has to be provided supporting its validity with respect to given policies. For example, visa applicants must demonstrate that they have sufficient funds to visit a foreign country. Such evidence can be expressed as a Cyberlogic proof, possibly combined with non-logical data (e.g., a digitally signed document). A key issue is how to construct and communicate such evidence/proofs. It turns out that attestation modalities makes it challenging to use established proof-theoretic methods such as focusing. Our first contribution is the refinement of Cyberlogic proof theory with knowledge operators which can be used to represent knowledge bases local to one or more principals. Our second contribution is the identification of a fragment of Cyberlogic, called Cyberlogic programs, enabling the specification of ETs. Our third contribution is a sound and complete proof system for Cyberlogic programs enabling proof search similar to search in logic programming. Our final contribution is a proof certificate format for Cyberlogic programs inspired by Foundational Proof Certificates as a means to communicate evidence and check its validity.