ABSTRACT. The vast majority of modal theorem provers implement modal tableau, or backwards proof search in (cut-free) sequent calculi. The design of suitable calculi is highly non-trivial, and employs nested sequents, labelled sequents and/or specifically designated transitional formulae. Theorem provers for first-order logic, on the other hand, are by and large based on resolution. In this paper, we present a resolution system for preference-based modal logics, specifically Burgess' system S. Our main technical results are soundness and completeness. Conceptually, we argue that resolution-based systems are not more difficult to design than cut-free sequent calculi but their purely syntactic nature makes them much better suited for implementation in automated reasoning systems.

ABSTRACT. If a conclusion follows from a set of axioms, then its justification is a minimal subset of axioms for which the entailment holds. An entailment can have several justifications. Such justifications are commonly used for the purpose of debugging of incorrect entailments in Description Logic ontologies. Recently a number of SAT-based methods have been proposed that can enumerate all justifications for entailments in light-weight ontologies languages, such as EL. These methods work by encoding EL inferences by propositional Horn clauses, and finding minimal models that correspond to justifications using SAT solvers. In this paper, we propose a new procedure for enumeration of justifications that uses resolution with answer literals instead of SAT solvers. In comparison to SAT-based methods, our procedure can enumerate justifications in any user-defined order that extends the set inclusion relation. The procedure is easy to implement and, like resolution, can be parametrized with ordering and selection strategies. We have implemented this procedure in PULi---a new Java-based Proof Utility Library, and performed an empirical comparison of (several strategies of) our procedure and other SAT-based tools on popular EL ontologies. The experiments show that our procedure provides a comparable, if not better performance than those highly optimized tools. For example, using one of the strategies, we were able for the first time to compute all justifications for all concept subsumptions in one of the largest commonly used medical ontology Snomed CT.

ABSTRACT. A tableau calculus is proposed, based on a compressed representation of clauses, where literals sharing a similar shape may be merged. The inferences applied on these literals are fused when possible, which reduces the size of the proof. It is shown that the obtained proof procedure is sound, refutationally complete and allows to reduce the size of the tableau by an exponential factor. The approach is compatible with all usual refinements of tableaux.

ABSTRACT. The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., purely propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction operation to eliminate universal literals from clauses. We apply QUP to an abstraction of the QBF where certain universal quantifiers are converted into existential ones. This way, we obtain a generalization of QRAT which we call QRAT+. The resulting redundancy property in QRAT+ based on QUP is more powerful than the one in QRAT based on UP. We report on proof theoretical improvements and on experimental results to illustrate the benefits of using QRAT+ for QBF preprocessing.

ABSTRACT. We prove that extended resolution, a well-known proof system introduced by Tseitin, polynomially simulates DRAT, the standard proof system in modern SAT solving. Our simulation procedure takes as input a DRAT proof and transforms it into an extended-resolution proof whose size is only polynomial with respect to the original proof. Based on our simulation, we implemented a tool that transforms DRAT proofs into extended-resolution proofs. We ran our tool on several benchmark formulas to estimate the increase in size caused by our simulation in practice. Finally, as a side note, we show how blocked-clause addition, a generalization of the extension rule from extended-resolution, can be used to replace the addition of resolution asymmetric tautologies in DRAT without introducing new variables.

ABSTRACT. The successes of machine learning in recent years triggered a fast growing range of applications. In important settings, including safety critical applications, accurate predictions do not suffice; one expects the machine learning model to also explain the predicions made, in forms understandable by humans. Recent work proposed explainable models based on decision sets which can be viewed as unordered sets of rules, respecting some sort of rule non-overlap constraint. This paper investigates existing solutions for computing decision sets and identifies a number of drawbacks, related with rule overlap and succinctness of explanations, the accuracy of achieved results, but also the efficiency of proposed approaches. To address these drawbacks, the paper develops novel SAT-based solutions for learning decision sets. Experimental results on computing decision sets for representative datasets demonstrate that SAT enables solutions that are not only the most efficient, but also offer stronger guarantees in terms of rule non-overlap.

ABSTRACT. We give a constructive account of Kripke-Curry's method which was used to establish the decidability of Implicational Relevance Logic (R->). To sustain our approach, we mechanize this method in axiom-free Coq, abstracting away from the specific features of R-> to keep only the essential ingredients of the technique. In particular we show how to replace Kripke/Dickson's lemma by a constructive form of Ramsey's theorem based on the notion of almost full relation. We also explain how to replace König's lemma with an inductive form of Brouwer's Fan theorem. We instantiate our abstract proof to get a constructive decision procedure for R-> and discuss potential applications to other logical decidability problems.

ABSTRACT. We present a framework in Isabelle for verifying asymptotic time complexity of imperative programs. We build upon an extension of Imperative HOL and its separation logic to include running time. In addition to the basic arguments, our framework is able to handle advanced techniques for time complexity analysis, such as the use of the Akra-Bazzi theorem and amortized analysis. Various automation is built and incorporated into the auto2 prover to reason about separation logic with time credits, and to derive asymptotic behavior of functions. As case studies, we verify the asymptotic time complexity (in addition to functional correctness) of imperative algorithms and data structures such as median of medians selection, Karatsuba's algorithm, and splay trees.

ABSTRACT. We present a formalization of the first half of Bachmair and Ganzinger's chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi. Our work clarifies several of the fine points in the chapter's text, emphasizing the value of formal proofs in the field of automated reasoning.

ABSTRACT. We consider the problem of solving floating-point constraints obtained from software verification. We present UppSAT - an new implementation of a systematic approximation refinement framework as an abstract SMT solver. Provided with an approximation and a decision procedure (implemented in an off-the-shelf SMT solver), UppSAT yields an approximating SMT solver. Additionally, UppSAT includes a library of predefined approximation components which can be combined and extended to define new encodings, orderings and solving strategies. We propose that UppSAT can be used as a sandbox for easy and flexible exploration of new approximations. To substantiate this, we explore several approximations of floating-point arithmetic. Approximations can be viewed as a composition of an encoding into a target theory, a precision ordering, and a number of strategies for model reconstruction and precision (or approximation) refinement. We present encodings of floating-point arithmetic into reduced precision floating-point arithmetic, real-arithmetic, and fixed-point arithmetic (encoded into the theory of bit-vectors in practice). In an experimental evaluation we compare the advantages and disadvantages of approximating solvers obtained by combining various encodings and decision procedures (based on existing, state-of-the-art SMT solvers for floating-point, real, and bit-vector arithmetic).

ABSTRACT. We introduce a new theory of algebraic datatypes where selector symbols can be shared between multiple constructors, thereby reducing the number of terms considered by current SMT-based solving approaches. We show the satisfiability problem for the traditional theory of algebraic datatypes can be reduced to problems where selectors are mapped to shared symbols based on a transformation provided in this paper. The use of shared selectors addresses a key bottleneck for an SMT-based enumerative approach to the Syntax-Guided Synthesis (SyGuS) problem. Our experimental evaluation of an implementation of the new theory in the solver CVC4 on syntax-guided synthesis and other domains shows evidence that the use of shared selectors improves state-of-the-art SMT-based approaches for datatype constraints.

ABSTRACT. We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over-)approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system.

ABSTRACT. Optique (http://optique-project.eu) was an EU FP7 project that ran from November 2012 to
October 2016. The main objective was to test the idea of “Ontology Based Data Access” (OBDA)
on real industrial applications. Concretely: to support the work of geologists and
geophysicists in the oil & gas company Statoil, and the work of turbine engineers at Siemens
AG. This line of work now continues in the nationally funded ‘Centre for Research-based
Innovation’ SIRIUS (http://sirius-labs.no) at the University of Oslo, with participation from
the Universities of Oxford and Trondheim, as well a large number of participating companies.

The software produced by the project features elaborate user interfaces, and no ∀ or ∃ can
be seen on the surface. Still, most of the functionality is controlled by an ontology, which
is nothing more than a set of axioms in a particular description logic. As a consequence, a
variety of reasoning tasks takes place under the hood, all the way from query optimisation,
via entity alignment and up to the user interface control code. This talk presents a
selection of these problems, both solved and as-yet unsolved.

Though logic and reasoning are close to the hearts of many of the researchers involved, the
success of the project was also dependent on other factors: interdisciplinary communication,
usability considerations, and many pragmatic compromises, to name some. And sometimes, these
would again lead to ‘nice’ research. The talk also covers some of these extra-logical aspects
of the project.

ABSTRACT. Given two or more well-founded (terminating) binary relations, when can one be sure that their union is likewise well-founded? We suggest new conditions for an arbitrary number of relations, generalising known conditions for two relations. We also provide counterexamples to several potential weakenings. All proofs have been machine checked.

ABSTRACT. In this paper we investigate Hughes' combinatorial proofs as notion of proof identity for classical logic. We show for various syntactic formalisms, including sequent calculus, analytic tableaux and resolution, how they can be translated into combinatorial proofs, and which notion of identity they enforce. This allows, in particular, to compare proofs that are given in different formalisms.

ABSTRACT. Logical frameworks allow the specification of deductive systems using the same logical machinery. Linear logical frameworks have been successfully used for the specification of a number of computational, logics and proof systems. Its success lies on the fact that formulas can be distinguished as linear, which behave intuitively as resources, and unbounded, which behave intuitionistically. Commutative subexponentials enhance the expressiveness of linear logic frameworks by allowing the distinction of multiple contexts. These contexts may behave as multisets of formulas or sets of formulas. Motivated by applications in distributed systems and in type-logical grammar, we propose a linear logical framework containing both commutative and non-commutative subexponentials. Non-commutative subexponentials can be used to specify contexts which behave as lists, not multisets, of formulas. In addition, motivated by our applications in type-logical grammar, where the weakenening rule is disallowed, we investigate the proof theory of formulas that can only contract, but not weaken. In fact, our contraction is non-local. We demonstrate that under some conditions such formulas may be treated as unbounded formulas, which behave intuitionistically.

ABSTRACT. This paper presents a uniform substitution calculus for differential game logic (dGL). Church's uniform substitutions substitute a term or formula for a function or predicate symbol everywhere. After generalizing them to differential game logic and allowing for the substitution of hybrid games for game symbols, uniform substitutions make it possible to only use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting axiomatization adopts only a finite number of ordinary dGL formulas as axioms, which uniform substitutions instantiate soundly. This paper proves the soundness of uniform substitution for the monotone modal logic dGL. The resulting axiomatization admits a straightforward modular implementation of dGL in theorem provers.

ABSTRACT. Theories are an essential structuring principle that enable modularity, encapsulation, and reuse in formal libraries and programs (called classes there). Similar effects can be achieved by dependent record types. While the former forms a separate language layer, the latter is a normal part of the type theory. This overlap in functionality can render different systems non-interoperable and lead to duplication of work.

We present a type-theoretic calculus and implementation of a variant of record types that for a wide class of formal languages naturally corresponds to theories. Moreover, we can now elegantly obtain a contravariant functor that reflects the theory level into the object level: for each theory we obtain the type of its models and for every theory morphism a function between the corresponding types. In particular this allows shallow – and thus structure-preserving – encodings of mathematical knowledge and program specifications while allowing the use of object-level features on models, e.g. equality and quantification.

ABSTRACT. Array access out of bounds is a typical programming error. From the '70s, static analysis has been used to identify where such error actually occurs at runtime, through abstract interpretation into linear constraints. However, feasibility and scalability to modern object-oriented code has not been established yet. This article builds on previous work on linear constraints and shows that the result does not scale, when polyhedra implement the linear constraints, while the more abstract zones scale to the analysis of medium-size applications. Moreover, this article formalises the inclusion of symbolic expressions in the constraints and shows that this improves its precision. Expressions are automatically selected on-demand. The resulting analysis applies to code with dynamic memory allocation and arrays held in expressions. It is sound, also in the presence of arbitrary side-effects. It is fully defined in the abstract interpretation framework and does not use any code instrumentation. Its proof of correctness, its implementation inside the commercial Julia analyzer and experiments on third-party code complete the work.

ABSTRACT. Automated theorem provers are routinely used in program analysis and verification for checking program properties. These properties are translated from program fragments to formulas expressed in the logic supported by the theorem prover. Such translations can be complex and require deep knowledge of how theorem provers work in order for the prover to succeed on the translated formulas. Our previous work introduced FOOL, a modification of first-order logic that extends it with syntactical constructs resembling features of programming languages. One can express program properties directly in FOOL and leave translations to plain first-order logic to the theorem prover. In this paper we present a FOOL encoding of the next state relations of imperative programs. Based on this encoding we implement a translation of imperative programs annotated with their pre- and post-conditions to partial correctness properties of these programs. We present experimental results which demonstrate that program properties translated using our method can be efficiently checked by the first-order theorem prover Vampire.

ABSTRACT. We introduce an automatic method for producing stateful ML programs together with proofs of correctness from monadic functions in HOL. Our mechanism supports references, exceptions, and I/O operations, and can generate functions manipulating local state, which can then be encapsulated for use in a pure context. We apply this approach to several non-trivial examples, including the type inferencer and register allocator of the otherwise pure CakeML compiler, which now benefits from better runtime performance. This development has been carried out in the HOL4 theorem prover.

ABSTRACT. Earlier work showed that automatic verification of GMP's algorithms using Why3 exceeds the current capabilities of automatic solvers. To complete this verification, numerous cut indications had to be supplied by the user, slowing the project to a crawl. This paper shows how we have extended Why3 with a framework for proofs by reflection, with minimal impact on the trusted computing base. This framework makes it easy to write dedicated decision procedures that make full use of Why3's imperative features and are formally verified. We evaluate how much work could have been saved when verifying GMP's algorithms, had this framework been available. This approach opens the way to efficiently tackling the further verification of GMP's algorithms.