58 research outputs found
On synthesizing Skolem functions for first order logic formulae
Skolem functions play a central role in logic, from eliminating quantifiers
in first order logic formulas to providing functional implementations of
relational specifications. While classical results in logic are only interested
in their existence, the question of how to effectively compute them is also
interesting, important and useful for several applications. In the restricted
case of Boolean propositional logic formula, this problem of synthesizing
Boolean Skolem functions has been addressed in depth, with various recent work
focussing on both theoretical and practical aspects of the problem. However,
there are few existing results for the general case, and the focus has been on
heuristical algorithms.
In this article, we undertake an investigation into the computational
hardness of the problem of synthesizing Skolem functions for first order logic
formula. We show that even under reasonable assumptions on the signature of the
formula, it is impossible to compute or synthesize Skolem functions. Then we
determine conditions on theories of first order logic which would render the
problem computable. Finally, we show that several natural theories satisfy
these conditions and hence do admit effective synthesis of Skolem functions
Towards Verifying Nonlinear Integer Arithmetic
We eliminate a key roadblock to efficient verification of nonlinear integer
arithmetic using CDCL SAT solvers, by showing how to construct short resolution
proofs for many properties of the most widely used multiplier circuits. Such
short proofs were conjectured not to exist. More precisely, we give n^{O(1)}
size regular resolution proofs for arbitrary degree 2 identities on array,
diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs
for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result
First Experiments with a Flexible Infrastructure for Normative Reasoning
A flexible infrastructure for normative reasoning is outlined. A small-scale
demonstrator version of the envisioned system has been implemented in the proof
assistant Isabelle/HOL by utilising the first authors universal logical
reasoning approach based on shallow semantical embeddings in meta-logic HOL.
The need for such a flexible reasoning infrastructure is motivated and
illustrated with a contrary-to-duty example scenario selected from the General
Data Protection Regulation.Comment: 9 pages, 4 figure
Temporal Stream Logic: Synthesis beyond the Bools
Reactive systems that operate in environments with complex data, such as
mobile apps or embedded controllers with many sensors, are difficult to
synthesize. Synthesis tools usually fail for such systems because the state
space resulting from the discretization of the data is too large. We introduce
TSL, a new temporal logic that separates control and data. We provide a
CEGAR-based synthesis approach for the construction of implementations that are
guaranteed to satisfy a TSL specification for all possible instantiations of
the data processing functions. TSL provides an attractive trade-off for
synthesis. On the one hand, synthesis from TSL, unlike synthesis from standard
temporal logics, is undecidable in general. On the other hand, however,
synthesis from TSL is scalable, because it is independent of the complexity of
the handled data. Among other benchmarks, we have successfully synthesized a
music player Android app and a controller for an autonomous vehicle in the Open
Race Car Simulator (TORCS.
Model-Checking Speculation-Dependent Security Properties: Abstracting and Reducing Processor Models for Sound and Complete Verification
Spectre and Meltdown attacks in modern microprocessors represent a new class of attacks that have been difficult to deal with. They underline vulnerabilities in hardware design that have been going unnoticed for years. This shows the weakness of the state-of-the-art verification process and design practices. These attacks are OS-independent, and they do not exploit any software vulnerabilities. Moreover, they violate all security assumptions ensured by standard security procedures, (e.g., address space isolation), and, as a result, every security mechanism built upon these guarantees. These vulnerabilities allow the attacker to retrieve leaked data without accessing the secret directly. Indeed, they make use of covert channels, which are mechanisms of hidden communication that convey sensitive information without any visible information flow between the malicious party and the victim. The root cause of this type of side-channel attacks lies within the speculative and out-of-order execution of modern high-performance microarchitectures. Since modern processors are hard to verify with standard formal verification techniques, we present a methodology that shows how to transform a realistic model of a speculative and out-of-order processor into an abstract one. Following related formal verification approaches, we simplify the model under consideration by abstraction and refinement steps. We also present an approach to formally verify the abstract model using a standard model checker. The theoretical flow, reliant on established formal verification results, is introduced and a sketch of proof is provided for soundness and correctness. Finally, we demonstrate the feasibility of our approach, by applying it on a pipelined DLX RISC-inspired processor architecture. We show preliminary experimental results to support our claim, performing Bounded Model-Checking with a state-of-the-art model checker
Incomplete SMT techniques for solving non-linear formulas over the integers
We present new methods for solving the Satisfiability Modulo Theories problem over the theory of QuantifierFree Non-linear Integer Arithmetic, SMT(QF-NIA), which consists of deciding the satisfiability of ground formulas with integer polynomial constraints. Following previous work, we propose to solve SMT(QF-NIA)
instances by reducing them to linear arithmetic: non-linear monomials are linearized by abstracting them
with fresh variables and by performing case splitting on integer variables with finite domain. For variables
that do not have a finite domain, we can artificially introduce one by imposing a lower and an upper bound
and iteratively enlarge it until a solution is found (or the procedure times out).
The key for the success of the approach is to determine, at each iteration, which domains have to be
enlarged. Previously, unsatisfiable cores were used to identify the domains to be changed, but no clue was
obtained as to how large the new domains should be. Here, we explain two novel ways to guide this process by
analyzing solutions to optimization problems: (i) to minimize the number of violated artificial domain bounds,
solved via a Max-SMT solver, and (ii) to minimize the distance with respect to the artificial domains, solved
via an Optimization Modulo Theories (OMT) solver. Using this SMT-based optimization technology allows
smoothly extending the method to also solve Max-SMT problems over non-linear integer arithmetic. Finally,
we leverage the resulting Max-SMT(QF-NIA) techniques to solve ∃∀ formulas in a fragment of quantified
non-linear arithmetic that appears commonly in verification and synthesis applications.Peer ReviewedPostprint (author's final draft
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