3,211 research outputs found
Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints
The monadic shallow linear Horn fragment is well-known to be decidable and
has many application, e.g., in security protocol analysis, tree automata, or
abstraction refinement. It was a long standing open problem how to extend the
fragment to the non-Horn case, preserving decidability, that would, e.g.,
enable to express non-determinism in protocols. We prove decidability of the
non-Horn monadic shallow linear fragment via ordered resolution further
extended with dismatching constraints and discuss some applications of the new
decidable fragment.Comment: 29 pages, long version of CADE-26 pape
Deduction with XOR Constraints in Security API Modelling
We introduce XOR constraints, and show how they enable a theorem prover to reason effectively about security critical subsystems which employ bitwise XOR. Our primary case study is the API of the IBM 4758 hardware security module. We also show how our technique can be applied to standard security protocols
Encapsulation for Practical Simplification Procedures
ACL2 was used to prove properties of two simplification procedures. The
procedures differ in complexity but solve the same programming problem that
arises in the context of a resolution/paramodulation theorem proving system.
Term rewriting is at the core of the two procedures, but details of the
rewriting procedure itself are irrelevant. The ACL2 encapsulate construct was
used to assert the existence of the rewriting function and to state some of its
properties. Termination, irreducibility, and soundness properties were
established for each procedure. The availability of the encapsulation mechanism
in ACL2 is considered essential to rapid and efficient verification of this
kind of algorithm.Comment: 6 page
HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories
We present a simple resolution proof system for higher-order constrained Horn
clauses (HoCHC) - a system of higher-order logic modulo theories - and prove
its soundness and refutational completeness w.r.t. the standard semantics. As
corollaries, we obtain the compactness theorem and semi-decidability of HoCHC
for semi-decidable background theories, and we prove that HoCHC satisfies a
canonical model property. Moreover a variant of the well-known translation from
higher-order to 1st-order logic is shown to be sound and complete for HoCHC in
standard semantics. We illustrate how to transfer decidability results for
(fragments of) 1st-order logic modulo theories to our higher-order setting,
using as example the Bernays-Schonfinkel-Ramsey fragment of HoCHC modulo a
restricted form of Linear Integer Arithmetic
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Automated Synthesis of a Finite Complexity Ordering for Saturation
We present in this paper a new procedure to saturate a set of clauses with
respect to a well-founded ordering on ground atoms such that A < B implies
Var(A) {\subseteq} Var(B) for every atoms A and B. This condition is satisfied
by any atom ordering compatible with a lexicographic, recursive, or multiset
path ordering on terms. Our saturation procedure is based on a priori ordered
resolution and its main novelty is the on-the-fly construction of a finite
complexity atom ordering. In contrast with the usual redundancy, we give a new
redundancy notion and we prove that during the saturation a non-redundant
inference by a priori ordered resolution is also an inference by a posteriori
ordered resolution. We also prove that if a set S of clauses is saturated with
respect to an atom ordering as described above then the problem of whether a
clause C is entailed from S is decidable
A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds
We study the problem of obtaining lower bounds for polynomial calculus (PC)
and polynomial calculus resolution (PCR) on proof degree, and hence by
[Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03]
established that if the clause-variable incidence graph of a CNF formula F is a
good enough expander, then proving that F is unsatisfiable requires high PC/PCR
degree. We further develop the techniques in [AR03] to show that if one can
"cluster" clauses and variables in a way that "respects the structure" of the
formula in a certain sense, then it is sufficient that the incidence graph of
this clustered version is an expander. As a corollary of this, we prove that
the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree
when restricted to constant-degree expander graphs. This answers an open
question in [Razborov '02], and also implies that the standard CNF encoding of
the FPHP formulas require exponential proof size in polynomial calculus
resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as
shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when
restricted to bounded-degree expanders.Comment: Full-length version of paper to appear in Proceedings of the 30th
Annual Computational Complexity Conference (CCC '15), June 201
- …