30,688 research outputs found
Term Reduction Using Directed Congruence Closure
Many problems in computer science can be described in terms of reduction rules that tell how to transform terms. Problems that can be handled in this way include interpreting programs, implementing abstract data types, and proving certain kinds of theorems. A terms is said to have a normal form if it can be transformed, using the reduction rules, into a term to which no further reduction rules apply. In this paper, we extend the Congruence Closure Algorithm, an algorithm for finding the consequences of a finite set of equations, to develop Directed Congruence Closure, a technique for finding the normal form of a term provided the reduction rules satisfy the conditions for a regular term rewriting system. This technique is particularly efficient because it inherits, from the Congruence Closure Algorithm, the ability to remember all objects that have already been proved equivalent
Extending SMTCoq, a Certified Checker for SMT (Extended Abstract)
This extended abstract reports on current progress of SMTCoq, a communication
tool between the Coq proof assistant and external SAT and SMT solvers. Based on
a checker for generic first-order certificates implemented and proved correct
in Coq, SMTCoq offers facilities both to check external SAT and SMT answers and
to improve Coq's automation using such solvers, in a safe way. Currently
supporting the SAT solver zChaff, and the SMT solver veriT for the combination
of the theories of congruence closure and linear integer arithmetic, SMTCoq is
meant to be extendable with a reasonable amount of effort: we present work in
progress to support the SMT solver CVC4 and the theory of bit vectors.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
Automatic Abstraction for Congruences
One approach to verifying bit-twiddling algorithms is to derive invariants between the bits that constitute the variables of a program. Such invariants can often be described with systems of congruences where in each equation , (unknown variable m)\vec{c}\vec{x}$ is a vector of propositional variables (bits). Because of the low-level nature of these invariants and the large number of bits that are involved, it is important that the transfer functions can be derived automatically. We address this problem, showing how an analysis for bit-level congruence relationships can be decoupled into two parts: (1) a SAT-based abstraction (compilation) step which can be automated, and (2) an interpretation step that requires no SAT-solving. We exploit triangular matrix forms to derive transfer functions efficiently, even in the presence of large numbers of bits. Finally we propose program transformations that improve the analysis results
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations
A partial algebra construction of Gr\"atzer and Schmidt from
"Characterizations of congruence lattices of abstract algebras" (Acta Sci.
Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to
a well-known fact that every finite distributive lattice is representable, seen
as a special case of the Finite Lattice Representation Problem.
The construction of this proof brings together Birkhoff's representation
theorem for finite distributive lattices, an emphasis on boolean lattices when
representing finite lattices, and a perspective based on inequalities of
partially ordered sets. It may be possible to generalize the techniques used in
this approach.
Other than the aforementioned representation theorem only elementary tools
are used for the two theorems of this note. In particular there is no reliance
on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and
Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice
to be representable (see William J. DeMeo), such as the closure method
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