2,569 research outputs found
The satisfiability problem for Boolean set theory with a choice correspondence (Extended version)
Given a set of alternatives, a choice (correspondence) on is a
contractive map defined on a family of nonempty subsets of .
Semantically, a choice associates to each menu a nonempty
subset comprising all elements of that are deemed
selectable by an agent. A choice on is total if its domain is the powerset
of minus the empty set, and partial otherwise. According to the theory of
revealed preferences, a choice is rationalizable if it can be retrieved from a
binary relation on by taking all maximal elements of each menu. It is
well-known that rationalizable choices are characterized by the satisfaction of
suitable axioms of consistency, which codify logical rules of selection within
menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes
choices rationalizable by a transitive relation. Here we study the
satisfiability problem for unquantified formulae of an elementary fragment of
set theory involving a choice function symbol , the Boolean set
operators and the singleton, the equality and inclusion predicates, and the
propositional connectives. In particular, we consider the cases in which the
interpretation of satisfies any combination of two specific axioms
of consistency, whose conjunction is equivalent to WARP. In two cases we prove
that the related satisfiability problem is NP-complete, whereas in the
remaining cases we obtain NP-completeness under the additional assumption that
the number of choice terms is constant
The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence
Given a set U of alternatives, a choice (correspondence) on U is a
contractive map c defined on a family Omega of nonempty subsets of U.
Semantically, a choice c associates to each menu A in Omega a nonempty subset
c(A) of A comprising all elements of A that are deemed selectable by an agent.
A choice on U is total if its domain is the powerset of U minus the empty set,
and partial otherwise. According to the theory of revealed preferences, a
choice is rationalizable if it can be retrieved from a binary relation on U by
taking all maximal elements of each menu. It is well-known that rationalizable
choices are characterized by the satisfaction of suitable axioms of
consistency, which codify logical rules of selection within menus. For
instance, WARP (Weak Axiom of Revealed Preference) characterizes choices
rationalizable by a transitive relation. Here we study the satisfiability
problem for unquantified formulae of an elementary fragment of set theory
involving a choice function symbol c, the Boolean set operators and the
singleton, the equality and inclusion predicates, and the propositional
connectives. In particular, we consider the cases in which the interpretation
of c satisfies any combination of two specific axioms of consistency, whose
conjunction is equivalent to WARP. In two cases we prove that the related
satisfiability problem is NP-complete, whereas in the remaining cases we obtain
NP-completeness under the additional assumption that the number of choice terms
is constant.Comment: In Proceedings GandALF 2017, arXiv:1709.01761. "extended" version at
arXiv:1708.0612
An exactly solvable random satisfiability problem
We introduce a new model for the generation of random satisfiability
problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt
and Zecchina, which is a variant of the famous K-SAT model: it is extended to
q-state variables and relates to a different choice of the statistical
ensemble. The model has an exactly solvable statistic: the critical exponents
and scaling functions of the SAT/UNSAT transition are calculable at zero
temperature, with no need of replicas, also with exact finite-size corrections.
We also introduce an exact duality of the model, and show an analogy of
thermodynamic properties with the Random Energy Model of disordered spin
systems theory. Relations with Error-Correcting Codes are also discussed.Comment: 31 pages, 1 figur
Languages of Dot-depth One over Infinite Words
Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
Pruning Processes and a New Characterization of Convex Geometries
We provide a new characterization of convex geometries via a multivariate
version of an identity that was originally proved by Maneva, Mossel and
Wainwright for certain combinatorial objects arising in the context of the
k-SAT problem. We thus highlight the connection between various
characterizations of convex geometries and a family of removal processes
studied in the literature on random structures.Comment: 14 pages, 3 figures; the exposition has changed significantly from
previous versio
Applying Formal Methods to Networking: Theory, Techniques and Applications
Despite its great importance, modern network infrastructure is remarkable for
the lack of rigor in its engineering. The Internet which began as a research
experiment was never designed to handle the users and applications it hosts
today. The lack of formalization of the Internet architecture meant limited
abstractions and modularity, especially for the control and management planes,
thus requiring for every new need a new protocol built from scratch. This led
to an unwieldy ossified Internet architecture resistant to any attempts at
formal verification, and an Internet culture where expediency and pragmatism
are favored over formal correctness. Fortunately, recent work in the space of
clean slate Internet design---especially, the software defined networking (SDN)
paradigm---offers the Internet community another chance to develop the right
kind of architecture and abstractions. This has also led to a great resurgence
in interest of applying formal methods to specification, verification, and
synthesis of networking protocols and applications. In this paper, we present a
self-contained tutorial of the formidable amount of work that has been done in
formal methods, and present a survey of its applications to networking.Comment: 30 pages, submitted to IEEE Communications Surveys and Tutorial
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
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