45,218 research outputs found
From Many-Valued Consequence to Many-Valued Connectives
Given a consequence relation in many-valued logic, what connectives can be
defined? For instance, does there always exist a conditional operator
internalizing the consequence relation, and which form should it take? In this
paper, we pose this question in a multi-premise multi-conclusion setting for
the class of so-called intersective mixed consequence relations, which extends
the class of Tarskian relations. Using computer-aided methods, we answer
extensively for 3-valued and 4-valued logics, focusing not only on conditional
operators, but on what we call Gentzen-regular connectives (including negation,
conjunction, and disjunction). For arbitrary N-valued logics, we state
necessary and sufficient conditions for the existence of such connectives in a
multi-premise multi-conclusion setting. The results show that mixed consequence
relations admit all classical connectives, and among them pure consequence
relations are those that admit no other Gentzen-regular connectives.
Conditionals can also be found for a broader class of intersective mixed
consequence relations, but with the exclusion of order-theoretic consequence
relations.Comment: Updated version [corrections of an incorrect claim in first version;
two bib entries added
Calibrating Generative Models: The Probabilistic Chomsky-SchĂŒtzenberger Hierarchy
A probabilistic ChomskyâSchĂŒtzenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
The complexity of finite-valued CSPs
We study the computational complexity of exact minimisation of
rational-valued discrete functions. Let be a set of rational-valued
functions on a fixed finite domain; such a set is called a finite-valued
constraint language. The valued constraint satisfaction problem,
, is the problem of minimising a function given as
a sum of functions from . We establish a dichotomy theorem with respect
to exact solvability for all finite-valued constraint languages defined on
domains of arbitrary finite size.
We show that every constraint language either admits a binary
symmetric fractional polymorphism in which case the basic linear programming
relaxation solves any instance of exactly, or
satisfies a simple hardness condition that allows for a
polynomial-time reduction from Max-Cut to
On Role Logic
We present role logic, a notation for describing properties of relational
structures in shape analysis, databases, and knowledge bases. We construct role
logic using the ideas of de Bruijn's notation for lambda calculus, an encoding
of first-order logic in lambda calculus, and a simple rule for implicit
arguments of unary and binary predicates. The unrestricted version of role
logic has the expressive power of first-order logic with transitive closure.
Using a syntactic restriction on role logic formulas, we identify a natural
fragment RL^2 of role logic. We show that the RL^2 fragment has the same
expressive power as two-variable logic with counting C^2 and is therefore
decidable. We present a translation of an imperative language into the
decidable fragment RL^2, which allows compositional verification of programs
that manipulate relational structures. In addition, we show how RL^2 encodes
boolean shape analysis constraints and an expressive description logic.Comment: 20 pages. Our later SAS 2004 result builds on this wor
Normality Operators and Classical Collapse
In this paper, we extend the expressive power of the logics K3, LP and FDE with anormality operator, which is able to express whether a for-mula is assigned a classical truth value or not. We then establish classical recapture theorems for the resulting logics. Finally, we compare the approach via normality operator with the classical collapse approach devisedby Jc Beall
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