974 research outputs found
Quantifier-free logic for nondeterministic theories
AbstractWe develop a quantifier-free logic for deriving consequences of multialgebraic theories. Multialgebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set-valued operations. Formulae are sequents over atoms allowing one to state set-inclusion or identity of 1-element sets (determinacy). We introduce a sound and weakly complete Rasiowa–Sikorski (R–S) logic for proving multialgebraic tautologies. We then extend this system for proving consequences of specifications based on translation of finite theories into logical formulae. Finally, we show how such a translation may be avoided—introduction of the specific cut rules leads to a sound and strongly complete Gentzen system for proving directly consequences of specifications. Besides giving examples of the general techniques of R–S and the specific cut rules, we improve the earlier logics for multialgebras by providing means to handle empty carriers (as well as empty result-sets) without the use of quantifiers, and to derive consequences of theories without translation into another format and without using general cut
Relativized Propositional Calculus
Proof systems for the Relativized Propositional Calculus are defined and
compared.Comment: 8 page
Bounds on the Automata Size for Presburger Arithmetic
Automata provide a decision procedure for Presburger arithmetic. However,
until now only crude lower and upper bounds were known on the sizes of the
automata produced by this approach. In this paper, we prove an upper bound on
the the number of states of the minimal deterministic automaton for a
Presburger arithmetic formula. This bound depends on the length of the formula
and the quantifiers occurring in the formula. The upper bound is established by
comparing the automata for Presburger arithmetic formulas with the formulas
produced by a quantifier elimination method. We also show that our bound is
tight, even for nondeterministic automata. Moreover, we provide optimal
automata constructions for linear equations and inequations
Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality
principles that are equivalent to the axiom of choice in set theory. Among
these are the principle asserting that every family of sets has a
-maximal subfamily with the finite intersection property and the
principle asserting that if is a property of finite character then every
set has a -maximal subset of which holds. We show that these
principles and their variations have a wide range of strengths in the context
of second-order arithmetic, from being equivalent to to being
weaker than and incomparable with . In
particular, we identify a choice principle that, modulo induction,
lies strictly below the atomic model theorem principle and
implies the omitting partial types principle
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