78 research outputs found
Bounded-analytic sequent calculi and embeddings for hypersequent logics
A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory
Theorems of Alternatives for Substructural Logics
A theorem of alternatives provides a reduction of validity in a substructural
logic to validity in its multiplicative fragment. Notable examples include a
theorem of Arnon Avron that reduces the validity of a disjunction of
multiplicative formulas in the R-mingle logic RM to the validity of a linear
combination of these formulas, and Gordan's theorem for solutions of linear
systems over the real numbers, that yields an analogous reduction for validity
in Abelian logic A. In this paper, general conditions are provided for
axiomatic extensions of involutive uninorm logic without additive constants to
admit a theorem of alternatives. It is also shown that a theorem of
alternatives for a logic can be used to establish (uniform) deductive
interpolation and completeness with respect to a class of dense totally ordered
residuated lattices
Models for Substructural Arithmetics
This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R
Admissibility in Finitely Generated Quasivarieties
Checking the admissibility of quasiequations in a finitely generated (i.e.,
generated by a finite set of finite algebras) quasivariety Q amounts to
checking validity in a suitable finite free algebra of the quasivariety, and is
therefore decidable. However, since free algebras may be large even for small
sets of small algebras and very few generators, this naive method for checking
admissibility in \Q is not computationally feasible. In this paper,
algorithms are introduced that generate a minimal (with respect to a multiset
well-ordering on their cardinalities) finite set of algebras such that the
validity of a quasiequation in this set corresponds to admissibility of the
quasiequation in Q. In particular, structural completeness (validity and
admissibility coincide) and almost structural completeness (validity and
admissibility coincide for quasiequations with unifiable premises) can be
checked. The algorithms are illustrated with a selection of well-known finitely
generated quasivarieties, and adapted to handle also admissibility of rules in
finite-valued logics
Poset products as relational models
We introduce a relational semantics based on poset products, and provide
sufficient conditions guaranteeing its soundness and completeness for various
substructural logics. We also demonstrate that our relational semantics unifies
and generalizes two semantics already appearing in the literature: Aguzzoli,
Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and
Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz
logic. As a consequence of our general theory, we recover the soundness and
completeness results of these prior studies in a uniform fashion, and extend
them to infinitely-many other substructural logics
Decidability and Complexity in Weakening and Contraction Hypersequent Substructural Logics
We establish decidability for the infinitely many axiomatic extensions of the commutative Full Lambek logic with weakening FLew (i.e. IMALLW) that have a cut-free hypersequent proof calculus. Specifically: every analytic structural rule exten- sion of HFLew. Decidability for the corresponding extensions of its contraction counterpart FLec was established recently but their computational complexity was left unanswered. In the second part of this paper, we introduce just enough on length functions for well-quasi-orderings and the fast-growing complexity classes to obtain complexity upper bounds for both the weakening and contraction extensions. A specific instance of this result yields the first complexity bound for the prominent fuzzy logic MTL (monoidal t-norm based logic) providing an answer to a long- standing open problem
Interpolation and the Exchange Rule
It was proved by Maksimova in 1977 that exactly eight varieties of Heyting
algebras have the amalgamation property, and hence exactly eight axiomatic
extensions of intuitionistic propositional logic have the deductive
interpolation property. The prevalence of the deductive interpolation property
for axiomatic extensions of substructural logics and the amalgamation property
for varieties of pointed residuated lattices, their equivalent algebraic
semantics, is far less well understood, however. Taking as our starting point a
formulation of intuitionistic propositional logic as the full Lambek calculus
with exchange, weakening, and contraction, we investigate the role of the
exchange rule--algebraically, the commutativity law--in determining the scope
of these properties. First, we show that there are continuum-many varieties of
idempotent semilinear residuated lattices that have the amalgamation property
and contain non-commutative members, and hence continuum-many axiomatic
extensions of the corresponding logic that have the deductive interpolation
property in which exchange is not derivable. We then show that, in contrast,
exactly sixty varieties of commutative idempotent semilinear residuated
lattices have the amalgamation property, and hence exactly sixty axiomatic
extensions of the corresponding logic with exchange have the deductive
interpolation property. From this latter result, it follows also that there are
exactly sixty varieties of commutative idempotent semilinear residuated
lattices whose first-order theories have a model completion
Models for Substructural Arithmetics
This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R
Goal-directed proof theory
This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape
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