761 research outputs found
Towards Automated Reasoning in Herbrand Structures
Herbrand structures have the advantage, computationally speaking, of being guided by the definability of all elements in them. A salient feature of the logics induced by them is that they internally
exhibit the induction scheme, thus providing a congenial, computationally-oriented framework for
formal inductive reasoning. Nonetheless, their enhanced expressivity renders any effective proof
system for them incomplete. Furthermore, the fact that they are not compact poses yet another prooftheoretic challenge. This paper offers several layers for coping with the inherent incompleteness and
non-compactness of these logics. First, two types of infinitary proof system are introduced—one
of infinite width and one of infinite height—which manipulate infinite sequents and are sound and
complete for the intended semantics. The restriction of these systems to finite sequents induces a
completeness result for finite entailments. Then, in search of effectiveness, two finite approximations
of these systems are presented and explored. Interestingly, the approximation of the infinite-width
system via an explicit induction scheme turns out to be weaker than the effective cyclic fragment of the
infinite-height system
Introducing Quantified Cuts in Logic with Equality
Cut-introduction is a technique for structuring and compressing formal
proofs. In this paper we generalize our cut-introduction method for the
introduction of quantified lemmas of the form (for
quantifier-free ) to a method generating lemmas of the form . Moreover, we extend the original method to predicate
logic with equality. The new method was implemented and applied to the TSTP
proof database. It is shown that the extension of the method to handle equality
and quantifier-blocks leads to a substantial improvement of the old algorithm
On the Expressivity and Applicability of Model Representation Formalisms
A number of first-order calculi employ an explicit model representation
formalism for automated reasoning and for detecting satisfiability. Many of
these formalisms can represent infinite Herbrand models. The first-order
fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism
used in the approximation refinement calculus. Our first result is a finite
model property for MSLH clause sets. Therefore, MSLH clause sets cannot
represent models of clause sets with inherently infinite models. Through a
translation to tree automata, we further show that this limitation also applies
to the linear fragments of implicit generalizations, which is the formalism
used in the model-evolution calculus, to atoms with disequality constraints,
the formalisms used in the non-redundant clause learning calculus (NRCL), and
to atoms with membership constraints, a formalism used for example in decision
procedures for algebraic data types. Although these formalisms cannot represent
models of clause sets with inherently infinite models, through an additional
approximation step they can. This is our second main result. For clause sets
including the definition of an equivalence relation with the help of an
additional, novel approximation, called reflexive relation splitting, the
approximation refinement calculus can automatically show satisfiability through
the MSLH clause set formalism.Comment: 15 page
On the Expressivity and Applicability of Model Representation Formalisms
A number of first-order calculi employ an explicit model representation formalism for automated reasoning and for detecting satisfiability. Many of these formalisms can represent infinite Herbrand models. The first-order fragment of monadic, shallow, linear, Horn (MSLH) clauses, is such a formalism used in the approximation refinement calculus. Our first result is a finite model property for MSLH clause sets. Therefore, MSLH clause sets cannot represent models of clause sets with inherently infinite models. Through a translation to tree automata, we further show that this limitation also applies to the linear fragments of implicit generalizations, which is the formalism used in the model-evolution calculus, to atoms with disequality constraints, the formalisms used in the non-redundant clause learning calculus (NRCL), and to atoms with membership constraints, a formalism used for example in decision procedures for algebraic data types. Although these formalisms cannot represent models of clause sets with inherently infinite models, through an additional approximation step they can. This is our second main result. For clause sets including the definition of an equivalence relation with the help of an additional, novel approximation, called reflexive relation splitting, the approximation refinement calculus can automatically show satisfiability through the MSLH clause set formalism
On the incorporation of interval-valued fuzzy sets into the Bousi-Prolog system: declarative semantics, implementation and applications
In this paper we analyse the benefits of incorporating interval-valued fuzzy
sets into the Bousi-Prolog system. A syntax, declarative semantics and im-
plementation for this extension is presented and formalised. We show, by using
potential applications, that fuzzy logic programming frameworks enhanced with
them can correctly work together with lexical resources and ontologies in order
to improve their capabilities for knowledge representation and reasoning
Deciding First-Order Satisfiability when Universal and Existential Variables are Separated
We introduce a new decidable fragment of first-order logic with equality,
which strictly generalizes two already well-known ones -- the
Bernays-Sch\"onfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The
defining principle is the syntactic separation of universally quantified
variables from existentially quantified ones at the level of atoms. Thus, our
classification neither rests on restrictions on quantifier prefixes (as in the
BSR case) nor on restrictions on the arity of predicate symbols (as in the
monadic case). We demonstrate that the new fragment exhibits the finite model
property and derive a non-elementary upper bound on the computing time required
for deciding satisfiability in the new fragment. For the subfragment of prenex
sentences with the quantifier prefix the
satisfiability problem is shown to be complete for NEXPTIME. Finally, we
discuss how automated reasoning procedures can take advantage of our results.Comment: Extended version of our LICS 2016 conference paper, 23 page
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