12,424 research outputs found
Deduction modulo theory
This paper is a survey on Deduction modulo theor
The Complexity of Reasoning with FODD and GFODD
Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a
knowledge representation that is useful in mechanizing decision theoretic
planning in relational domains. GFODDs generalize function-free first order
logic and include numerical values and numerical generalizations of existential
and universal quantification. Previous work presented heuristic inference
algorithms for GFODDs and implemented these heuristics in systems for decision
theoretic planning. In this paper, we study the complexity of the computational
problems addressed by such implementations. In particular, we study the
evaluation problem, the satisfiability problem, and the equivalence problem for
GFODDs under the assumption that the size of the intended model is given with
the problem, a restriction that guarantees decidability. Our results provide a
complete characterization placing these problems within the polynomial
hierarchy. The same characterization applies to the corresponding restriction
of problems in first order logic, giving an interesting new avenue for
efficient inference when the number of objects is bounded. Our results show
that for formulas, and for corresponding GFODDs, evaluation and
satisfiability are complete, and equivalence is
complete. For formulas evaluation is complete, satisfiability
is one level higher and is complete, and equivalence is
complete.Comment: A short version of this paper appears in AAAI 2014. Version 2
includes a reorganization and some expanded proof
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
Cut-Simulation and Impredicativity
We investigate cut-elimination and cut-simulation in impredicative
(higher-order) logics. We illustrate that adding simple axioms such as Leibniz
equations to a calculus for an impredicative logic -- in our case a sequent
calculus for classical type theory -- is like adding cut. The phenomenon
equally applies to prominent axioms like Boolean- and functional
extensionality, induction, choice, and description. This calls for the
development of calculi where these principles are built-in instead of being
treated axiomatically.Comment: 21 page
On Irrelevance and Algorithmic Equality in Predicative Type Theory
Dependently typed programs contain an excessive amount of static terms which
are necessary to please the type checker but irrelevant for computation. To
separate static and dynamic code, several static analyses and type systems have
been put forward. We consider Pfenning's type theory with irrelevant
quantification which is compatible with a type-based notion of equality that
respects eta-laws. We extend Pfenning's theory to universes and large
eliminations and develop its meta-theory. Subject reduction, normalization and
consistency are obtained by a Kripke model over the typed equality judgement.
Finally, a type-directed equality algorithm is described whose completeness is
proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author,
titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement
Distributed First Order Logic
Distributed First Order Logic (DFOL) has been introduced more than ten years
ago with the purpose of formalising distributed knowledge-based systems, where
knowledge about heterogeneous domains is scattered into a set of interconnected
modules. DFOL formalises the knowledge contained in each module by means of
first-order theories, and the interconnections between modules by means of
special inference rules called bridge rules. Despite their restricted form in
the original DFOL formulation, bridge rules have influenced several works in
the areas of heterogeneous knowledge integration, modular knowledge
representation, and schema/ontology matching. This, in turn, has fostered
extensions and modifications of the original DFOL that have never been
systematically described and published. This paper tackles the lack of a
comprehensive description of DFOL by providing a systematic account of a
completely revised and extended version of the logic, together with a sound and
complete axiomatisation of a general form of bridge rules based on Natural
Deduction. The resulting DFOL framework is then proposed as a clear formal tool
for the representation of and reasoning about distributed knowledge and bridge
rules
Virtual Evidence: A Constructive Semantics for Classical Logics
This article presents a computational semantics for classical logic using
constructive type theory. Such semantics seems impossible because classical
logic allows the Law of Excluded Middle (LEM), not accepted in constructive
logic since it does not have computational meaning. However, the apparently
oracular powers expressed in the LEM, that for any proposition P either it or
its negation, not P, is true can also be explained in terms of constructive
evidence that does not refer to "oracles for truth." Types with virtual
evidence and the constructive impossibility of negative evidence provide
sufficient semantic grounds for classical truth and have a simple computational
meaning. This idea is formalized using refinement types, a concept of
constructive type theory used since 1984 and explained here. A new axiom
creating virtual evidence fully retains the constructive meaning of the logical
operators in classical contexts.
Key Words: classical logic, constructive logic, intuitionistic logic,
propositions-as-types, constructive type theory, refinement types, double
negation translation, computational content, virtual evidenc
Conjunctive Query Answering for the Description Logic SHIQ
Conjunctive queries play an important role as an expressive query language
for Description Logics (DLs). Although modern DLs usually provide for
transitive roles, conjunctive query answering over DL knowledge bases is only
poorly understood if transitive roles are admitted in the query. In this paper,
we consider unions of conjunctive queries over knowledge bases formulated in
the prominent DL SHIQ and allow transitive roles in both the query and the
knowledge base. We show decidability of query answering in this setting and
establish two tight complexity bounds: regarding combined complexity, we prove
that there is a deterministic algorithm for query answering that needs time
single exponential in the size of the KB and double exponential in the size of
the query, which is optimal. Regarding data complexity, we prove containment in
co-NP
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