582 research outputs found
A herbrandized functional interpretation of classical first-order logic
We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio
Formalizing Termination Proofs under Polynomial Quasi-interpretations
Usual termination proofs for a functional program require to check all the
possible reduction paths. Due to an exponential gap between the height and size
of such the reduction tree, no naive formalization of termination proofs yields
a connection to the polynomial complexity of the given program. We solve this
problem employing the notion of minimal function graph, a set of pairs of a
term and its normal form, which is defined as the least fixed point of a
monotone operator. We show that termination proofs for programs reducing under
lexicographic path orders (LPOs for short) and polynomially quasi-interpretable
can be optimally performed in a weak fragment of Peano arithmetic. This yields
an alternative proof of the fact that every function computed by an
LPO-terminating, polynomially quasi-interpretable program is computable in
polynomial space. The formalization is indeed optimal since every
polynomial-space computable function can be computed by such a program. The
crucial observation is that inductive definitions of minimal function graphs
under LPO-terminating programs can be approximated with transfinite induction
along LPOs.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Incompleteness of a first-order Gödel logic and some temporal logics of programs
It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows
Cut Elimination for a Logic with Induction and Co-induction
Proof search has been used to specify a wide range of computation systems. In
order to build a framework for reasoning about such specifications, we make use
of a sequent calculus involving induction and co-induction. These proof
principles are based on a proof theoretic (rather than set-theoretic) notion of
definition. Definitions are akin to logic programs, where the left and right
rules for defined atoms allow one to view theories as "closed" or defining
fixed points. The use of definitions and free equality makes it possible to
reason intentionally about syntax. We add in a consistent way rules for pre and
post fixed points, thus allowing the user to reason inductively and
co-inductively about properties of computational system making full use of
higher-order abstract syntax. Consistency is guaranteed via cut-elimination,
where we give the first, to our knowledge, cut-elimination procedure in the
presence of general inductive and co-inductive definitions.Comment: 42 pages, submitted to the Journal of Applied Logi
Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types
Type systems certify program properties in a compositional way. From a bigger
program one can abstract out a part and certify the properties of the resulting
abstract program by just using the type of the part that was abstracted away.
Termination and productivity are non-trivial yet desired program properties,
and several type systems have been put forward that guarantee termination,
compositionally. These type systems are intimately connected to the definition
of least and greatest fixed-points by ordinal iteration. While most type
systems use conventional iteration, we consider inflationary iteration in this
article. We demonstrate how this leads to a more principled type system, with
recursion based on well-founded induction. The type system has a prototypical
implementation, MiniAgda, and we show in particular how it certifies
productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317
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