135 research outputs found
Syntactic Proofs for Yablo’s Paradoxes in Temporal Logic
Temporal logic is of importance in theoretical computer science for its application in formal verification, to state requirements of hardware or software systems. Linear temporal logic is an appropriate logical environment to formalize Yablo’s paradox which is seemingly non-self-referential and basically has a sequential structure. We give a brief review of Yablo’s paradox and its various versions. Formalization of these paradoxes yields some theorems in Linear Temporal Logic (LTL) for which we give syntactic proofs using an appropriate axiomatization of LTL
A Step-indexed Semantics of Imperative Objects
Step-indexed semantic interpretations of types were proposed as an
alternative to purely syntactic proofs of type safety using subject reduction.
The types are interpreted as sets of values indexed by the number of
computation steps for which these values are guaranteed to behave like proper
elements of the type. Building on work by Ahmed, Appel and others, we introduce
a step-indexed semantics for the imperative object calculus of Abadi and
Cardelli. Providing a semantic account of this calculus using more
`traditional', domain-theoretic approaches has proved challenging due to the
combination of dynamically allocated objects, higher-order store, and an
expressive type system. Here we show that, using step-indexing, one can
interpret a rich type discipline with object types, subtyping, recursive and
bounded quantified types in the presence of state
Principals in Programming Languages: Technical Results
This is the companion technical report for ``Principals in Programming Languages'' [20]. See that document for a more readable version of these results. In this paper, we describe two variants of the simply typed -calculus extended with a notion of {\em principal}. The results are languages in which intuitive statements like ``the client must call to obtain a file handle'' can be phrased and proven formally. The first language is a two-agent calculus with references and recursive types, while the second language explores the possibility of multiple agents with varying amounts of type information. We use these calculi to give syntactic proofs of some type abstraction results that traditionally require semantic arguments
Cut-elimination for the modal Grzegorczyk logic via non-well-founded proofs
We present a sequent calculus for the modal Grzegorczyk logic Grz allowing
non-well-founded proofs and obtain the cut-elimination theorem for it by
constructing a continuous cut-elimination mapping acting on these proofs.Comment: WOLLIC'17, 12 pages, 1 appendi
Proof Theory and Ordered Groups
Ordering theorems, characterizing when partial orders of a group extend to
total orders, are used to generate hypersequent calculi for varieties of
lattice-ordered groups (l-groups). These calculi are then used to provide new
proofs of theorems arising in the theory of ordered groups. More precisely: an
analytic calculus for abelian l-groups is generated using an ordering theorem
for abelian groups; a calculus is generated for l-groups and new decidability
proofs are obtained for the equational theory of this variety and extending
finite subsets of free groups to right orders; and a calculus for representable
l-groups is generated and a new proof is obtained that free groups are
orderable
Proofs Without Syntax
"[M]athematicians care no more for logic than logicians for mathematics."
Augustus de Morgan, 1868.
Proofs are traditionally syntactic, inductively generated objects. This paper
presents an abstract mathematical formulation of propositional calculus
(propositional logic) in which proofs are combinatorial (graph-theoretic),
rather than syntactic. It defines a *combinatorial proof* of a proposition P as
a graph homomorphism h : C -> G(P), where G(P) is a graph associated with P and
C is a coloured graph. The main theorem is soundness and completeness: P is
true iff there exists a combinatorial proof h : C -> G(P).Comment: Appears in Annals of Mathematics, 2006. 5 pages + references. Version
1 is submitted version; v3 is final published version (in two-column format
rather than Annals style). Changes for v2: dualised definition of
combinatorial truth, thereby shortening some subsequent proofs; added
references; corrected typos; minor reworking of some sentences/paragraphs;
added comments on polynomial-time correctness (referee request). Changes for
v3: corrected two typos, reworded one sentence, repeated a citation in Notes
sectio
Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets
In the context of strategic games, we provide an axiomatic proof of the
statement Common knowledge of rationality implies that the players will choose
only strategies that survive the iterated elimination of strictly dominated
strategies. Rationality here means playing only strategies one believes to be
best responses. This involves looking at two formal languages. One is
first-order, and is used to formalise optimality conditions, like avoiding
strictly dominated strategies, or playing a best response. The other is a modal
fixpoint language with expressions for optimality, rationality and belief.
Fixpoints are used to form expressions for common belief and for iterated
elimination of non-optimal strategies.Comment: 16 pages, Proc. 11th International Workshop on Computational Logic in
Multi-Agent Systems (CLIMA XI). To appea
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