Estimation of the length of interactions in arena game semantics

We estimate the maximal length of interactions between strategies in HO/N game semantics, in the spirit of the work by Schwichtenberg and Beckmann for the length of reduction in simply typed lambdacalculus. Because of the operational content of game semantics, the bounds presented here also apply to head linear reduction on lambda-terms and to the execution of programs by abstract machines (PAM/KAM), including in presence of computational effects such as non-determinism or ground type references. The proof proceeds by extracting from the games model a combinatorial rewriting rule on trees of natural numbers, which can then be analyzed independently of game semantics or lambda-calculus.Comment: Foundations of Software Science and Computational Structures 14th International Conference, FOSSACS 2011, Saarbr\"ucken : Germany (2011

Some Turing-Complete Extensions of First-Order Logic

We introduce a natural Turing-complete extension of first-order logic FO. The extension adds two novel features to FO. The first one of these is the capacity to add new points to models and new tuples to relations. The second one is the possibility of recursive looping when a formula is evaluated using a semantic game. We first define a game-theoretic semantics for the logic and then prove that the expressive power of the logic corresponds in a canonical way to the recognition capacity of Turing machines. Finally, we show how to incorporate generalized quantifiers into the logic and argue for a highly natural connection between oracles and generalized quantifiers.Comment: In Proceedings GandALF 2014, arXiv:1408.556

The Geometry of Concurrent Interaction: Handling Multiple Ports by Way of Multiple Tokens (Long Version)

We introduce a geometry of interaction model for Mazza's multiport interaction combinators, a graph-theoretic formalism which is able to faithfully capture concurrent computation as embodied by process algebras like the $\pi$-calculus. The introduced model is based on token machines in which not one but multiple tokens are allowed to traverse the underlying net at the same time. We prove soundness and adequacy of the introduced model. The former is proved as a simulation result between the token machines one obtains along any reduction sequence. The latter is obtained by a fine analysis of convergence, both in nets and in token machines

Team Semantics and Recursive Enumerability

It is well known that dependence logic captures the complexity class NP, and it has recently been shown that inclusion logic captures P on ordered models. These results demonstrate that team semantics offers interesting new possibilities for descriptive complexity theory. In order to properly understand the connection between team semantics and descriptive complexity, we introduce an extension D* of dependence logic that can define exactly all recursively enumerable classes of finite models. Thus D* provides an approach to computation alternative to Turing machines. The essential novel feature in D* is an operator that can extend the domain of the considered model by a finite number of fresh elements. Due to the close relationship between generalized quantifiers and oracles, we also investigate generalized quantifiers in team semantics. We show that monotone quantifiers of type (1) can be canonically eliminated from quantifier extensions of first-order logic by introducing corresponding generalized dependence atoms

Wave-Style Token Machines and Quantum Lambda Calculi

Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum lambda-calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

A Praxical Solution of the Symbol Grounding Problem

Peer reviewe

The Geometry of Synchronization (Long Version)

We graft synchronization onto Girard's Geometry of Interaction in its most concrete form, namely token machines. This is realized by introducing proof-nets for SMLL, an extension of multiplicative linear logic with a specific construct modeling synchronization points, and of a multi-token abstract machine model for it. Interestingly, the correctness criterion ensures the absence of deadlocks along reduction and in the underlying machine, this way linking logical and operational properties.Comment: 26 page

A logical basis for constructive systems

The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article elaborates some basic complexity theory for the CoL framework. Then it proves soundness and completeness for the deductive system CL12 with respect to the semantics of CoL, including the version of the latter based on polynomial time computability instead of computability-in-principle. CL12 is a sequent calculus system, where the meaning of a sequent intuitively can be characterized as "the succedent is algorithmically reducible to the antecedent", and where formulas are built from predicate letters, function letters, variables, constants, identity, negation, parallel and choice connectives, and blind and choice quantifiers. A case is made that CL12 is an adequate logical basis for constructive applied theories, including complexity-oriented ones