Complete proof systems for algebraic simply-typed terms (original) (raw)
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Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds. This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones. We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.
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The proof search method is a traditionally established way to prove the completeness theorem for various logics. The purpose of this paper is to show that this method can be adapted to linear logic. First we prove the completeness theorem for a certain fragment of intuitionistic linear logic, called naive linear logic, with respect to naive phase semantics, i.e., phase semantics without any closure condition, using the proof search method in a certain labelled sequent system. Then the completeness of the (rudimentary) classical linear logic can be obtained as a direct corollary by a Kolmogorov-G\"odel style double negation interpretation. To apply the proof search method for the full system of linear logic, we generalize the notion of branch in the standard proof search method to that of OR-iree, and give a proof of the completeness theorem for intuitionistic (classical, resp.) linear logic with respect to intuitionistic (classical, resp.
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[Attached is an accepted version.] In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.
1977
I t is proved that in the general case of arbitrary context-free schemes a program is (partially) correct with respect to given initial and final assertions if and only if a suitable finite system of intermediate assertions can be found. Assertions are allowed from the extended state space V x V. This result contrasts with the results of [2], where it is proved that if assertions are taken from the original state space V. then in the general case an infinite system of intermediate assertions is needed. The extension of the state space allows a unification in the relational framework of [2], of the (essence of the) results of [2], and of [4], 151 and [6], and provides a semantic counterpart of the use of auxiliary variables.
Solutions of implication constraints yield type inference for more general algebraic data types
2005
We study an extension of Hindley/Milner with a user-programmable constraint domain and a general form of algebraic data types which unifies common forms such as existential types, the combination of type classes and existential types and the more recent extension of guarded recursive data types. We can furthermore express some novel variants such as a limited form of dependent types. Type inference is reduced to solving of implication constraints. We identify sufficient conditions in terms of solutions of implication constraints under which inference is sound and complete. One of our main technical achievements is a generic implication solver which is parameterized in the solver of the underlying constraint domain. Our results show that our system is practical and greatly extends the expressive power of languages such as Haskell.