This work is an attempt to lay foundations for a theory of interactive computation and bring logic and theory of computing closer together. It semantically introduces a logic of computability and sets a program for studying various aspects of that logic. The intuitive notion of computational problems is formalized as a certain new, procedural-rule-free sort of games between the machine and the environment, and computability is understood as existence of an interactive Turing machine that wins the game against any possible (...) environment. The formalism used as a specification language for computational problems, called the universal language, is a non-disjoint union of the formalisms of classical, intuitionistic and linear logics, with logical operators interpreted as certain—most basic and natural—operations on problems. Validity of a formula is understood as being “always computable”, and the set of all valid formulas is called the universal logic. The name “universal” is related to the potential of this logic to integrate, on the basis of one semantics, classical, intuitionistic and linear logics, with their seemingly unrelated or even antagonistic philosophies. In particular, the classical notion of truth turns out to be nothing but computability restricted to the formulas of the classical fragment of the universal language, which makes classical logic a natural syntactic fragment of the universal logic. The same appears to be the case for intuitionistic and linear logics . Unlike classical logic, these two do not have a good concept of truth, and the notion of computability restricted to the corresponding two fragments of the universal language, based on the intuitions that it formalizes, can well qualify as “intuitionistic truth” and “linear-logic truth”. The paper also provides illustrations of potential applications of the universal logic in knowledgebase, resourcebase and planning systems, as well as constructive applied theories. The author has tried to make this article easy to read. It is fully self-contained and can be understood without any specialized knowledge of any particular subfield of logic or computer science. (shrink)

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of reducibility — is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity.

Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, "truth" means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with (...) respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic-based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings—an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems. (shrink)

This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible — and hence most natural — sense, disjunction and conjunction as deterministic-choice combinations of problems , and “absurd” as a computational problem of universal strength.

Within the program of finding axiomatizations for various parts of computability logic, it was proven earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting’s intuitionistic calculus. That sort of reduction permits unlimited reusage of the computational resource represented by the antecedent. An at least equally basic and natural sort of algorithmic reduction, however, is the one that does not allow such reusage. The present article shows that turning the logic of the first sort of (...) reduction into the logic of the second sort of reduction takes nothing more than just deleting the contraction rule from its Gentzen-style axiomatization. The first sort of interactive reduction is also shown to come in three natural versions. While those three versions are very different from each other, their logical behaviors turn out to be indistinguishable, with that common behavior being precisely captured by implicative intuitionistic logic. Among the other contributions of the present article is an informal introduction of a series of new — finite and bounded — versions of recurrence operations and the associated reduction operations. (shrink)

I present a semantics for the language of first-order additive-multiplicative linear logic, i.e. the language of classical first-order logic with two sorts of disjunction and conjunction. The semantics allows us to capture intuitions often associated with linear logic or constructivism such as sentences = games, SENTENCES = resources or sentences = problems, where “truth” means existence of an effective winning strategy.The paper introduces a decidable first-order logic ET in the above language and gives a proof of its soundness and completeness (...) with respect to this semantics. Allowing noneffective strategies in the latter is shown to lead to classical logic.The semantics presented here is very similar to Blass's game semantics. Although there is no straightforward reduction between the two corresponding notions of validity, my completeness proof can likely be adapted to the logic induced by Blass's semantics to show its decidability, which was the main problem left open in Blass's paper.The reader needs to be familiar with classical logic and arithmetic. (shrink)

Hájek and Montagna proved that the modal propositional logic ILM is the logic of -conservativity over sound theories containing I (PA with induction restricted to formulas). I give a simpler proof of the same fact.

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${\neg}$ , parallel conjunction ${\wedge}$ , parallel disjunction ${\vee}$ , branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the present) Part I containing preliminaries and a soundness proof, and (the forthcoming) Part II containing a completeness proof.

The paper introduces a semantics for the language of classical first order logic supplemented with the additional operators and . This semantics understands formulas as tasks. An agent , working as a slave for its master , can carry out the task αβ if it can carry out any one of the two tasks α, β, depending on which of them was requested by the master; similarly, it can carry out xα if it can carry out α for any particular (...) value for x selected by the master; an agent can carry out α→β if it can carry out β as long as it has, as a slave , an agent who carries out α; finally, carrying out P, where P is an atomic formula, simply means making P true; in particular, is a task that no agent can carry out. When restricted to the language of classical logic, the meaning of formulas is isomorphic to their classical meaning, which makes our semantics a conservative extension of classical semantics.This semantics can claim to be a formalization of the resource philosophy associated with linear logic, if resources are understood as agents carrying out tasks. The classical operators of our language correspond to the multiplicative operators of linear logic, while and correspond to the additive conjunction and universal quantifier, respectively.Our formalism may also have a potential to be used in AI as an alternative logic of planning and action. Its main appeal is that it is immune to the frame problem and the knowledge preconditions problem.The paper axiomatically defines a logic L in the above language and proves its soundness and completeness with respect to the task semantics in the following intuitive sense: L α iff α can be carried out by an agent who has nothing but its intelligence for carrying out tasks. This logic is shown to be semidecidable in the full language and decidable when the classical quantifier is forbidden in it. (shrink)

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\neg}}$$\end{document}, parallel conjunction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\wedge}}$$\end{document}, parallel disjunction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\vee}}$$\end{document}, branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the previous) (...) Part I containing preliminaries and a soundness proof, and (the present) Part II containing a completeness proof. (shrink)

The paper introduces a semantics for the language of propositional additive-multiplicative linear logic. It understands formulas as tasks that are to be accomplished by an agent (machine, robot) working as a slave for its master (user, environment). This semantics can claim to be a formalization of the resource philosophy associated with linear logic when resources are understood as agents accomplishing tasks. I axiomatically define a decidable logic TSKp and prove its soundness and completeness with respect to the task semantics in (...) the following intuitive sense: iff can be accomplished by an agent who has nothing but its intelligence (that is, no physical resources or external sources of information) for accomplishing tasks. (shrink)

The present paper constructs three new systems of clarithmetic : CLA8, CLA9 and CLA10. System CLA8 is shown to be sound and extensionally complete with respect to PA-provably recursive time computability. This is in the sense that an arithmetical problem A has a τ-time solution for some PA-provably recursive function τ iff A is represented by some theorem of CLA8. System CLA9 is shown to be sound and intensionally complete with respect to constructively PA-provable computability. This is in the sense (...) that a sentence X is a theorem of CLA9 iff, for some particular machine M, PA proves that M computes X. And system CLA10 is shown to be sound and intensionally complete with respect to not-necessarily-constructively PA-provable computability. This means that a sentence X is a theorem of CLA10 iff PA proves that X is computable, even if PA does not “know” of any particular machine M that computes X. (shrink)

Cirquent calculus is a novel proof theory permitting component-sharing between logical expressions. Using it, the predecessor article ‘Elementary-base cirquent calculus I: Parallel and choice connectives’ built the sound and complete axiomatization $\textbf{CL16}$ of a propositional fragment of computability logic. The atoms of the language of $\textbf{CL16}$ represent elementary, i.e. moveless, games and the logical vocabulary consists of negation, parallel connectives and choice connectives. The present paper constructs the first-order version $\textbf{CL17}$ of $\textbf{CL16}$, also enjoying soundness and completeness. The language of (...) $\textbf{CL17}$ augments that of $\textbf{CL16}$ by including choice quantifiers. Unlike classical predicate calculus, $\textbf{CL17}$ turns out to be decidable. (shrink)