Game Logic is an excellent setting to study proofs-about-programs via the interpretation of those proofs as programs, because constructive proofs for games correspond to effective winning strategies to follow in response to the opponent's actions. We thus develop Constructive Game Logic which extends Parikh's Game Logic (GL) with constructivity and with first-order programs a la Pratt's first-order dynamic logic (DL). Our major contributions include:
1) a novel realizability semantics capturing the adversarial dynamics of games, 2) a natural deduction calculus and operational semantics describing the computational meaning of strategies via proof-terms, and 3) theoretical results including soundness of the proof calculus w.r.t. realizability semantics, progress and preservation of the operational semantics of proofs, and Existence Properties on support of the extraction of computational artifacts from game proofs.
Together, these results provide the most general account of a Curry-Howard interpretation for any program logic to date, and the first at all for Game Logic.
We present constructive provability logic, an intuitionstic modal logic that validates the Löb rule of Gödel and Löb's provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provability logic to justify stratified negation in logic programming within an intuitionstic and structural proof theory.
We extend the constructive differential game logic (CdGL) of hybrid games with a refinement connective that relates two hybrid games. We use this connective to prove a folk theorem relating hybrid games to hybrid systems.
This article presents a computational semantics for classical logic using constructive type theory. Such semantics seems impossible because classical logic allows the Law of Excluded Middle (LEM), not accepted in constructive logic since it does not have computational meaning. However, the apparently oracular powers expressed in the LEM, that for any proposition P either it or its negation, not P, is true can also be explained in terms of constructive evidence that does not refer to "oracles for truth." Types with virtual evidence and the constructive impossibility of negative evidence provide…
Hybrid games are models which combine discrete, continuous, and adversarial dynamics. Game logic enables proving (classical) existence of winning strategies. We introduce constructive differential game logic (CdGL) for hybrid games, where proofs that a player can win the game correspond to computable winning strategies. This is the logical foundation for synthesis of correct control and monitoring code for safety-critical cyber-physical systems. Our contributions include novel static and dynamic semantics as well as soundness and consistency.
We present the intuitionistic version of PUC-Logic. After that, we present a constructive approach to Lewis' counterfactual abstraction to show that it does not require the classical absurd rule.
We present deductive systems for various modal logics that can be obtained from the constructive variant of the normal modal logic CK by adding combinations of the axioms d, t, b, 4, and 5. This includes the constructive variants of the standard modal logics K4, S4, and S5. We use for our presentation the formalism of nested sequents and give a syntactic proof of cut elimination.
We argue that the notion of epistemic \emph{possible worlds} in constructivism (intuitionism) is not as the same as it is in classic view, and there are possibilities, called non-predetermined worlds, which are ignored in (classic) Epistemic Logic. Regarding non-predetermined possibilities, we propose a constructive epistemic logic and prove soundness and completeness theorems for it. We extend the proposed logic by adding a public announcement operator. To declare the significance of our work, we formulate the well-known Surprise Exam Paradox, $\mathbf{SEP}$, via the proposed constructive epi…
We define a family of propositional constructive modal logics corresponding each to a different classical modal system. The logics are defined in the style of Wijesekera's constructive modal logic, and are both proof-theoretically and semantically motivated. On the one hand, they correspond to the single-succedent restriction of standard sequent calculi for classical modal logics. On the other hand, they are obtained by incorporating the hereditariness of intuitionistic Kripke models into the classical satisfaction clauses for modal formulas. We show that, for the considered classical logics, …
The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas. Surprisingly, this logic is constructive and so far this has been proved by cut-elimination for ad-hoc sequent calculi. Here we use the methods of natural deduction and Curry-Howard correspondence to provide a simple computational interpretation of the logic.
Duality for Constructive Modal Logics: from Sahqlvist to Goldblatt-Thomason
Mathematical Institute, University of Bern（伯尔尼大学数学研究所）；School of Computer Science, University of Birimingham（伯明翰大学计算机学院）；School of Computing, The Australian National University（澳大利亚国立大学计算学院）
Dummett's logic LC is intuitionistic logic extended with Dummett's axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and second-order Dummett's logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the viewpoint of logic, Dummett's axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quan…
Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possi…
We study a conservative extension of classical propositional logic distinguishing between four modes of statement: a proposition may be affirmed or denied, and it may be strong or classical. Proofs of strong propositions must be constructive in some sense, whereas proofs of classical propositions proceed by contradiction. The system, in natural deduction style, is shown to be sound and complete with respect to a Kripke semantics. We develop the system from the perspective of the propositions-as-types correspondence by deriving a term assignment system with confluent reduction. The proof of str…
We develop a sound, complete and practically implementable tableaux-based decision method for constructive satisfiability testing and model synthesis in the fragment ATL+ of the full Alternating time temporal logic ATL*. The method extends in an essential way a previously developed tableaux-based decision method for ATL and works in 2EXPTIME, which is the optimal worst case complexity of the satisfiability problem for ATL+ . We also discuss how suitable parametrizations and syntactic restrictions on the class of input ATL+ formulae can reduce the complexity of the satisfiability problem.
Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results…
The logics CS4 and IS4 are intuitionistic variants of the modal logic S4. Whether the finite model property holds for each of these logics has been a long-standing open problem. In this paper we introduce two logics closely related to IS4: GS4, obtained by adding the Godel-Dummett axiom to IS4, and S4I, obtained by reversing the roles of the modal and intuitionistic relations. We then prove that CS4, GS4, and S4I all enjoy the finite model property.
A modified realisability interpretation of infinitary logic is formalised and proved sound in constructive type theory (CTT). The logic considered subsumes first order logic. The interpretation makes it possible to extract programs with simplified types and to incorporate and reason about them in CTT.
The two major systems of formal verification are model checking and algebraic model-based testing. Model checking is based on some form of temporal logic such as linear temporal logic (LTL) or computation tree logic (CTL). One powerful and realistic logic being used is CTL, which is capable of expressing most interesting properties of processes such as liveness and safety. Model-based testing is based on some operational semantics of processes (such as traces, failures, or both) and its associated preorders. The most fine-grained preorder beside bisimulation (mostly of theoretical importance) …
In this paper we present a constructive proof of cut elimination for a system of full second order logic with the structural rules absorbed and using sets instead of sequences. The standard problem of the cutrank growth is avoided by using a new parameter for the induction, the cutweight. This technique can also be applied to first order logic.
In this paper, we develop a game-theoretic account of concurrent separation logic. To every execution trace of the Code confronted to the Environment, we associate a specification game where Eve plays for the Code, and Adam for the Environment. The purpose of Eve and Adam is to decompose every intermediate machine state of the execution trace into three pieces: one piece for the Code, one piece for the Environment, and one piece for the available shared resources. We establish the soundness of concurrent separation logic by interpreting every derivation tree of the logic as a winning strategy …
We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov's Principle and Weak König's Lemma and discuss non-standard semantics a…
An old dream of concurrency theory and programming language semantics has been to uncover the fundamental synchronization mechanisms which regulate situations as different as game semantics for higher-order programs, and Hoare logic for concurrent programs with shared memory and locks. In this paper, we establish a deep and unexpected connection between two recent lines of work on concurrent separation logic (CSL) and on template game semantics for differential linear logic (DiLL). Thanks to this connection, we reformulate in the purely conceptual style of template games for DiLL the asynchron…
In this paper we provide two new semantics for proofs in the constructive modal logics CK and CD. The first semantics is given by extending the syntax of combinatorial proofs for propositional intuitionistic logic, in which proofs are factorised in a linear fragment (arena net) and a parallel weakening-contraction fragment (skew fibration). In particular we provide an encoding of modal formulas by means of directed graphs (modal arenas), and an encoding of linear proofs as modal arenas equipped with vertex partitions satisfying topological criteria. The second semantics is given by means of wi…
In this paper, we introduce a general family of sequent-style calculi over the modal language and its fragments to capture the essence of all constructively acceptable systems. Calling these calculi \emph{constructive}, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules, i.e., there is a polynomial time algorithm that reads a proof of the premise of a Visser's rule and provides a proof for its conclusion. As a positive application, we show the feasible admissibility of Visser's rules in several sequent calculi…
The idea of the Semantic Web is to annotate Web content and services with computer interpretable descriptions with the aim to automatize many tasks currently performed by human users. In the context of Web services, one of the most interesting tasks is their composition. In this paper we formalize this problem in the framework of a constructive description logic. In particular we propose a declarative service specification language and a calculus for service composition. We show by means of an example how this calculus can be used to define composed Web services and we discuss the problem of a…
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, wh…
C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves…
Classical logic is embedded into constructive logic, through a definition of the classical connectives and quantifiers in terms of the constructive ones.
Game logic is a dynamic modal logic which models strategic two person games; it contains propositional dynamic logic (PDL) as a fragment. We propose an interpretation of game logic based on stochastic effectivity functions. A definition of these functions is proposed, and some algebraic properties of effectivity functions such as congruences are investigated. The relationship to stochastic relations is characterized through a deduction system. Logical and behavioral equivalence of game models is investigated. Finally the completion of models receives some attention.
Game semantics and winning strategies offer a potential conceptual bridge between semantics and proof systems of logics. We illustrate this link for hybrid logic -- an extension of modal logic that allows for explicit reference to worlds within the language. The main result is that the systematic search of winning strategies over all models can be finitized and thus reformulated as a proof system.
Logics of non-sense allow a third truth value to express propositions that are \emph{nonsense}. These logics are ideal formalisms to understand how errors are handled in programs and how they propagate throughout the programs once they appear. In this paper, we give a Hintikkan game semantics for logics of non-sense and prove its correctness. We also discuss how a known solution method in game theory, the iterated elimination of strictly dominated strategies, relates to semantic games for logics of nonsense. Finally, we extend the logics of nonsense only by means of semantic games, developing …
Inductions and game semantics are two useful extensions to traditional logic programming. To be specific, inductions can capture a wider class of provable formulas in logic programming. Adopting game semantics can make logic programming more interactive. In this paper, we propose an execution model for a logic language with these features. This execution model follows closely the reasoning process in real life.
Parikh's game logic is a PDL-like fixpoint logic interpreted on monotone neighbourhood frames that represent the strategic power of players in determined two-player games. Game logic translates into a fragment of the monotone $μ$-calculus, which in turn is expressively equivalent to monotone modal automata. Parity games and automata are important tools for dealing with the combinatorial complexity of nested fixpoints in modal fixpoint logics, such as the modal $μ$-calculus. In this paper, we (1) discuss the semantics a of game logic over neighbourhood structures in terms of parity games, and (…
"Church's thesis" ($\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\mathbb{N} \to \mathbb{N}$ is computable, i.e. definable in a model of computation. $\mathsf{CT}$ is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak König's Lemma and the fan theorem, respectively. Recently, $\mathsf{CT}$ was proved consistent for (univalent) constructive type theory. Since neither Weak König's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, i…
This is a survey of λ-calculi that, through the Curry-Howard isomorphism, correspond to constructive modal logics. We cover the prehistory of the subject and then concentrate on the developments that took place in the 1990s and early 2000s. We discuss logical aspects, modal λ-calculi, and their categorical semantics. The logics underlying the calculi surveyed are constructive versions of K, K4, S4, and LTL.
Church's Higher Order Logic is a basis for influential proof assistants -- HOL and PVS. Church's logic has a simple set-theoretic semantics, making it trustworthy and extensible. We factor HOL into a constructive core plus axioms of excluded middle and choice. We similarly factor standard set theory, ZFC, into a constructive core, IZF, and axioms of excluded middle and choice. Then we provide the standard set-theoretic semantics in such a way that the constructive core of HOL is mapped into IZF. We use the disjunction, numerical existence and term existence properties of IZF to provide a progr…
We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke-models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler-Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player (duplicator) does not have a trivial optimal strategy. Thus, unlike the Adler-Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a nonelementary succinctn…
The Poison Game is a two-player game played on a graph in which one player can influence which edges the other player is able to traverse. It operationalizes the notion of existence of credulously admissible sets in an argumentation framework or, in graph-theoretic terminology, the existence of non-trivial semi-kernels. We develop a modal logic (poison modal logic, PML) tailored to represent winning positions in such a game, thereby identifying the precise modal reasoning that underlies the notion of credulous admissibility in argumentation. We study model-theoretic and decidability properties…
We propose an interpretation of the first-order answer set programming (FOASP) in terms of intuitionistic proof theory. It is obtained by two polynomial translations between FOASP and the bounded-arity fragment of the Sigma_1 level of the Mints hierarchy in first-order intuitionistic logic. It follows that Sigma_1 formulas using predicates of fixed arity (in particular unary) is of the same strength as FOASP. Our construction reveals a close similarity between constructive provability and stable entailment, or equivalently, between the construction of an answer set and an intuitionistic refuta…
In 1953, Kuhn showed that every sequential game has a Nash equilibrium by showing that a procedure, named ``backward induction'' in game theory, yields a Nash equilibrium. It actually yields Nash equilibria that define a proper subclass of Nash equilibria. In 1965, Selten named this proper subclass subgame perfect equilibria. In game theory, payoffs are rewards usually granted at the end of a game. Although traditional game theory mainly focuses on real-valued payoffs that are implicitly ordered by the usual total order over the reals, works of Simon or Blackwell already involved partially ord…
This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator $Φ^{aggr}_P$ for aggregate programs, independently proposed Pelov et al. This operator allows us to closely tie the computational complexity of the answer set checking and answer sets existence problems to the cost of checking a solution of the aggregates in the program. Finally, we relate th…
Logic programming with fixed-point definitions is a useful extension of traditional logic programming. Fixed-point definitions can capture simple model checking problems and closed-world assumptions. Its operational semantics is typically based on intuitionistic provability. We extend the operational semantics of these languages with game semantics. This extended semantics has several interesting aspects: in particular, it gives a logical status to the $read$ predicate.
I overview the work of the Tbilisi school on intuitionistic modal logics of well-founded/scattered structures and its connections with contemporary theoretical computer science. Fixed-point theorems and their consequences are of particular interest.
We refine HO/N game semantics with an additional notion of pointer (mu-pointers) and extend it to first-order classical logic with completeness results. We use a Church style extension of Parigot's lambda-mu-calculus to represent proofs of first-order classical logic. We present some relations with Krivine's classical realizability and applications to type isomorphisms.
