[Paper Review] Finitary semantics of linear logic and higher-order model-checking

Charles Grellois, Paul-André Melliès|arXiv (Cornell University)|Feb 18, 2015

Logic, programming, and type systems13 references4 citations

TL;DR

This paper establishes a finitary semantics for linear logic using prime-algebraic lattices to provide a compositional, denotational proof of the decidability of the selection problem in higher-order model-checking. By interpreting higher-order recursion schemes in a finite model, the authors show that for any MSO formula, a witness-generating higher-order recursion scheme can be algorithmically constructed, resolving the selection problem via finite parity games and intersection types.

ABSTRACT

In this paper, we explain how the connection between higher-order model-checking and linear logic recently exhibited by the authors leads to a new and conceptually enlightening proof of the selection problem originally established by Carayol and Serre using collapsible pushdown automata. The main idea is to start from an infinitary and colored relational semantics of the lambdaY-calculus already formulated, and to replace it by its finitary counterpart based on finite prime-algebraic lattices. Given a higher-order recursion scheme G, the finiteness of its interpretation in the model enables us to associate to any MSO formula phi a new higher-order recursion scheme G_phi resolving the selection problem.

Motivation & Objective

• To establish a finitary, qualitative semantics for higher-order recursion schemes using prime-algebraic lattices.
• To provide a denotational proof of the decidability of the selection problem in higher-order model-checking.
• To connect linear logic semantics with automata-theoretic constructions in higher-order model-checking.
• To show that the interpretation of higher-order recursion schemes in the finitary model yields finite representations of accepting run-trees for alternating parity automata.
• To demonstrate that memoryless winning strategies in finite parity games correspond to finite, pointer-based derivation trees in an intersection type system.

Proposed method

• Replacing the infinitary, colored relational semantics of linear logic with a finitary semantics based on prime-algebraic lattices.
• Constructing a parameterized fixpoint operator in the finitary model to interpret recursive types in the $λ Y$-calculus.
• Defining a colored intersection type system with backtracking pointers to represent finite derivation trees of sequents.
• Reducing the problem of checking whether a state is accepting to solving a finite parity game whose nodes are sequents in the type system.
• Using winning strategies in the finite parity game to construct a new higher-order recursion scheme that generates a witness run-tree for the automaton.
• Annotating terminals and non-terminals in the original HORS with intersection types from the derivation tree to build the witness scheme $σ_{\varphi}$.

Experimental results

Research questions

• RQ1 Can the selection problem in higher-order model-checking be solved using a finitary, compositional semantics based on linear logic?
• RQ2 Does the finitary semantics of linear logic based on prime-algebraic lattices provide a decidable framework for higher-order model-checking?
• RQ3 How can the existence of a winning strategy in a finite parity game be used to construct a witness-generating higher-order recursion scheme?
• RQ4 What is the precise correspondence between derivations in the colored intersection type system and accepting run-trees of alternating parity automata?
• RQ5 Can the finiteness of the interpretation in the Scott model be leveraged to ensure decidability of the selection problem?

Key findings

• The selection problem is decidable: for any higher-order recursion scheme $σ$ and MSO formula $φ$, a new HORS $σ_{\varphi}$ generating a witness run-tree can be algorithmically constructed.
• The finitary semantics of linear logic via prime-algebraic lattices ensures that only finitely many intersection types and contexts occur in derivations, enabling decidability.
• Checking whether a state $q_0$ is accepting for an alternating parity automaton over the tree generated by $σ$ reduces to checking whether $q_0 \in [\![\sigma]\!]_{\mathcal{A}}$, which is decidable due to finiteness.
• Memoryless winning strategies in the finite parity game correspond to finite, pointer-based representations of derivation trees in the intersection type system.
• The construction of $σ_{\varphi}$ is compositional and based on annotating the original HORS with intersection types from the derivation tree, preserving the structure of the original scheme.
• The approach provides a new, conceptual proof of the selection problem using linear logic semantics, distinct from the automata-theoretic proof by Carayol and Serre.