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Support is Search

This paper demonstrates that Sandqvist's base-extension semantics for intuitionistic propositional logic admits a constructive, computational interpretation where support in a fixed base corresponds exactly to proof-search in a second-order hereditary Harrop logic program.

Original authors: Alexander V. Gheorghiu

Published 2026-03-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander V. Gheorghiu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: What is this paper about?

Imagine you are trying to understand what it means for a statement to be "true" in a world where truth isn't just about matching reality (like a photo of a tree), but about how we can prove it. This is the world of Intuitionistic Logic.

In this world, a statement is only "supported" (valid) if we have a concrete method to prove it. The paper asks a specific question: If we have a specific set of rules (a "Base"), how do we actually check if a complex statement is supported?

The author, Alexander Gheorghiu, discovers a surprising answer: Checking if a statement is supported is exactly the same thing as running a computer program to find a solution.

In short: Support = Search.

The Characters and the Setting

To understand the paper, let's break down the jargon into a story.

1. The "Base" (The Rulebook)

Imagine a Base is like a Rulebook for a specific game.

It contains simple rules about basic facts (e.g., "If it rains, the grass is wet").
In this paper, the Rulebook can be very complex. It can have rules about rules (e.g., "If you can prove that 'if A then B', then you can conclude C"). This is called a "higher-level" system.

2. "Support" (The Verdict)

Usually, to say a statement is "supported" by a Rulebook, you have to check it against every possible way the Rulebook could be expanded.

The Old Way (The Realist Trap): Imagine checking a statement by looking at every single possible future version of the universe where the rules might change. This is like trying to count every grain of sand on every beach on Earth. It's impossible and feels "unreal" because it assumes a completed infinity.
The Problem: The paper argues that this "checking everything" approach betrays the spirit of the logic, which is supposed to be about what we can actually do (construct), not what exists in some abstract infinity.

3. The "Search" (The Detective)

The paper proposes a new way to look at "Support." Instead of checking every possible universe, we act like a Detective or a Computer Program.

We take the statement we want to prove.
We feed it into our Rulebook.
We try to search for a path that leads from the rules to the conclusion.

The "Aha!" Moment: The Magic Translation

The core discovery of the paper is a translation tool.

The author shows that you can take any logical statement (like "If it rains, then the grass is wet, OR the sprinkler is on") and translate it into a Logic Program (a set of instructions for a computer).

The Translation: The complex logical rules are rewritten as "clauses" (instructions).
The Twist: The scary parts of the logic (like "for every possible extension of the rules") turn out to be just variables in the computer program.
- Analogy: Imagine a form that asks, "For every possible person who might enter the room..."
- The Old View: You have to wait for every person in the world to walk through the door to check the form.
- The New View: You just pick one random person (a "fresh variable"), ask them to fill out the form, and if they can do it, the form is valid. You don't need to wait for everyone; you just need to show the process works for a generic person.

The "Continuation-Passing" Analogy

The paper mentions a technical term called Continuation-Passing Style (CPS). Here is a simple metaphor for that:

Imagine you are ordering a pizza.

Normal Style: You say, "I want a pizza." The chef makes it and hands it to you.
CPS Style: You say, "I want a pizza. Here is a box. When you make the pizza, put it in this box and hand the box to me."

In this paper, the logical rules are written in this "CPS" style. Instead of saying "This is true," the rules say, "If you can prove this part, pass the result to this next step."

The author realized that the weird, complex definitions of "Support" in the original logic were actually just these "pass the result" instructions all along. Once you see them as instructions, they become a search algorithm.

Why Does This Matter?

It Saves the Philosophy: The original logic was built on "Anti-Realism" (truth is what we can prove). But the old way of defining it sounded "Realist" (truth is what exists in all possible worlds). This paper proves that the logic is truly Anti-Realist because it can be executed as a step-by-step search. You don't need to see the whole universe; you just need to run the search.
It's Practical: Because "Support" is now "Search," we can write computer programs to do it. We can build tools that automatically check if a logical argument is valid by running a search, rather than doing abstract math.
The "Classical" Mystery: The paper notes that this magic only works for Intuitionistic logic (which allows for "higher-level" rules). If you try to do this with Classical logic (the standard logic of most math), the "search" breaks down. This leaves an open mystery: How do we make a "search-based" version of classical logic?

Summary in One Sentence

This paper proves that checking if a logical statement is valid in a specific rulebook is not about staring into an infinite void of possibilities, but is simply a matter of running a search algorithm to find a proof, turning abstract philosophy into concrete computer code.

1. Problem Statement

The paper addresses a foundational gap in Base-Extension Semantics (B-eS) for Intuitionistic Propositional Logic (IPL), developed by Sandqvist.

The Global vs. Local Question: Sandqvist's original work established a global completeness theorem: a formula $\phi$ is intuitionistically valid if and only if it is "supported" ( $\Vdash_B \phi$ ) in every possible base $B$ . However, the paper identifies a lack of understanding regarding the local question: For a fixed base $B$ , what exactly does the support relation $\Vdash_B \phi$ correspond to computationally?
The Realist Tension: The standard definition of support involves universal quantification over all base extensions ( $C \supseteq B$ ) and atoms. Read in a standard model-theoretic way, this implies a "completed totality" of infinite sets, which contradicts the anti-realist (constructivist) foundations of proof-theoretic semantics. The semantics appears to rely on non-constructive, global truths rather than local, manifestable proof procedures.
The Challenge: The authors aim to provide a fully constructive, computational interpretation of $\Vdash_B \phi$ that avoids appealing to completed infinities, thereby vindicating the anti-realist stance on its own terms.

2. Methodology

The authors employ a computational approach by encoding logical formulae into Second-Order Hereditary Harrop (HH) logic programs.

Atomic Systems as Logic Programs:
- The paper treats "bases" ( $B$ ) not as abstract sets of rules, but as programs in a logic programming language.
- It utilizes Higher-Level Atomic Systems, where rules can discharge hypotheses (e.g., rules of the form $[A] \vdash B \implies C$ ). These are mapped to Hereditary Harrop formulae (specifically, definite clauses with nested implications).
- To handle infinite bases (required for completeness) without assuming completed infinities, the authors introduce Second-Order Quantifiers ( $\forall X$ ) over atomic variables. This allows schematic rules (e.g., "for all atoms $X$ ") to be represented finitely.
Continuation-Passing Style (CPS) Encoding:
- The core technical innovation is reading the semantic clauses of support in Continuation-Passing Style.
- Instead of defining support directly, the authors define an encoding function $\llbracket \phi \rrbracket$ that maps a formula $\phi$ to a logic-programming goal.
- Key Encodings:
  - Implication ( $\to$ ): $\llbracket \phi \to \psi \rrbracket = \llbracket \phi \rrbracket \supset \llbracket \psi \rrbracket$
  - Conjunction ( $\land$ ): Encoded as $\forall X ((\llbracket \phi \rrbracket \supset \llbracket \psi \rrbracket \supset X) \supset X)$ . This treats conjunction as a continuation that takes a proof of both components to produce a result.
  - Disjunction ( $\lor$ ): Encoded as $\forall X ((\llbracket \phi \rrbracket \supset X) \supset (\llbracket \psi \rrbracket \supset X) \supset X)$ . This mirrors the CPS definition of disjunction, where a proof of $\phi \lor \psi$ is a function that, given continuations for both $\phi$ and $\psi$ , produces a result.
  - Falsum ( $\bot$ ): Encoded as $\forall X (X)$ .
Operational Semantics:
- The paper defines an operational judgment relation $\vdash_O$ (proof-search) for these logic programs.
- Crucially, the universal quantifiers in the encoding are handled via an eigenvariable discipline (freshness constraints). When proving $\forall X G$ , the system picks a fresh atom $A$ and proves $G[A/X]$ . This avoids the need to check "all" atoms, adhering to constructive principles.

3. Key Contributions

Local Correspondence Theorem: The paper establishes a precise equivalence between semantic support and operational proof-search:
$\Vdash_B \phi \iff B \vdash_O \llbracket \phi \rrbracket$
This means that checking if a formula is supported in a base is exactly equivalent to running a proof-search procedure (resolution) against that base.
Constructive Interpretation of Quantifiers: The paper demonstrates that the universal quantifiers over base extensions and atoms in the semantic definition are not ranging over a completed domain. Instead, they are governed by eigenvariable disciplines in the proof-search procedure. This resolves the tension between the semantics and anti-realist philosophy.
Rejection of "Flattening": Previous local approaches (e.g., Gheorghiu & Pym) relied on "flattening" formulae into atoms, which obscured the logical structure. This paper preserves the structure of $\phi$ by encoding it directly as a goal, maintaining a direct correspondence between logical connectives and proof-search steps.

4. Results

Theorem 4.6 (Soundness and Completeness): For any base $B$ and formula $\phi$ , $\phi$ is supported in $B$ if and only if the logic program $B$ successfully resolves the goal $\llbracket \phi \rrbracket$ .
Proof Strategy: The proof proceeds by simultaneous induction on the structure of $\phi$ $ϕ$ .
- Soundness: If $\Vdash_B \phi$ , then the operational semantics can derive $\llbracket \phi \rrbracket$ . This relies on the monotonicity of the support relation and the operational rules (Augment, Backchain, Generic).
- Completeness: If $B \vdash_O \llbracket \phi \rrbracket$ , then $\Vdash_B \phi$ . This relies on the substitution lemma and the fact that the operational rules (specifically the handling of fresh atoms) mimic the semantic requirements for base extensions.
Role of Higher-Order Rules: The result depends critically on the presence of higher-level atomic rules (rules that discharge hypotheses). If the system is restricted to first-level rules (Horn clauses), the CPS encoding fails, and the semantics collapses to classical logic rather than intuitionistic logic.

5. Significance

Philosophical Vindication: The paper successfully defends the anti-realist foundations of proof-theoretic semantics. It shows that the semantics is not just "anti-realist in aspiration" but "anti-realist in execution." The meaning of logical constants is fully realized through a computational search procedure, not by reference to abstract, completed models.
Computational Transparency: It provides a concrete algorithm for determining support. "Support is Search." This opens the door for implementing base-extension semantics in automated reasoning tools and modeling tasks.
Unification of Logic and Computation: By mapping intuitionistic logic directly to second-order Hereditary Harrop logic programming, the paper bridges the gap between proof-theoretic semantics and the theory of uniform proofs.
Future Directions: The authors note that while this works for intuitionistic logic, a similar constructive account for classical logic (which arises from first-level atomic systems) remains an open problem, as the necessary CPS encoding is not available in that restricted setting.

In summary, the paper transforms the abstract, potentially non-constructive definition of base-extension semantics into a concrete, executable proof-search procedure, proving that the "meaning" of a formula in a base is exactly its computability as a logic-programming goal.